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The Size of Sets — Explore equations

The Size of Sets

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Equation guide

Every distinct expression in The Size of Sets appears once below with navigable MathML, its conventional reading, its meaning here, and links to every occurrence.

398 expressions

Expression 1

f(1)f(2)f(3)f(4)f(5)f(6)f(7)…−⌈02⌉⌈12⌉−⌈22⌉⌈32⌉−⌈42⌉⌈52⌉−⌈62⌉…01−12−23…

Conventional reading: array, row 1: f open parenthesis 1 close parenthesis; f open parenthesis 2 close parenthesis; f open parenthesis 3 close parenthesis; f open parenthesis 4 close parenthesis; f open parenthesis 5 close parenthesis; f open parenthesis 6 close parenthesis; f open parenthesis 7 close parenthesis; and so on; row 2: minus the ceiling of the fraction 0 over 2 end ceiling; the ceiling of the fraction 1 over 2 end ceiling; minus the ceiling of the fraction 2 over 2 end ceiling; the ceiling of the fraction 3 over 2 end ceiling; minus the ceiling of the fraction 4 over 2 end ceiling; the ceiling of the fraction 5 over 2 end ceiling; minus the ceiling of the fraction 6 over 2 end ceiling; and so on; row 3: 0; 1; minus 1; 2; minus 2; 3; and so on

Meaning here: This multirow mathematical object presents the displayed values or cases in source order.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 146, column 1

Expression 3

N≺℘(N)

Conventional reading: the cardinality of the natural numbers is smaller than the cardinality of the power set of the natural numbers

Meaning here: This asserts that the first displayed set has strictly smaller cardinality than the second.

1 occurrence
  1. Occurrence 1: comparing-size.tex, line 50, column 1

Expression 6

h(n)=000…0⏟n 0's

Conventional reading: h open parenthesis n close parenthesis equals zero zero zero and so on zero, underbraced to indicate n zeros

Meaning here: This defines h of n as the finite binary string consisting of exactly n zeros; the underbrace records its length.

1 occurrence
  1. Occurrence 1: reduction.tex, line 96, column 1

Expression 7

f(x)={x}

Conventional reading: f open parenthesis x close parenthesis equals open brace x close brace

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: comparing-size.tex, line 61, column 9

Expression 9

Q

Conventional reading: the rational numbers

Meaning here: This denotes the named number system, distinguished set, or infinite value used in the surrounding statement.

1 occurrence
  1. Occurrence 1: pairing.tex, line 71, column 11

Expression 12

f:Z+→Z+

Conventional reading: f from the positive integers to the positive integers

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 125, column 15

Expression 17

g(n)=f(n+1)

Conventional reading: g open parenthesis n close parenthesis equals f open parenthesis n plus 1 close parenthesis

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 200, column 64

Expression 21

N∖A

Conventional reading: the natural numbers set minus A

Meaning here: This denotes a set obtained by the displayed union, intersection, or set-difference operation.

1 occurrence
  1. Occurrence 1: pairing.tex, line 94, column 14

Expression 22

12345678910…⟨0,0⟩⟨0,1⟩⟨0,2⟩⟨0,3⟩⟨0,4⟩…

Conventional reading: position 1: the tuple 0 comma 0; position 2: blank; position 3: the tuple 0 comma 1; position 4: blank; position 5: the tuple 0 comma 2; position 6: blank; position 7: the tuple 0 comma 3; position 8: blank; position 9: the tuple 0 comma 4; position 10: blank; and the pattern continues

Meaning here: This partial position table places the pairs with first coordinate zero in odd-numbered positions and intentionally leaves every even position blank.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 24, column 1

Expression 24

0123…⟨0,0⟩⟨0,0,0⟩⟨0,0,1⟩⟨0,0,2⟩⟨0,0,3⟩…⟨0,1⟩⟨0,1,0⟩⟨0,1,1⟩⟨0,1,2⟩⟨0,1,3⟩…⟨1,0⟩⟨1,0,0⟩⟨1,0,1⟩⟨1,0,2⟩⟨1,0,3⟩…⟨0,2⟩⟨0,2,0⟩⟨0,2,1⟩⟨0,2,2⟩⟨0,2,3⟩…⋮⋮⋮⋮⋮⋱

Conventional reading: array, row 1: blank position; 0; 1; 2; 3; and so on; row 2: the tuple 0 comma 0; the tuple 0 comma 0 comma 0; the tuple 0 comma 0 comma 1; the tuple 0 comma 0 comma 2; the tuple 0 comma 0 comma 3; and so on; row 3: the tuple 0 comma 1; the tuple 0 comma 1 comma 0; the tuple 0 comma 1 comma 1; the tuple 0 comma 1 comma 2; the tuple 0 comma 1 comma 3; and so on; row 4: the tuple 1 comma 0; the tuple 1 comma 0 comma 0; the tuple 1 comma 0 comma 1; the tuple 1 comma 0 comma 2; the tuple 1 comma 0 comma 3; and so on; row 5: the tuple 0 comma 2; the tuple 0 comma 2 comma 0; the tuple 0 comma 2 comma 1; the tuple 0 comma 2 comma 2; the tuple 0 comma 2 comma 3; and so on; row 6: vertical ellipsis; vertical ellipsis; vertical ellipsis; vertical ellipsis; vertical ellipsis; diagonal ellipsis

Meaning here: This multirow mathematical object presents the displayed values or cases in source order.

1 occurrence
  1. Occurrence 1: zig-zag.tex, line 94, column 1

Expression 25

℘(N)

Conventional reading: the power set of the natural numbers

Meaning here: This expression involves a power set, the set of all subsets of the indicated set.

12 occurrences
  1. Occurrence 1: non-enumerability.tex, line 203, column 11
  2. Occurrence 2: non-enumerability-alt.tex, line 15, column 3
  3. Occurrence 3: non-enumerability-alt.tex, line 115, column 1
  4. Occurrence 4: reduction-alt.tex, line 22, column 1
  5. Occurrence 5: reduction-alt.tex, line 23, column 6
  6. Occurrence 6: reduction-alt.tex, line 24, column 1
  7. Occurrence 7: reduction-alt.tex, line 26, column 18
  8. Occurrence 8: reduction-alt.tex, line 30, column 13
  9. Occurrence 9: reduction-alt.tex, line 31, column 4
  10. Occurrence 10: reduction-alt.tex, line 49, column 31
  11. Occurrence 11: reduction-alt.tex, line 82, column 7
  12. Occurrence 12: reduction-alt.tex, line 84, column 49

Expression 26

21·(2·2+1)−1=2·5−1=9

Conventional reading: 2 to the power 1 times open parenthesis 2 times 2 plus 1 close parenthesis minus 1 equals 2 times 5 minus 1 equals 9

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 97, column 24

Expression 27

A=∅

Conventional reading: A equals the empty set

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

6 occurrences
  1. Occurrence 1: enumerability.tex, line 270, column 18
  2. Occurrence 2: enumerability.tex, line 275, column 38
  3. Occurrence 3: equinumerous-sets.tex, line 72, column 15
  4. Occurrence 4: equinumerous-sets.tex, line 73, column 31
  5. Occurrence 5: enumerability-alt.tex, line 59, column 42
  6. Occurrence 6: enumerability-alt.tex, line 78, column 50

Expression 30

1010101…

Conventional reading: one zero one zero one zero one, continuing indefinitely

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

1 occurrence
  1. Occurrence 1: reduction-alt.tex, line 59, column 53

Expression 32

1=20

Conventional reading: 1 equals 2 to the power 0

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 77, column 1

Expression 33

0123…0s0(0)s0(1)s0(2)s0(3)…1s1(0)s1(1)s1(2)s1(3)…2s2(0)s2(1)s2(2)s2(3)…3s3(0)s3(1)s3(2)s3(3)…⋮⋮⋮⋮⋮⋱

Conventional reading: array, row 1: blank position; 0; 1; 2; 3; and so on; row 2: 0; s sub 0 open parenthesis 0 close parenthesis; s sub 0 open parenthesis 1 close parenthesis; s sub 0 open parenthesis 2 close parenthesis; s sub 0 open parenthesis 3 close parenthesis; and so on; row 3: 1; s sub 1 open parenthesis 0 close parenthesis; s sub 1 open parenthesis 1 close parenthesis; s sub 1 open parenthesis 2 close parenthesis; s sub 1 open parenthesis 3 close parenthesis; and so on; row 4: 2; s sub 2 open parenthesis 0 close parenthesis; s sub 2 open parenthesis 1 close parenthesis; s sub 2 open parenthesis 2 close parenthesis; s sub 2 open parenthesis 3 close parenthesis; and so on; row 5: 3; s sub 3 open parenthesis 0 close parenthesis; s sub 3 open parenthesis 1 close parenthesis; s sub 3 open parenthesis 2 close parenthesis; s sub 3 open parenthesis 3 close parenthesis; and so on; row 6: vertical ellipsis; vertical ellipsis; vertical ellipsis; vertical ellipsis; vertical ellipsis; diagonal ellipsis

Meaning here: This array lists zero-indexed binary sequences by rows and their entries by columns; bold cells mark the main diagonal.

1 occurrence
  1. Occurrence 1: non-enumerability-alt.tex, line 67, column 1

Expression 35

n

Conventional reading: n

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

23 occurrences
  1. Occurrence 1: enumerability.tex, line 106, column 26
  2. Occurrence 2: enumerability.tex, line 107, column 40
  3. Occurrence 3: enumerability.tex, line 180, column 24
  4. Occurrence 4: enumerability.tex, line 237, column 24
  5. Occurrence 5: zig-zag.tex, line 39, column 5
  6. Occurrence 6: zig-zag.tex, line 75, column 5
  7. Occurrence 7: zig-zag.tex, line 111, column 49
  8. Occurrence 8: pairing.tex, line 30, column 8
  9. Occurrence 9: pairing.tex, line 42, column 49
  10. Occurrence 10: non-enumerability.tex, line 92, column 33
  11. Occurrence 11: non-enumerability.tex, line 119, column 46
  12. Occurrence 12: comparing-size.tex, line 117, column 22
  13. Occurrence 13: comparing-size.tex, line 133, column 11
  14. Occurrence 14: enumerability-alt.tex, line 27, column 56
  15. Occurrence 15: enumerability-alt.tex, line 31, column 59
  16. Occurrence 16: enumerability-alt.tex, line 50, column 20
  17. Occurrence 17: enumerability-alt.tex, line 131, column 22
  18. Occurrence 18: non-enumerability-alt.tex, line 64, column 57
  19. Occurrence 19: non-enumerability-alt.tex, line 66, column 40
  20. Occurrence 20: non-enumerability-alt.tex, line 83, column 26
  21. Occurrence 21: non-enumerability-alt.tex, line 94, column 58
  22. Occurrence 22: non-enumerability-alt.tex, line 135, column 5
  23. Occurrence 23: non-enumerability-alt.tex, line 149, column 12

Expression 36

h(0,0)=20(2·0+1)−1=0

Conventional reading: h open parenthesis 0 comma 0 close parenthesis equals 2 to the power 0 times open parenthesis 2 times 0 plus 1 close parenthesis minus 1 equals 0

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 96, column 24

Expression 38

Z={1,…,n}

Conventional reading: Z equals open brace 1 comma and so on comma n close brace

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 262, column 13

Expression 42

x∈A

Conventional reading: x is an element of A

Meaning here: This expression states membership or nonmembership in the displayed set.

9 occurrences
  1. Occurrence 1: equinumerous-sets.tex, line 43, column 26
  2. Occurrence 2: equinumerous-sets.tex, line 74, column 64
  3. Occurrence 3: equinumerous-sets.tex, line 78, column 31
  4. Occurrence 4: comparing-size.tex, line 74, column 29
  5. Occurrence 5: comparing-size.tex, line 81, column 35
  6. Occurrence 6: comparing-size.tex, line 83, column 15
  7. Occurrence 7: comparing-size.tex, line 87, column 46
  8. Occurrence 8: comparing-size.tex, line 119, column 34
  9. Occurrence 9: comparing-size.tex, line 134, column 67

Expression 43

g:N2→N

Conventional reading: g from the natural numbers to the power 2 to the natural numbers

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: pairing.tex, line 19, column 1

Expression 46

(n+m)th

Conventional reading: the open parenthesis n plus m close parenthesis-th triangular number

Meaning here: This identifies the triangular number at index n plus m: the open parenthesis n plus m close parenthesis-th triangular number.

1 occurrence
  1. Occurrence 1: pairing.tex, line 42, column 1

Expression 48

f

Conventional reading: f

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

22 occurrences
  1. Occurrence 1: enumerability.tex, line 97, column 47
  2. Occurrence 2: enumerability.tex, line 144, column 29
  3. Occurrence 3: enumerability.tex, line 154, column 23
  4. Occurrence 4: enumerability.tex, line 222, column 24
  5. Occurrence 5: enumerability.tex, line 223, column 35
  6. Occurrence 6: enumerability.tex, line 245, column 40
  7. Occurrence 7: pairing.tex, line 53, column 30
  8. Occurrence 8: pairing.tex, line 53, column 65
  9. Occurrence 9: pairing.tex, line 108, column 26
  10. Occurrence 10: reduction.tex, line 69, column 4
  11. Occurrence 11: reduction.tex, line 75, column 7
  12. Occurrence 12: reduction.tex, line 76, column 22
  13. Occurrence 13: reduction.tex, line 92, column 63
  14. Occurrence 14: equinumerous-sets.tex, line 47, column 41
  15. Occurrence 15: equinumerous-sets.tex, line 77, column 53
  16. Occurrence 16: equinumerous-sets.tex, line 98, column 12
  17. Occurrence 17: enumerability-alt.tex, line 48, column 34
  18. Occurrence 18: enumerability-alt.tex, line 115, column 12
  19. Occurrence 19: enumerability-alt.tex, line 117, column 1
  20. Occurrence 20: reduction-alt.tex, line 72, column 4
  21. Occurrence 21: reduction-alt.tex, line 78, column 7
  22. Occurrence 22: reduction-alt.tex, line 79, column 22

Expression 49

f(n)=n

Conventional reading: f open parenthesis n close parenthesis equals n

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 120, column 28

Expression 50

f(n)=g(n−1)

Conventional reading: f open parenthesis n close parenthesis equals g open parenthesis n minus 1 close parenthesis

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 203, column 23

Expression 51

f(N)=s

Conventional reading: f open parenthesis N close parenthesis equals s

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: reduction-alt.tex, line 71, column 6

Expression 52

k∈N

Conventional reading: k is an element of the natural numbers

Meaning here: This expression states membership or nonmembership in the displayed set.

1 occurrence
  1. Occurrence 1: zig-zag.tex, line 73, column 50

Expression 55

A∩B=C∩D=∅

Conventional reading: A intersect B equals C intersect D equals the empty set

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: equinumerous-sets.tex, line 102, column 55

Expression 56

Z1={1,2,3,4,5,6,…}Z2={2,4,6,…}Z3={1,2,5,}Z4={3,4,5,6,…}⋮⋱

Conventional reading: array, row 1: Z sub 1 equals open brace; 1 comma; 2 comma; 3 comma; 4 comma; 5 comma; 6 comma; and so on close brace; row 2: Z sub 2 equals open brace; blank position; 2 comma; blank position; 4 comma; blank position; 6 comma; and so on close brace; row 3: Z sub 3 equals open brace; 1 comma; 2 comma; blank position; blank position; 5; blank position; close brace; row 4: Z sub 4 equals open brace; blank position; blank position; 3 comma; 4 comma; 5 comma; 6 comma; and so on close brace; row 5: vertical ellipsis; blank position; blank position; blank position; blank position; diagonal ellipsis

Meaning here: This multirow mathematical object presents the displayed values or cases in source order.

1 occurrence
  1. Occurrence 1: non-enumerability.tex, line 186, column 1

Expression 59

⋃i=1∞Ai

Conventional reading: the union, from i equals 1 to infinity, of A sub i

Meaning here: This denotes the union of the indexed family A sub i over every positive-integer index i.

1 occurrence
  1. Occurrence 1: pairing.tex, line 102, column 46

Expression 61

k(k+1)/2

Conventional reading: k times open parenthesis k plus 1 close parenthesis divided by 2

Meaning here: This denotes the value of the displayed function or indexed construction at its argument.

1 occurrence
  1. Occurrence 1: pairing.tex, line 35, column 1

Expression 63

f¯:Z+→Z+

Conventional reading: f with an overline from the positive integers to the positive integers

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: non-enumerability.tex, line 210, column 42

Expression 65

h:N2→N

Conventional reading: h from the natural numbers to the power 2 to the natural numbers

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 87, column 14

Expression 66

f:N→N×N

Conventional reading: f from the natural numbers to the natural numbers cross the natural numbers

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: zig-zag.tex, line 73, column 5

Expression 68

IdA:A→A

Conventional reading: the identity function on A from A to A

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: equinumerous-sets.tex, line 42, column 38

Expression 71

g(s)=s(1)

Conventional reading: g open parenthesis s close parenthesis equals s open parenthesis 1 close parenthesis

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: reduction.tex, line 89, column 41

Expression 73

x=g(D)

Conventional reading: x equals g open parenthesis D close parenthesis

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: comparing-size.tex, line 146, column 26

Expression 75

fi:Z+→Z+

Conventional reading: f sub i from the positive integers to the positive integers

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: non-enumerability.tex, line 209, column 57

Expression 76

012345…01357911…126101418……24122028………382440…………41648……………532………………⋮⋮⋮⋮⋮⋮⋮⋱

Conventional reading: array, row 1: blank position; 0; 1; 2; 3; 4; 5; and so on; row 2: 0; 1; 3; 5; 7; 9; 11; and so on; row 3: 1; 2; 6; 10; 14; 18; and so on; and so on; row 4: 2; 4; 12; 20; 28; and so on; and so on; and so on; row 5: 3; 8; 24; 40; and so on; and so on; and so on; and so on; row 6: 4; 16; 48; and so on; and so on; and so on; and so on; and so on; row 7: 5; 32; and so on; and so on; and so on; and so on; and so on; and so on; row 8: vertical ellipsis; vertical ellipsis; vertical ellipsis; vertical ellipsis; vertical ellipsis; vertical ellipsis; vertical ellipsis; diagonal ellipsis

Meaning here: This multirow mathematical object presents the displayed values or cases in source order.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 51, column 1

Expression 77

g:Z+→Z+

Conventional reading: g from the positive integers to the positive integers

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 125, column 49

Expression 81

s¯(k)=sk(k)

Conventional reading: s with an overline open parenthesis k close parenthesis equals s sub k open parenthesis k close parenthesis

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: non-enumerability.tex, line 121, column 1

Expression 82

2n·(2m+1)

Conventional reading: 2 to the power n times open parenthesis 2 m plus 1 close parenthesis

Meaning here: This denotes the displayed indexed object or power, as determined by its subscript or superscript.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 80, column 31

Expression 88

f(n)=2n andg(n)=2n+1

Conventional reading: f open parenthesis n close parenthesis equals 2 n and then g open parenthesis n close parenthesis equals 2 n plus 1

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: enumerability-alt.tex, line 86, column 1

Expression 89

f∘g:Z+→B

Conventional reading: f composed after g from the positive integers to B

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: equinumerous-sets.tex, line 76, column 27

Expression 91

g:Bω→B

Conventional reading: g from the set of binary digits to the power omega to B

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: reduction.tex, line 87, column 40

Expression 92

℘(N×N)

Conventional reading: the power set of the natural numbers cross the natural numbers

Meaning here: This expression involves a power set, the set of all subsets of the indicated set.

1 occurrence
  1. Occurrence 1: reduction-alt.tex, line 114, column 7

Expression 93

A¯≠g(x)

Conventional reading: A with an overline is not equal to g open parenthesis x close parenthesis

Meaning here: This asserts that the two displayed mathematical objects are not equal.

1 occurrence
  1. Occurrence 1: comparing-size.tex, line 83, column 57

Expression 94

g(i+1)

Conventional reading: g open parenthesis i plus 1 close parenthesis

Meaning here: This denotes the value of the displayed function or indexed construction at its argument.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 235, column 33

Expression 96

4×32+1=7

Conventional reading: the fraction 4 cross 3 over 2 plus 1 equals 7

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: pairing.tex, line 44, column 48

Expression 98

g(⟨n,m⟩)

Conventional reading: g open parenthesis the tuple n comma m close parenthesis

Meaning here: This denotes the value of g when its single argument is the ordered pair n comma m.

1 occurrence
  1. Occurrence 1: pairing.tex, line 40, column 43

Expression 99

g(n)=n−1

Conventional reading: g open parenthesis n close parenthesis equals n minus 1

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 121, column 40

Expression 100

s¯(1)

Conventional reading: s with an overline open parenthesis 1 close parenthesis

Meaning here: This denotes the value of the displayed function or indexed construction at its argument.

1 occurrence
  1. Occurrence 1: non-enumerability.tex, line 112, column 1

Expression 103

z∈Z

Conventional reading: z is an element of the integers

Meaning here: This expression states membership or nonmembership in the displayed set.

1 occurrence
  1. Occurrence 1: pairing.tex, line 72, column 41

Expression 110

4

Conventional reading: 4

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

8 occurrences
  1. Occurrence 1: enumerability.tex, line 45, column 57
  2. Occurrence 2: enumerability.tex, line 47, column 32
  3. Occurrence 3: enumerability.tex, line 54, column 52
  4. Occurrence 4: enumerability.tex, line 61, column 30
  5. Occurrence 5: enumerability.tex, line 137, column 52
  6. Occurrence 6: pairing-alt.tex, line 76, column 21
  7. Occurrence 7: non-enumerability.tex, line 199, column 5
  8. Occurrence 8: enumerability-alt.tex, line 96, column 50

Expression 112

v

Conventional reading: v

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

1 occurrence
  1. Occurrence 1: pairing.tex, line 30, column 50

Expression 113

N={0,…,n}

Conventional reading: N equals open brace 0 comma and so on comma n close brace

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 254, column 46

Expression 115

g(x)≠A¯

Conventional reading: g open parenthesis x close parenthesis is not equal to A with an overline

Meaning here: This asserts that the two displayed mathematical objects are not equal.

1 occurrence
  1. Occurrence 1: comparing-size.tex, line 90, column 3

Expression 116

3

Conventional reading: 3

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

10 occurrences
  1. Occurrence 1: enumerability.tex, line 49, column 8
  2. Occurrence 2: enumerability.tex, line 49, column 13
  3. Occurrence 3: enumerability.tex, line 50, column 3
  4. Occurrence 4: enumerability.tex, line 53, column 13
  5. Occurrence 5: enumerability.tex, line 54, column 47
  6. Occurrence 6: enumerability.tex, line 55, column 50
  7. Occurrence 7: enumerability.tex, line 61, column 8
  8. Occurrence 8: pairing.tex, line 33, column 25
  9. Occurrence 9: non-enumerability.tex, line 198, column 53
  10. Occurrence 10: non-enumerability-alt.tex, line 151, column 5

Expression 117

d(n)≠sn(n)

Conventional reading: d open parenthesis n close parenthesis is not equal to s sub n open parenthesis n close parenthesis

Meaning here: This asserts that the two displayed mathematical objects are not equal.

1 occurrence
  1. Occurrence 1: non-enumerability-alt.tex, line 93, column 48

Expression 125

A≺℘(A)

Conventional reading: A has smaller cardinality than the power set of A

Meaning here: This asserts that the first displayed set has strictly smaller cardinality than the second.

1 occurrence
  1. Occurrence 1: comparing-size.tex, line 57, column 1

Expression 127

s2

Conventional reading: s sub 2

Meaning here: This denotes the displayed indexed object or power, as determined by its subscript or superscript.

6 occurrences
  1. Occurrence 1: non-enumerability.tex, line 75, column 8
  2. Occurrence 2: non-enumerability.tex, line 82, column 47
  3. Occurrence 3: non-enumerability.tex, line 108, column 1
  4. Occurrence 4: non-enumerability.tex, line 113, column 59
  5. Occurrence 5: non-enumerability.tex, line 128, column 23
  6. Occurrence 6: non-enumerability-alt.tex, line 96, column 15

Expression 128

A

Conventional reading: A

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

98 occurrences
  1. Occurrence 1: enumerability.tex, line 28, column 44
  2. Occurrence 2: enumerability.tex, line 29, column 30
  3. Occurrence 3: enumerability.tex, line 29, column 65
  4. Occurrence 4: enumerability.tex, line 30, column 49
  5. Occurrence 5: enumerability.tex, line 31, column 19
  6. Occurrence 6: enumerability.tex, line 44, column 58
  7. Occurrence 7: enumerability.tex, line 69, column 6
  8. Occurrence 8: enumerability.tex, line 74, column 11
  9. Occurrence 9: enumerability.tex, line 75, column 30
  10. Occurrence 10: enumerability.tex, line 78, column 25
  11. Occurrence 11: enumerability.tex, line 95, column 49
  12. Occurrence 12: enumerability.tex, line 95, column 64
  13. Occurrence 13: enumerability.tex, line 98, column 22
  14. Occurrence 14: enumerability.tex, line 99, column 51
  15. Occurrence 15: enumerability.tex, line 105, column 4
  16. Occurrence 16: enumerability.tex, line 108, column 67
  17. Occurrence 17: enumerability.tex, line 115, column 9
  18. Occurrence 18: enumerability.tex, line 165, column 16
  19. Occurrence 19: enumerability.tex, line 169, column 49
  20. Occurrence 20: enumerability.tex, line 173, column 36
  21. Occurrence 21: enumerability.tex, line 182, column 25
  22. Occurrence 22: enumerability.tex, line 191, column 25
  23. Occurrence 23: enumerability.tex, line 209, column 7
  24. Occurrence 24: enumerability.tex, line 213, column 57
  25. Occurrence 25: enumerability.tex, line 217, column 59
  26. Occurrence 26: enumerability.tex, line 220, column 4
  27. Occurrence 27: enumerability.tex, line 225, column 47
  28. Occurrence 28: enumerability.tex, line 229, column 4
  29. Occurrence 29: enumerability.tex, line 237, column 11
  30. Occurrence 30: enumerability.tex, line 239, column 14
  31. Occurrence 31: enumerability.tex, line 240, column 33
  32. Occurrence 32: enumerability.tex, line 244, column 60
  33. Occurrence 33: enumerability.tex, line 253, column 7
  34. Occurrence 34: enumerability.tex, line 259, column 1
  35. Occurrence 35: enumerability.tex, line 259, column 27
  36. Occurrence 36: enumerability.tex, line 269, column 62
  37. Occurrence 37: non-enumerability.tex, line 26, column 53
  38. Occurrence 38: non-enumerability.tex, line 29, column 58
  39. Occurrence 39: non-enumerability.tex, line 30, column 16
  40. Occurrence 40: non-enumerability.tex, line 30, column 60
  41. Occurrence 41: non-enumerability.tex, line 35, column 27
  42. Occurrence 42: non-enumerability.tex, line 37, column 17
  43. Occurrence 43: non-enumerability.tex, line 40, column 4
  44. Occurrence 44: non-enumerability.tex, line 40, column 67
  45. Occurrence 45: reduction.tex, line 34, column 19
  46. Occurrence 46: reduction.tex, line 35, column 4
  47. Occurrence 47: reduction.tex, line 37, column 20
  48. Occurrence 48: reduction.tex, line 43, column 38
  49. Occurrence 49: reduction.tex, line 44, column 39
  50. Occurrence 50: equinumerous-sets.tex, line 30, column 3
  51. Occurrence 51: equinumerous-sets.tex, line 58, column 26
  52. Occurrence 52: equinumerous-sets.tex, line 70, column 26
  53. Occurrence 53: equinumerous-sets.tex, line 96, column 42
  54. Occurrence 54: equinumerous-sets.tex, line 107, column 14
  55. Occurrence 55: comparing-size.tex, line 27, column 1
  56. Occurrence 56: comparing-size.tex, line 37, column 1
  57. Occurrence 57: comparing-size.tex, line 44, column 20
  58. Occurrence 58: comparing-size.tex, line 45, column 6
  59. Occurrence 59: comparing-size.tex, line 57, column 38
  60. Occurrence 60: comparing-size.tex, line 82, column 28
  61. Occurrence 61: comparing-size.tex, line 117, column 50
  62. Occurrence 62: comparing-size.tex, line 120, column 67
  63. Occurrence 63: comparing-size.tex, line 133, column 39
  64. Occurrence 64: comparing-size.tex, line 144, column 19
  65. Occurrence 65: schroder-bernstein.tex, line 14, column 34
  66. Occurrence 66: schroder-bernstein.tex, line 15, column 16
  67. Occurrence 67: schroder-bernstein.tex, line 15, column 26
  68. Occurrence 68: schroder-bernstein.tex, line 30, column 49
  69. Occurrence 69: schroder-bernstein.tex, line 31, column 28
  70. Occurrence 70: schroder-bernstein.tex, line 31, column 67
  71. Occurrence 71: enumerability-alt.tex, line 27, column 38
  72. Occurrence 72: enumerability-alt.tex, line 30, column 36
  73. Occurrence 73: enumerability-alt.tex, line 31, column 24
  74. Occurrence 74: enumerability-alt.tex, line 32, column 27
  75. Occurrence 75: enumerability-alt.tex, line 40, column 32
  76. Occurrence 76: enumerability-alt.tex, line 41, column 1
  77. Occurrence 77: enumerability-alt.tex, line 47, column 62
  78. Occurrence 78: enumerability-alt.tex, line 49, column 21
  79. Occurrence 79: enumerability-alt.tex, line 59, column 9
  80. Occurrence 80: enumerability-alt.tex, line 60, column 21
  81. Occurrence 81: enumerability-alt.tex, line 60, column 38
  82. Occurrence 82: enumerability-alt.tex, line 60, column 67
  83. Occurrence 83: enumerability-alt.tex, line 78, column 17
  84. Occurrence 84: enumerability-alt.tex, line 79, column 58
  85. Occurrence 85: enumerability-alt.tex, line 127, column 14
  86. Occurrence 86: non-enumerability-alt.tex, line 26, column 50
  87. Occurrence 87: non-enumerability-alt.tex, line 29, column 27
  88. Occurrence 88: non-enumerability-alt.tex, line 29, column 47
  89. Occurrence 89: non-enumerability-alt.tex, line 29, column 59
  90. Occurrence 90: non-enumerability-alt.tex, line 30, column 55
  91. Occurrence 91: non-enumerability-alt.tex, line 35, column 54
  92. Occurrence 92: non-enumerability-alt.tex, line 39, column 17
  93. Occurrence 93: non-enumerability-alt.tex, line 40, column 16
  94. Occurrence 94: reduction-alt.tex, line 35, column 7
  95. Occurrence 95: reduction-alt.tex, line 35, column 58
  96. Occurrence 96: reduction-alt.tex, line 38, column 12
  97. Occurrence 97: reduction-alt.tex, line 44, column 38
  98. Occurrence 98: reduction-alt.tex, line 45, column 39

Expression 129

⟨1,2⟩

Conventional reading: the tuple 1 comma 2

Meaning here: This denotes the displayed ordered tuple, whose coordinate order is significant.

1 occurrence
  1. Occurrence 1: pairing.tex, line 44, column 22

Expression 130

f:℘(N)→Bω

Conventional reading: f from the power set of the natural numbers to the set of binary digits to the power omega

Meaning here: This declares f as a function from subsets of the natural numbers to infinite binary sequences.

1 occurrence
  1. Occurrence 1: reduction-alt.tex, line 39, column 49

Expression 133

s¯(k)≠sk(k)

Conventional reading: s with an overline open parenthesis k close parenthesis is not equal to s sub k open parenthesis k close parenthesis

Meaning here: This asserts that the two displayed mathematical objects are not equal.

1 occurrence
  1. Occurrence 1: non-enumerability.tex, line 125, column 6

Expression 137

1234…1s1(1)s1(2)s1(3)s1(4)…2s2(1)s2(2)s2(3)s2(4)…3s3(1)s3(2)s3(3)s3(4)…4s4(1)s4(2)s4(3)s4(4)…⋮⋮⋮⋮⋮⋱

Conventional reading: array, row 1: blank position; 1; 2; 3; 4; and so on; row 2: 1; s sub 1 open parenthesis 1 close parenthesis; s sub 1 open parenthesis 2 close parenthesis; s sub 1 open parenthesis 3 close parenthesis; s sub 1 open parenthesis 4 close parenthesis; and so on; row 3: 2; s sub 2 open parenthesis 1 close parenthesis; s sub 2 open parenthesis 2 close parenthesis; s sub 2 open parenthesis 3 close parenthesis; s sub 2 open parenthesis 4 close parenthesis; and so on; row 4: 3; s sub 3 open parenthesis 1 close parenthesis; s sub 3 open parenthesis 2 close parenthesis; s sub 3 open parenthesis 3 close parenthesis; s sub 3 open parenthesis 4 close parenthesis; and so on; row 5: 4; s sub 4 open parenthesis 1 close parenthesis; s sub 4 open parenthesis 2 close parenthesis; s sub 4 open parenthesis 3 close parenthesis; s sub 4 open parenthesis 4 close parenthesis; and so on; row 6: vertical ellipsis; vertical ellipsis; vertical ellipsis; vertical ellipsis; vertical ellipsis; diagonal ellipsis

Meaning here: This array lists one-indexed binary sequences by rows and their entries by columns; bold cells mark the main diagonal.

1 occurrence
  1. Occurrence 1: non-enumerability.tex, line 64, column 1

Expression 139

Z+

Conventional reading: the positive integers

Meaning here: This denotes the named number system, distinguished set, or infinite value used in the surrounding statement.

20 occurrences
  1. Occurrence 1: size-of-sets-complete.tex, line 13, column 53
  2. Occurrence 2: enumerability.tex, line 15, column 20
  3. Occurrence 3: enumerability.tex, line 95, column 30
  4. Occurrence 4: enumerability.tex, line 119, column 47
  5. Occurrence 5: enumerability.tex, line 133, column 1
  6. Occurrence 6: enumerability.tex, line 219, column 17
  7. Occurrence 7: enumerability.tex, line 225, column 33
  8. Occurrence 8: enumerability.tex, line 230, column 8
  9. Occurrence 9: non-enumerability.tex, line 20, column 28
  10. Occurrence 10: non-enumerability.tex, line 29, column 45
  11. Occurrence 11: non-enumerability.tex, line 154, column 4
  12. Occurrence 12: non-enumerability.tex, line 154, column 35
  13. Occurrence 13: non-enumerability.tex, line 155, column 53
  14. Occurrence 14: reduction.tex, line 58, column 1
  15. Occurrence 15: reduction.tex, line 100, column 19
  16. Occurrence 16: comparing-size.tex, line 111, column 38
  17. Occurrence 17: comparing-size.tex, line 118, column 6
  18. Occurrence 18: size-of-sets-complete.tex, line 46, column 26
  19. Occurrence 19: enumerability-alt.tex, line 53, column 4
  20. Occurrence 20: non-enumerability-alt.tex, line 17, column 3

Expression 140

n∈N

Conventional reading: n is an element of the natural numbers

Meaning here: This expression states membership or nonmembership in the displayed set.

9 occurrences
  1. Occurrence 1: enumerability.tex, line 201, column 19
  2. Occurrence 2: enumerability.tex, line 255, column 6
  3. Occurrence 3: zig-zag.tex, line 115, column 39
  4. Occurrence 4: zig-zag.tex, line 119, column 54
  5. Occurrence 5: comparing-size.tex, line 132, column 17
  6. Occurrence 6: non-enumerability-alt.tex, line 79, column 42
  7. Occurrence 7: non-enumerability-alt.tex, line 94, column 5
  8. Occurrence 8: non-enumerability-alt.tex, line 128, column 5
  9. Occurrence 9: reduction-alt.tex, line 57, column 5

Expression 141

g:Bω→B

Conventional reading: g from the set of binary digits to the power omega to the set of binary digits

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: reduction.tex, line 88, column 69

Expression 143

s¯(k)=0

Conventional reading: s with an overline open parenthesis k close parenthesis equals 0

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: non-enumerability.tex, line 124, column 39

Expression 146

IdN(n)=n

Conventional reading: the identity function on the natural numbers open parenthesis n close parenthesis equals n

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: enumerability-alt.tex, line 71, column 52

Expression 147

{0,…,n−1}

Conventional reading: open brace 0 comma and so on comma n minus 1 close brace

Meaning here: This denotes the unordered initial segment of natural numbers from 0 through n minus 1.

1 occurrence
  1. Occurrence 1: enumerability-alt.tex, line 31, column 32

Expression 149

0

Conventional reading: 0

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

38 occurrences
  1. Occurrence 1: enumerability.tex, line 189, column 16
  2. Occurrence 2: pairing.tex, line 33, column 15
  3. Occurrence 3: pairing-alt.tex, line 19, column 35
  4. Occurrence 4: pairing-alt.tex, line 70, column 34
  5. Occurrence 5: pairing-alt.tex, line 82, column 47
  6. Occurrence 6: non-enumerability.tex, line 44, column 14
  7. Occurrence 7: non-enumerability.tex, line 55, column 50
  8. Occurrence 8: non-enumerability.tex, line 77, column 20
  9. Occurrence 9: non-enumerability.tex, line 80, column 59
  10. Occurrence 10: non-enumerability.tex, line 83, column 40
  11. Occurrence 11: non-enumerability.tex, line 90, column 55
  12. Occurrence 12: non-enumerability.tex, line 91, column 1
  13. Occurrence 13: non-enumerability.tex, line 91, column 66
  14. Occurrence 14: non-enumerability.tex, line 93, column 21
  15. Occurrence 15: non-enumerability.tex, line 104, column 51
  16. Occurrence 16: non-enumerability.tex, line 105, column 64
  17. Occurrence 17: non-enumerability.tex, line 114, column 25
  18. Occurrence 18: non-enumerability.tex, line 122, column 1
  19. Occurrence 19: non-enumerability.tex, line 122, column 22
  20. Occurrence 20: non-enumerability.tex, line 130, column 29
  21. Occurrence 21: non-enumerability.tex, line 131, column 14
  22. Occurrence 22: reduction.tex, line 22, column 23
  23. Occurrence 23: reduction.tex, line 60, column 46
  24. Occurrence 24: reduction.tex, line 90, column 25
  25. Occurrence 25: reduction.tex, line 91, column 54
  26. Occurrence 26: enumerability-alt.tex, line 28, column 9
  27. Occurrence 27: enumerability-alt.tex, line 49, column 30
  28. Occurrence 28: enumerability-alt.tex, line 51, column 6
  29. Occurrence 29: non-enumerability-alt.tex, line 44, column 53
  30. Occurrence 30: non-enumerability-alt.tex, line 64, column 20
  31. Occurrence 31: non-enumerability-alt.tex, line 77, column 51
  32. Occurrence 32: non-enumerability-alt.tex, line 81, column 60
  33. Occurrence 33: non-enumerability-alt.tex, line 82, column 16
  34. Occurrence 34: non-enumerability-alt.tex, line 82, column 61
  35. Occurrence 35: non-enumerability-alt.tex, line 84, column 11
  36. Occurrence 36: non-enumerability-alt.tex, line 92, column 64
  37. Occurrence 37: non-enumerability-alt.tex, line 150, column 17
  38. Occurrence 38: reduction-alt.tex, line 63, column 44

Expression 151

⟨0,0⟩,⟨0,1⟩,⟨1,0⟩,⟨0,2⟩,⟨1,1⟩,⟨2,0⟩,⟨0,3⟩,⟨1,2⟩,⟨2,1⟩,⟨3,0⟩,…

Conventional reading: the tuple 0 comma 0 comma the tuple 0 comma 1 comma the tuple 1 comma 0 comma the tuple 0 comma 2 comma the tuple 1 comma 1 comma the tuple 2 comma 0 comma the tuple 0 comma 3 comma the tuple 1 comma 2 comma the tuple 2 comma 1 comma the tuple 3 comma 0 comma and so on

Meaning here: This is an ordered list of distinct ordered pairs following Cantor's zig-zag enumeration.

1 occurrence
  1. Occurrence 1: zig-zag.tex, line 61, column 1

Expression 153

f(x,y)

Conventional reading: f open parenthesis x comma y close parenthesis

Meaning here: This denotes the value of the displayed function or indexed construction at its argument.

1 occurrence
  1. Occurrence 1: pairing.tex, line 54, column 41

Expression 156

m

Conventional reading: m

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

7 occurrences
  1. Occurrence 1: zig-zag.tex, line 39, column 19
  2. Occurrence 2: zig-zag.tex, line 75, column 19
  3. Occurrence 3: pairing.tex, line 30, column 22
  4. Occurrence 4: non-enumerability-alt.tex, line 65, column 5
  5. Occurrence 5: non-enumerability-alt.tex, line 66, column 54
  6. Occurrence 6: non-enumerability-alt.tex, line 135, column 26
  7. Occurrence 7: non-enumerability-alt.tex, line 135, column 37

Expression 157

A⊆N

Conventional reading: A is a subset of the natural numbers

Meaning here: This asserts that every member of the first displayed set also belongs to the second.

1 occurrence
  1. Occurrence 1: pairing.tex, line 93, column 45

Expression 161

N+=N∖{0}

Conventional reading: the positive natural numbers equal the natural numbers with zero removed

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: enumerability-alt.tex, line 72, column 59

Expression 162

Bk→B

Conventional reading: the set of binary digits to the power k to the set of binary digits

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: pairing.tex, line 82, column 24

Expression 163

f:℘(Z+)→Bω

Conventional reading: f from the power set of the positive integers to the set of binary digits to the power omega

Meaning here: This declares f as a function from subsets of the positive integers to infinite binary sequences.

1 occurrence
  1. Occurrence 1: reduction.tex, line 51, column 21

Expression 166

N≺A

Conventional reading: the cardinality of the natural numbers is smaller than the cardinality of A

Meaning here: This asserts that the first displayed set has strictly smaller cardinality than the second.

1 occurrence
  1. Occurrence 1: comparing-size.tex, line 45, column 35

Expression 168

f:A×B→N

Conventional reading: f from A cross B to the natural numbers

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: pairing.tex, line 107, column 5

Expression 169

1

Conventional reading: 1

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

53 occurrences
  1. Occurrence 1: enumerability.tex, line 43, column 28
  2. Occurrence 2: enumerability.tex, line 45, column 62
  3. Occurrence 3: enumerability.tex, line 47, column 27
  4. Occurrence 4: enumerability.tex, line 48, column 54
  5. Occurrence 5: enumerability.tex, line 48, column 59
  6. Occurrence 6: enumerability.tex, line 49, column 53
  7. Occurrence 7: enumerability.tex, line 53, column 3
  8. Occurrence 8: enumerability.tex, line 53, column 18
  9. Occurrence 9: enumerability.tex, line 54, column 37
  10. Occurrence 10: enumerability.tex, line 55, column 60
  11. Occurrence 11: enumerability.tex, line 61, column 3
  12. Occurrence 12: enumerability.tex, line 137, column 47
  13. Occurrence 13: enumerability.tex, line 187, column 32
  14. Occurrence 14: enumerability.tex, line 189, column 23
  15. Occurrence 15: pairing.tex, line 33, column 20
  16. Occurrence 16: pairing-alt.tex, line 76, column 11
  17. Occurrence 17: pairing-alt.tex, line 82, column 10
  18. Occurrence 18: pairing-alt.tex, line 82, column 68
  19. Occurrence 19: non-enumerability.tex, line 44, column 24
  20. Occurrence 20: non-enumerability.tex, line 55, column 60
  21. Occurrence 21: non-enumerability.tex, line 77, column 29
  22. Occurrence 22: non-enumerability.tex, line 81, column 1
  23. Occurrence 23: non-enumerability.tex, line 83, column 50
  24. Occurrence 24: non-enumerability.tex, line 90, column 46
  25. Occurrence 25: non-enumerability.tex, line 91, column 10
  26. Occurrence 26: non-enumerability.tex, line 92, column 4
  27. Occurrence 27: non-enumerability.tex, line 93, column 14
  28. Occurrence 28: non-enumerability.tex, line 105, column 1
  29. Occurrence 29: non-enumerability.tex, line 106, column 5
  30. Occurrence 30: non-enumerability.tex, line 114, column 51
  31. Occurrence 31: non-enumerability.tex, line 122, column 8
  32. Occurrence 32: non-enumerability.tex, line 122, column 57
  33. Occurrence 33: non-enumerability.tex, line 130, column 39
  34. Occurrence 34: non-enumerability.tex, line 131, column 24
  35. Occurrence 35: non-enumerability.tex, line 198, column 32
  36. Occurrence 36: reduction.tex, line 22, column 32
  37. Occurrence 37: reduction.tex, line 60, column 55
  38. Occurrence 38: reduction.tex, line 63, column 5
  39. Occurrence 39: reduction.tex, line 90, column 35
  40. Occurrence 40: reduction.tex, line 92, column 6
  41. Occurrence 41: enumerability-alt.tex, line 51, column 46
  42. Occurrence 42: enumerability-alt.tex, line 96, column 45
  43. Occurrence 43: non-enumerability-alt.tex, line 44, column 63
  44. Occurrence 44: non-enumerability-alt.tex, line 64, column 30
  45. Occurrence 45: non-enumerability-alt.tex, line 77, column 61
  46. Occurrence 46: non-enumerability-alt.tex, line 81, column 51
  47. Occurrence 47: non-enumerability-alt.tex, line 82, column 7
  48. Occurrence 48: non-enumerability-alt.tex, line 82, column 68
  49. Occurrence 49: non-enumerability-alt.tex, line 84, column 4
  50. Occurrence 50: non-enumerability-alt.tex, line 93, column 5
  51. Occurrence 51: non-enumerability-alt.tex, line 150, column 25
  52. Occurrence 52: reduction-alt.tex, line 63, column 53
  53. Occurrence 53: reduction-alt.tex, line 66, column 12

Expression 172

f(x)≠f(y)

Conventional reading: f open parenthesis x close parenthesis is not equal to f open parenthesis y close parenthesis

Meaning here: This asserts that the two displayed mathematical objects are not equal.

1 occurrence
  1. Occurrence 1: comparing-size.tex, line 63, column 8

Expression 175

N×N={⟨n,m⟩:n,m∈N}

Conventional reading: the natural numbers cross the natural numbers equals the set of ordered pairs n comma m such that n and m are natural numbers

Meaning here: This is set-builder notation specifying exactly which objects belong to the displayed set.

1 occurrence
  1. Occurrence 1: zig-zag.tex, line 18, column 1

Expression 180

f:℘(N)→Bω

Conventional reading: f from the power set of the natural numbers to the set of binary digits to the power omega

Meaning here: This declares f as a function from subsets of the natural numbers to infinite binary sequences.

1 occurrence
  1. Occurrence 1: reduction-alt.tex, line 52, column 21

Expression 181

12345678910…⟨0,0⟩⟨1,0⟩⟨0,1⟩⟨0,2⟩⟨1,1⟩⟨0,3⟩⟨0,4⟩⟨1,2⟩…

Conventional reading: position 1: the tuple 0 comma 0; position 2: the tuple 1 comma 0; position 3: the tuple 0 comma 1; position 4: blank; position 5: the tuple 0 comma 2; position 6: the tuple 1 comma 1; position 7: the tuple 0 comma 3; position 8: blank; position 9: the tuple 0 comma 4; position 10: the tuple 1 comma 2; and the pattern continues

Meaning here: This intermediate position table adds pairs with first coordinate one while preserving the still-unfilled positions.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 32, column 1

Expression 184

N2→N

Conventional reading: the natural numbers to the power 2 to the natural numbers

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 111, column 28

Expression 185

IdN:N→N

Conventional reading: the identity function on the natural numbers from the natural numbers to the natural numbers

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: enumerability-alt.tex, line 71, column 10

Expression 189

s1

Conventional reading: s sub 1

Meaning here: This denotes the displayed indexed object or power, as determined by its subscript or superscript.

7 occurrences
  1. Occurrence 1: non-enumerability.tex, line 75, column 1
  2. Occurrence 2: non-enumerability.tex, line 82, column 40
  3. Occurrence 3: non-enumerability.tex, line 107, column 65
  4. Occurrence 4: non-enumerability.tex, line 110, column 45
  5. Occurrence 5: non-enumerability.tex, line 111, column 1
  6. Occurrence 6: non-enumerability.tex, line 128, column 16
  7. Occurrence 7: non-enumerability-alt.tex, line 96, column 8

Expression 190

7

Conventional reading: 7

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

1 occurrence
  1. Occurrence 1: pairing.tex, line 16, column 50

Expression 191

N

Conventional reading: the natural numbers

Meaning here: This denotes the named number system, distinguished set, or infinite value used in the surrounding statement.

26 occurrences
  1. Occurrence 1: size-of-sets-complete.tex, line 14, column 64
  2. Occurrence 2: enumerability.tex, line 18, column 35
  3. Occurrence 3: enumerability.tex, line 121, column 17
  4. Occurrence 4: enumerability.tex, line 188, column 28
  5. Occurrence 5: enumerability.tex, line 191, column 15
  6. Occurrence 6: zig-zag.tex, line 85, column 36
  7. Occurrence 7: zig-zag.tex, line 92, column 30
  8. Occurrence 8: pairing.tex, line 92, column 13
  9. Occurrence 9: pairing.tex, line 93, column 28
  10. Occurrence 10: pairing.tex, line 95, column 61
  11. Occurrence 11: comparing-size.tex, line 129, column 50
  12. Occurrence 12: comparing-size.tex, line 133, column 58
  13. Occurrence 13: size-of-sets-complete.tex, line 44, column 56
  14. Occurrence 14: enumerability-alt.tex, line 15, column 16
  15. Occurrence 15: enumerability-alt.tex, line 37, column 12
  16. Occurrence 16: enumerability-alt.tex, line 42, column 56
  17. Occurrence 17: enumerability-alt.tex, line 52, column 55
  18. Occurrence 18: enumerability-alt.tex, line 92, column 4
  19. Occurrence 19: non-enumerability-alt.tex, line 16, column 30
  20. Occurrence 20: non-enumerability-alt.tex, line 21, column 9
  21. Occurrence 21: non-enumerability-alt.tex, line 29, column 17
  22. Occurrence 22: non-enumerability-alt.tex, line 120, column 4
  23. Occurrence 23: non-enumerability-alt.tex, line 120, column 32
  24. Occurrence 24: non-enumerability-alt.tex, line 121, column 44
  25. Occurrence 25: reduction-alt.tex, line 60, column 41
  26. Occurrence 26: reduction-alt.tex, line 131, column 48

Expression 196

2m+1

Conventional reading: 2 m plus 1

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 71, column 55

Expression 197

s¯

Conventional reading: s with an overline

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

13 occurrences
  1. Occurrence 1: non-enumerability.tex, line 80, column 40
  2. Occurrence 2: non-enumerability.tex, line 82, column 1
  3. Occurrence 3: non-enumerability.tex, line 84, column 22
  4. Occurrence 4: non-enumerability.tex, line 87, column 11
  5. Occurrence 5: non-enumerability.tex, line 104, column 9
  6. Occurrence 6: non-enumerability.tex, line 106, column 57
  7. Occurrence 7: non-enumerability.tex, line 113, column 31
  8. Occurrence 8: non-enumerability.tex, line 117, column 20
  9. Occurrence 9: non-enumerability.tex, line 123, column 16
  10. Occurrence 10: non-enumerability.tex, line 124, column 23
  11. Occurrence 11: non-enumerability.tex, line 129, column 10
  12. Occurrence 12: non-enumerability.tex, line 134, column 33
  13. Occurrence 13: non-enumerability.tex, line 141, column 33

Expression 199

s¯∈Bω

Conventional reading: s with an overline is an element of the set of binary digits to the power omega

Meaning here: This expression states membership or nonmembership in the displayed set.

1 occurrence
  1. Occurrence 1: non-enumerability.tex, line 131, column 37

Expression 202

f:N→B

Conventional reading: f from the natural numbers to the set of binary digits

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: reduction-alt.tex, line 132, column 54

Expression 203

n∈Z+

Conventional reading: n is an element of the positive integers

Meaning here: This expression states membership or nonmembership in the displayed set.

10 occurrences
  1. Occurrence 1: enumerability.tex, line 99, column 6
  2. Occurrence 2: enumerability.tex, line 216, column 58
  3. Occurrence 3: enumerability.tex, line 230, column 48
  4. Occurrence 4: enumerability.tex, line 262, column 44
  5. Occurrence 5: non-enumerability.tex, line 88, column 44
  6. Occurrence 6: non-enumerability.tex, line 159, column 54
  7. Occurrence 7: non-enumerability.tex, line 171, column 14
  8. Occurrence 8: reduction.tex, line 54, column 37
  9. Occurrence 9: equinumerous-sets.tex, line 79, column 38
  10. Occurrence 10: comparing-size.tex, line 113, column 59

Expression 205

(2m+1)

Conventional reading: open parenthesis 2 m plus 1 close parenthesis

Meaning here: This denotes the value of the displayed function or indexed construction at its argument.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 75, column 57

Expression 206

v+1

Conventional reading: v plus 1

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

1 occurrence
  1. Occurrence 1: pairing.tex, line 31, column 51

Expression 207

(n+1)

Conventional reading: open parenthesis n plus 1 close parenthesis

Meaning here: This denotes the value of the displayed function or indexed construction at its argument.

1 occurrence
  1. Occurrence 1: pairing.tex, line 31, column 1

Expression 208

s¯(n)=1−sn(n)

Conventional reading: s with an overline open parenthesis n close parenthesis equals 1 minus s sub n open parenthesis n close parenthesis

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: non-enumerability.tex, line 102, column 8

Expression 209

f:Z+→B

Conventional reading: f from the positive integers to the set of binary digits

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: reduction.tex, line 130, column 16

Expression 211

D={n∈N:n∉Nn}

Conventional reading: D equals the set of natural numbers n such that n is not an element of N sub n

Meaning here: This is set-builder notation specifying exactly which objects belong to the displayed set.

1 occurrence
  1. Occurrence 1: non-enumerability-alt.tex, line 123, column 1

Expression 216

g(i−1)

Conventional reading: g open parenthesis i minus 1 close parenthesis

Meaning here: This denotes the value of the displayed function or indexed construction at its argument.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 248, column 53

Expression 220

f(N1),f(N2),f(N3),…

Conventional reading: f open parenthesis N sub 1 close parenthesis comma f open parenthesis N sub 2 close parenthesis comma f open parenthesis N sub 3 close parenthesis comma and so on

Meaning here: This lists the successive function values f of N sub 1, f of N sub 2, and so on.

1 occurrence
  1. Occurrence 1: reduction-alt.tex, line 75, column 1

Expression 222

(Z+)∗

Conventional reading: the positive integers Kleene star: the set of all finite sequences of positive integers, including the empty sequence

Meaning here: This denotes the Kleene closure of the positive integers: all finite sequences of positive integers, including the empty sequence.

1 occurrence
  1. Occurrence 1: zig-zag.tex, line 123, column 13

Expression 229

{x}≠{y}

Conventional reading: open brace x close brace is not equal to open brace y close brace

Meaning here: This asserts that the two displayed mathematical objects are not equal.

1 occurrence
  1. Occurrence 1: comparing-size.tex, line 62, column 32

Expression 232

(Z+)n

Conventional reading: open parenthesis the positive integers close parenthesis to the power n

Meaning here: This denotes the displayed indexed object or power, as determined by its subscript or superscript.

1 occurrence
  1. Occurrence 1: zig-zag.tex, line 119, column 11

Expression 235

01234…0013610…124711……25812………3913…………414……………⋮⋮⋮⋮⋮…⋱

Conventional reading: array, row 1: blank position; 0; 1; 2; 3; 4; and so on; row 2: 0; 0; 1; 3; 6; 10; and so on; row 3: 1; 2; 4; 7; 11; and so on; and so on; row 4: 2; 5; 8; 12; and so on; and so on; and so on; row 5: 3; 9; 13; and so on; and so on; and so on; and so on; row 6: 4; 14; and so on; and so on; and so on; and so on; and so on; row 7: vertical ellipsis; vertical ellipsis; vertical ellipsis; vertical ellipsis; vertical ellipsis; and so on; diagonal ellipsis

Meaning here: This multirow mathematical object presents the displayed values or cases in source order.

1 occurrence
  1. Occurrence 1: zig-zag.tex, line 42, column 1

Expression 240

s¯(n)={1if sn(n)=00if sn(n)=1.

Conventional reading: s with an overline at n equals: 1 if s sub n open parenthesis n close parenthesis equals 0; 0 if s sub n open parenthesis n close parenthesis equals 1

Meaning here: This piecewise definition flips the nth diagonal bit of the nth sequence to construct the complementary sequence.

1 occurrence
  1. Occurrence 1: non-enumerability.tex, line 94, column 1

Expression 241

f(n)={n2if n is even−n+12if n is odd

Conventional reading: f open parenthesis n close parenthesis equals: n divided by 2 if n is even; negative open parenthesis n plus 1 close parenthesis divided by 2 if n is odd

Meaning here: This piecewise function maps even and odd natural-number inputs to integers in alternating order.

1 occurrence
  1. Occurrence 1: enumerability-alt.tex, line 118, column 1

Expression 243

D={g(B):B⊆A and g(B)∉B}

Conventional reading: D equals the set of g open parenthesis B close parenthesis such that B is a subset of A and g open parenthesis B close parenthesis is not an element of B

Meaning here: This is set-builder notation specifying exactly which objects belong to the displayed set.

1 occurrence
  1. Occurrence 1: comparing-size.tex, line 145, column 27

Expression 245

A≈N

Conventional reading: A is equinumerous with the natural numbers

Meaning here: This asserts that a bijection exists between the two displayed sets, so they have the same cardinality.

1 occurrence
  1. Occurrence 1: equinumerous-sets.tex, line 108, column 1

Expression 247

0123…0⟨0,0⟩⟨0,1⟩⟨0,2⟩⟨0,3⟩…1⟨1,0⟩⟨1,1⟩⟨1,2⟩⟨1,3⟩…2⟨2,0⟩⟨2,1⟩⟨2,2⟩⟨2,3⟩…3⟨3,0⟩⟨3,1⟩⟨3,2⟩⟨3,3⟩…⋮⋮⋮⋮⋮⋱

Conventional reading: array, row 1: blank position; 0; 1; 2; 3; and so on; row 2: 0; the tuple 0 comma 0; the tuple 0 comma 1; the tuple 0 comma 2; the tuple 0 comma 3; and so on; row 3: 1; the tuple 1 comma 0; the tuple 1 comma 1; the tuple 1 comma 2; the tuple 1 comma 3; and so on; row 4: 2; the tuple 2 comma 0; the tuple 2 comma 1; the tuple 2 comma 2; the tuple 2 comma 3; and so on; row 5: 3; the tuple 3 comma 0; the tuple 3 comma 1; the tuple 3 comma 2; the tuple 3 comma 3; and so on; row 6: vertical ellipsis; vertical ellipsis; vertical ellipsis; vertical ellipsis; vertical ellipsis; diagonal ellipsis

Meaning here: This multirow mathematical object presents the displayed values or cases in source order.

1 occurrence
  1. Occurrence 1: zig-zag.tex, line 22, column 1

Expression 250

g(n,m)=(n+m+1)(n+m)2+n

Conventional reading: g open parenthesis n comma m close parenthesis equals the fraction open parenthesis n plus m plus 1 close parenthesis times open parenthesis n plus m close parenthesis over 2, plus n

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: pairing.tex, line 37, column 1

Expression 251

010101…

Conventional reading: zero one zero one zero one, continuing indefinitely

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

1 occurrence
  1. Occurrence 1: reduction.tex, line 57, column 1

Expression 253

z/m

Conventional reading: z divided by m

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

1 occurrence
  1. Occurrence 1: pairing.tex, line 72, column 30

Expression 263

f(n)=2n andg(n)=2n−1

Conventional reading: f open parenthesis n close parenthesis equals 2 n and then g open parenthesis n close parenthesis equals 2 n minus 1

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 127, column 1

Expression 265

f(Z)

Conventional reading: f open parenthesis Z close parenthesis

Meaning here: This denotes the value of the displayed function or indexed construction at its argument.

1 occurrence
  1. Occurrence 1: reduction.tex, line 52, column 1

Expression 267

j(n,m)=2n3m

Conventional reading: j open parenthesis n comma m close parenthesis equals 2 to the power n times 3 to the power m

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 108, column 1

Expression 270

A¯

Conventional reading: A with an overline

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

7 occurrences
  1. Occurrence 1: comparing-size.tex, line 81, column 46
  2. Occurrence 2: comparing-size.tex, line 85, column 31
  3. Occurrence 3: comparing-size.tex, line 87, column 24
  4. Occurrence 4: comparing-size.tex, line 90, column 46
  5. Occurrence 5: comparing-size.tex, line 118, column 25
  6. Occurrence 6: comparing-size.tex, line 121, column 46
  7. Occurrence 7: comparing-size.tex, line 123, column 29

Expression 271

f(0)f(1)f(2)f(3)f(4)f(5)f(6)…⌈02⌉−⌈12⌉⌈22⌉−⌈32⌉⌈42⌉−⌈52⌉⌈62⌉…0−11−22−33…

Conventional reading: array, row 1: f open parenthesis 0 close parenthesis; f open parenthesis 1 close parenthesis; f open parenthesis 2 close parenthesis; f open parenthesis 3 close parenthesis; f open parenthesis 4 close parenthesis; f open parenthesis 5 close parenthesis; f open parenthesis 6 close parenthesis; and so on; row 2: the ceiling of the fraction 0 over 2 end ceiling; minus the ceiling of the fraction 1 over 2 end ceiling; the ceiling of the fraction 2 over 2 end ceiling; minus the ceiling of the fraction 3 over 2 end ceiling; the ceiling of the fraction 4 over 2 end ceiling; minus the ceiling of the fraction 5 over 2 end ceiling; the ceiling of the fraction 6 over 2 end ceiling; and so on; row 3: 0; minus 1; 1; minus 2; 2; minus 3; 3; and so on

Meaning here: This multirow mathematical object presents the displayed values or cases in source order.

1 occurrence
  1. Occurrence 1: enumerability-alt.tex, line 108, column 1

Expression 272

f(N)

Conventional reading: f open parenthesis N close parenthesis

Meaning here: This denotes the value of the displayed function or indexed construction at its argument.

1 occurrence
  1. Occurrence 1: reduction-alt.tex, line 53, column 1

Expression 273

12345678910…⟨0,0⟩⟨1,0⟩⟨0,1⟩⟨2,0⟩⟨0,2⟩⟨1,1⟩⟨0,3⟩⟨3,0⟩⟨0,4⟩⟨1,2⟩…

Conventional reading: array, row 1: 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; and so on; row 2: the tuple 0 comma 0; the tuple 1 comma 0; the tuple 0 comma 1; the tuple 2 comma 0; the tuple 0 comma 2; the tuple 1 comma 1; the tuple 0 comma 3; the tuple 3 comma 0; the tuple 0 comma 4; the tuple 1 comma 2; and so on

Meaning here: This multirow mathematical object presents the displayed values or cases in source order.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 41, column 1

Expression 274

h:Z+→A∪B

Conventional reading: h from the positive integers to A union B

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 168, column 12

Expression 275

si(1),si(2),si(3),…

Conventional reading: s sub i open parenthesis 1 close parenthesis comma s sub i open parenthesis 2 close parenthesis comma s sub i open parenthesis 3 close parenthesis comma and so on

Meaning here: This lists the successive entries of the indexed sequence s sub i.

1 occurrence
  1. Occurrence 1: non-enumerability.tex, line 58, column 1

Expression 277

A¯={x∈A:x∉g(x)}.

Conventional reading: A with an overline equals the set of elements x of A such that x is not an element of g open parenthesis x close parenthesis

Meaning here: This is set-builder notation specifying exactly which objects belong to the displayed set.

1 occurrence
  1. Occurrence 1: comparing-size.tex, line 78, column 3

Expression 280

B∗

Conventional reading: the binary digits Kleene star: the set of all finite binary strings, including the empty string

Meaning here: This denotes the Kleene closure of the binary digits: all finite binary strings, including the empty string.

1 occurrence
  1. Occurrence 1: pairing.tex, line 76, column 26

Expression 282

A≼℘(A)

Conventional reading: A has cardinality no greater than the power set of A

Meaning here: This asserts that an injection exists from the first displayed set into the second, so the first is no larger.

1 occurrence
  1. Occurrence 1: comparing-size.tex, line 63, column 42

Expression 283

A≈B

Conventional reading: A is equinumerous with B

Meaning here: This asserts that a bijection exists between the two displayed sets, so they have the same cardinality.

7 occurrences
  1. Occurrence 1: equinumerous-sets.tex, line 30, column 48
  2. Occurrence 2: equinumerous-sets.tex, line 46, column 26
  3. Occurrence 3: equinumerous-sets.tex, line 51, column 35
  4. Occurrence 4: equinumerous-sets.tex, line 58, column 4
  5. Occurrence 5: equinumerous-sets.tex, line 69, column 9
  6. Occurrence 6: schroder-bernstein.tex, line 26, column 7
  7. Occurrence 7: schroder-bernstein.tex, line 46, column 1

Expression 286

Bω

Conventional reading: the set of binary digits to the power omega

Meaning here: This denotes the displayed indexed object or power, as determined by its subscript or superscript.

38 occurrences
  1. Occurrence 1: non-enumerability.tex, line 14, column 48
  2. Occurrence 2: non-enumerability.tex, line 43, column 30
  3. Occurrence 3: non-enumerability.tex, line 48, column 1
  4. Occurrence 4: non-enumerability.tex, line 52, column 40
  5. Occurrence 5: non-enumerability.tex, line 54, column 50
  6. Occurrence 6: non-enumerability.tex, line 107, column 20
  7. Occurrence 7: non-enumerability.tex, line 128, column 1
  8. Occurrence 8: non-enumerability.tex, line 133, column 28
  9. Occurrence 9: non-enumerability.tex, line 136, column 4
  10. Occurrence 10: reduction.tex, line 21, column 39
  11. Occurrence 11: reduction.tex, line 24, column 59
  12. Occurrence 12: reduction.tex, line 25, column 43
  13. Occurrence 13: reduction.tex, line 28, column 24
  14. Occurrence 14: reduction.tex, line 31, column 4
  15. Occurrence 15: reduction.tex, line 75, column 46
  16. Occurrence 16: reduction.tex, line 77, column 49
  17. Occurrence 17: reduction.tex, line 79, column 44
  18. Occurrence 18: reduction.tex, line 80, column 22
  19. Occurrence 19: reduction.tex, line 111, column 24
  20. Occurrence 20: reduction.tex, line 124, column 16
  21. Occurrence 21: non-enumerability-alt.tex, line 14, column 48
  22. Occurrence 22: non-enumerability-alt.tex, line 44, column 12
  23. Occurrence 23: non-enumerability-alt.tex, line 58, column 1
  24. Occurrence 24: non-enumerability-alt.tex, line 62, column 41
  25. Occurrence 25: non-enumerability-alt.tex, line 100, column 1
  26. Occurrence 26: non-enumerability-alt.tex, line 100, column 54
  27. Occurrence 27: non-enumerability-alt.tex, line 101, column 36
  28. Occurrence 28: non-enumerability-alt.tex, line 101, column 57
  29. Occurrence 29: reduction-alt.tex, line 20, column 16
  30. Occurrence 30: reduction-alt.tex, line 24, column 37
  31. Occurrence 31: reduction-alt.tex, line 25, column 33
  32. Occurrence 32: reduction-alt.tex, line 29, column 38
  33. Occurrence 33: reduction-alt.tex, line 32, column 16
  34. Occurrence 34: reduction-alt.tex, line 78, column 46
  35. Occurrence 35: reduction-alt.tex, line 80, column 49
  36. Occurrence 36: reduction-alt.tex, line 82, column 41
  37. Occurrence 37: reduction-alt.tex, line 83, column 22
  38. Occurrence 38: reduction-alt.tex, line 109, column 24

Expression 289

g(x)⊆A

Conventional reading: g open parenthesis x close parenthesis is a subset of A

Meaning here: This asserts that every member of the first displayed set also belongs to the second.

1 occurrence
  1. Occurrence 1: comparing-size.tex, line 74, column 61

Expression 290

d(n)={1if sn(n)=00if sn(n)=1

Conventional reading: d open parenthesis n close parenthesis equals: 1 if s sub n open parenthesis n close parenthesis equals 0; 0 if s sub n open parenthesis n close parenthesis equals 1

Meaning here: This piecewise definition flips the nth diagonal bit of the nth binary string to construct d.

1 occurrence
  1. Occurrence 1: non-enumerability-alt.tex, line 85, column 1

Expression 294

j:N2→N+

Conventional reading: j maps ordered pairs of natural numbers to positive natural numbers

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 107, column 14

Expression 296

g(n)=n+1

Conventional reading: g open parenthesis n close parenthesis equals n plus 1

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: enumerability-alt.tex, line 73, column 40

Expression 297

f:N→Z

Conventional reading: f from the natural numbers to the integers

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: enumerability-alt.tex, line 101, column 46

Expression 299

sk(k)=1

Conventional reading: s sub k open parenthesis k close parenthesis equals 1

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: non-enumerability.tex, line 123, column 39

Expression 301

g(n,m)

Conventional reading: g open parenthesis n comma m close parenthesis

Meaning here: This denotes the value of the displayed function or indexed construction at its argument.

1 occurrence
  1. Occurrence 1: pairing.tex, line 40, column 21

Expression 302

f(Z1),f(Z2),f(Z3),…

Conventional reading: f open parenthesis Z sub 1 close parenthesis comma f open parenthesis Z sub 2 close parenthesis comma f open parenthesis Z sub 3 close parenthesis comma and so on

Meaning here: This lists the successive function values f of Z sub 1, f of Z sub 2, and so on.

1 occurrence
  1. Occurrence 1: reduction.tex, line 72, column 1

Expression 303

IdA(x)=x

Conventional reading: the identity function on A open parenthesis x close parenthesis equals x

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: equinumerous-sets.tex, line 43, column 1

Expression 304

N×N×N={⟨n,m,k⟩:n,m,k∈N}

Conventional reading: the natural numbers cubed equals the set of ordered triples n comma m comma k such that n, m, and k are natural numbers

Meaning here: This is set-builder notation specifying exactly which objects belong to the displayed set.

1 occurrence
  1. Occurrence 1: zig-zag.tex, line 81, column 1

Expression 306

g:{1,…,n}→A

Conventional reading: g from open brace 1 comma and so on comma n close brace to A

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 238, column 46

Expression 308

N×N×N

Conventional reading: the natural numbers cross the natural numbers cross the natural numbers

Meaning here: This denotes the Cartesian product of the displayed sets.

1 occurrence
  1. Occurrence 1: zig-zag.tex, line 84, column 13

Expression 312

4·(2m+1)

Conventional reading: 4 times open parenthesis 2 m plus 1 close parenthesis

Meaning here: This denotes the value of the displayed function or indexed construction at its argument.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 75, column 13

Expression 314

g

Conventional reading: g

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

16 occurrences
  1. Occurrence 1: enumerability.tex, line 234, column 26
  2. Occurrence 2: enumerability.tex, line 244, column 16
  3. Occurrence 3: enumerability.tex, line 249, column 21
  4. Occurrence 4: pairing.tex, line 29, column 24
  5. Occurrence 5: pairing.tex, line 47, column 15
  6. Occurrence 6: reduction.tex, line 44, column 9
  7. Occurrence 7: equinumerous-sets.tex, line 79, column 3
  8. Occurrence 8: comparing-size.tex, line 74, column 9
  9. Occurrence 9: comparing-size.tex, line 75, column 34
  10. Occurrence 10: comparing-size.tex, line 77, column 12
  11. Occurrence 11: comparing-size.tex, line 83, column 6
  12. Occurrence 12: comparing-size.tex, line 91, column 19
  13. Occurrence 13: comparing-size.tex, line 91, column 58
  14. Occurrence 14: comparing-size.tex, line 124, column 6
  15. Occurrence 15: comparing-size.tex, line 146, column 56
  16. Occurrence 16: reduction-alt.tex, line 45, column 9

Expression 315

Z={n∈Z+:s(n)=1}

Conventional reading: Z equals the set of positive integers n such that s open parenthesis n close parenthesis equals 1

Meaning here: This is set-builder notation specifying exactly which objects belong to the displayed set.

1 occurrence
  1. Occurrence 1: reduction.tex, line 65, column 1

Expression 319

N={0,…,n−1}

Conventional reading: N equals open brace 0 comma and so on comma n minus 1 close brace

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 265, column 15

Expression 320

2

Conventional reading: 2

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

13 occurrences
  1. Occurrence 1: enumerability.tex, line 43, column 53
  2. Occurrence 2: enumerability.tex, line 48, column 64
  3. Occurrence 3: enumerability.tex, line 49, column 3
  4. Occurrence 4: enumerability.tex, line 49, column 58
  5. Occurrence 5: enumerability.tex, line 53, column 8
  6. Occurrence 6: enumerability.tex, line 54, column 42
  7. Occurrence 7: enumerability.tex, line 55, column 55
  8. Occurrence 8: enumerability.tex, line 61, column 25
  9. Occurrence 9: enumerability.tex, line 189, column 30
  10. Occurrence 10: pairing-alt.tex, line 76, column 16
  11. Occurrence 11: pairing-alt.tex, line 76, column 64
  12. Occurrence 12: non-enumerability.tex, line 198, column 40
  13. Occurrence 13: non-enumerability-alt.tex, line 150, column 47

Expression 322

2·(2m+1)

Conventional reading: 2 times open parenthesis 2 m plus 1 close parenthesis

Meaning here: This denotes the value of the displayed function or indexed construction at its argument.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 73, column 41

Expression 331

x1

Conventional reading: x sub 1

Meaning here: This denotes the displayed indexed object or power, as determined by its subscript or superscript.

7 occurrences
  1. Occurrence 1: enumerability.tex, line 74, column 34
  2. Occurrence 2: enumerability.tex, line 78, column 47
  3. Occurrence 3: enumerability.tex, line 80, column 3
  4. Occurrence 4: non-enumerability.tex, line 40, column 14
  5. Occurrence 5: reduction.tex, line 36, column 56
  6. Occurrence 6: non-enumerability-alt.tex, line 39, column 27
  7. Occurrence 7: reduction-alt.tex, line 37, column 40

Expression 332

2N={2n:n∈N}={0,2,4,6,…}

Conventional reading: 2 times the natural numbers equals the set of 2 times n such that n is a natural number, which equals the set containing 0, 2, 4, 6, and so on

Meaning here: This is set-builder notation specifying exactly which objects belong to the displayed set.

1 occurrence
  1. Occurrence 1: reduction-alt.tex, line 58, column 19

Expression 336

B

Conventional reading: B

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

29 occurrences
  1. Occurrence 1: enumerability.tex, line 165, column 24
  2. Occurrence 2: enumerability.tex, line 173, column 65
  3. Occurrence 3: enumerability.tex, line 182, column 33
  4. Occurrence 4: reduction.tex, line 33, column 51
  5. Occurrence 5: reduction.tex, line 35, column 31
  6. Occurrence 6: reduction.tex, line 38, column 11
  7. Occurrence 7: reduction.tex, line 43, column 1
  8. Occurrence 8: reduction.tex, line 44, column 55
  9. Occurrence 9: reduction.tex, line 88, column 32
  10. Occurrence 10: reduction.tex, line 95, column 25
  11. Occurrence 11: equinumerous-sets.tex, line 30, column 35
  12. Occurrence 12: equinumerous-sets.tex, line 59, column 13
  13. Occurrence 13: equinumerous-sets.tex, line 85, column 22
  14. Occurrence 14: equinumerous-sets.tex, line 85, column 34
  15. Occurrence 15: equinumerous-sets.tex, line 96, column 4
  16. Occurrence 16: comparing-size.tex, line 27, column 30
  17. Occurrence 17: comparing-size.tex, line 37, column 28
  18. Occurrence 18: schroder-bernstein.tex, line 14, column 56
  19. Occurrence 19: schroder-bernstein.tex, line 14, column 64
  20. Occurrence 20: schroder-bernstein.tex, line 15, column 34
  21. Occurrence 21: schroder-bernstein.tex, line 30, column 56
  22. Occurrence 22: schroder-bernstein.tex, line 31, column 21
  23. Occurrence 23: schroder-bernstein.tex, line 32, column 4
  24. Occurrence 24: enumerability-alt.tex, line 127, column 22
  25. Occurrence 25: reduction-alt.tex, line 34, column 44
  26. Occurrence 26: reduction-alt.tex, line 36, column 16
  27. Occurrence 27: reduction-alt.tex, line 38, column 66
  28. Occurrence 28: reduction-alt.tex, line 44, column 1
  29. Occurrence 29: reduction-alt.tex, line 45, column 55

Expression 338

⟨n,m⟩∈N×N

Conventional reading: the tuple n comma m is an element of the natural numbers cross the natural numbers

Meaning here: This expression states membership or nonmembership in the displayed set.

1 occurrence
  1. Occurrence 1: zig-zag.tex, line 74, column 7

Expression 339

f:Z+→A

Conventional reading: f from the positive integers to A

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

12 occurrences
  1. Occurrence 1: enumerability.tex, line 105, column 48
  2. Occurrence 2: enumerability.tex, line 110, column 37
  3. Occurrence 3: enumerability.tex, line 166, column 52
  4. Occurrence 4: enumerability.tex, line 174, column 54
  5. Occurrence 5: enumerability.tex, line 195, column 28
  6. Occurrence 6: enumerability.tex, line 200, column 25
  7. Occurrence 7: enumerability.tex, line 216, column 10
  8. Occurrence 8: enumerability.tex, line 228, column 4
  9. Occurrence 9: enumerability.tex, line 260, column 1
  10. Occurrence 10: enumerability.tex, line 270, column 62
  11. Occurrence 11: enumerability.tex, line 276, column 19
  12. Occurrence 12: non-enumerability.tex, line 38, column 10

Expression 340

g(⟨0,0⟩)=0,g(⟨0,1⟩)=1,g(⟨1,0⟩)=2,…,g(⟨1,2⟩)=7,…

Conventional reading: g open parenthesis the tuple 0 comma 0 close parenthesis equals 0 comma g open parenthesis the tuple 0 comma 1 close parenthesis equals 1 comma g open parenthesis the tuple 1 comma 0 close parenthesis equals 2 comma and so on comma g open parenthesis the tuple 1 comma 2 close parenthesis equals 7 comma and so on

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: pairing.tex, line 20, column 1

Expression 342

Z¯={n∈Z+:n∉Zn}

Conventional reading: Z with an overline equals the set of positive integers n such that n is not an element of Z sub n

Meaning here: This is set-builder notation specifying exactly which objects belong to the displayed set.

1 occurrence
  1. Occurrence 1: non-enumerability.tex, line 161, column 1

Expression 343

2Z+={2,4,6,…}

Conventional reading: 2 times the positive integers equals the set containing 2, 4, 6, and so on

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: reduction.tex, line 55, column 54

Expression 344

N={n∈N:s(n)=1}

Conventional reading: N equals the set of natural numbers n such that s open parenthesis n close parenthesis equals 1

Meaning here: This is set-builder notation specifying exactly which objects belong to the displayed set.

1 occurrence
  1. Occurrence 1: reduction-alt.tex, line 68, column 1

Expression 345

N0={0,1,2,…}N1={1,3,5,…}N2={0,1,4,}N3={2,3,4,…}⋮⋱}

Conventional reading: array, row 1: N sub 0 equals open brace; 0 comma; 1 comma; 2 comma; blank position; blank position; blank position; and so on close brace; row 2: N sub 1 equals open brace; blank position; 1 comma; blank position; 3 comma; blank position; 5 comma; and so on close brace; row 3: N sub 2 equals open brace; 0 comma; 1 comma; blank position; blank position; 4; blank position; close brace; row 4: N sub 3 equals open brace; blank position; blank position; 2 comma; 3 comma; 4 comma; blank position; and so on close brace; row 5: blank position; vertical ellipsis; blank position; blank position; blank position; blank position; blank position; diagonal ellipsis

Meaning here: This multirow mathematical object presents the displayed values or cases in source order.

1 occurrence
  1. Occurrence 1: non-enumerability-alt.tex, line 139, column 1

Expression 346

f:A×B→N

Conventional reading: f from A cross B to the natural numbers

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: pairing.tex, line 52, column 14

Expression 347

℘(Z+)

Conventional reading: the power set of the positive integers

Meaning here: This expression involves a power set, the set of all subsets of the indicated set.

11 occurrences
  1. Occurrence 1: non-enumerability.tex, line 15, column 3
  2. Occurrence 2: non-enumerability.tex, line 149, column 1
  3. Occurrence 3: reduction.tex, line 20, column 11
  4. Occurrence 4: reduction.tex, line 23, column 31
  5. Occurrence 5: reduction.tex, line 24, column 20
  6. Occurrence 6: reduction.tex, line 26, column 17
  7. Occurrence 7: reduction.tex, line 29, column 1
  8. Occurrence 8: reduction.tex, line 30, column 1
  9. Occurrence 9: reduction.tex, line 48, column 14
  10. Occurrence 10: reduction.tex, line 79, column 7
  11. Occurrence 11: reduction.tex, line 81, column 45

Expression 348

A∪B≈C∪D

Conventional reading: A union B is equinumerous with C union D

Meaning here: This asserts that a bijection exists between the two displayed sets, so they have the same cardinality.

1 occurrence
  1. Occurrence 1: equinumerous-sets.tex, line 103, column 29

Expression 352

Z¯

Conventional reading: Z with an overline

Meaning here: This denotes the mathematical object or value identified by the displayed symbol in the surrounding statement.

10 occurrences
  1. Occurrence 1: non-enumerability.tex, line 159, column 21
  2. Occurrence 2: non-enumerability.tex, line 164, column 1
  3. Occurrence 3: non-enumerability.tex, line 166, column 21
  4. Occurrence 4: non-enumerability.tex, line 170, column 52
  5. Occurrence 5: non-enumerability.tex, line 174, column 34
  6. Occurrence 6: non-enumerability.tex, line 175, column 50
  7. Occurrence 7: non-enumerability.tex, line 195, column 6
  8. Occurrence 8: comparing-size.tex, line 112, column 7
  9. Occurrence 9: comparing-size.tex, line 116, column 3
  10. Occurrence 10: comparing-size.tex, line 122, column 26

Expression 353

A={a1,a2,…,an}.

Conventional reading: A equals open brace a sub 1 comma a sub 2 comma and so on comma a sub n close brace point

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: enumerability-alt.tex, line 23, column 1

Expression 354

h:Z+→Bω

Conventional reading: h from the positive integers to the set of binary digits to the power omega

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: reduction.tex, line 99, column 20

Expression 357

g(1)=f(1)

Conventional reading: g open parenthesis 1 close parenthesis equals f open parenthesis 1 close parenthesis

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 234, column 47

Expression 360

f(Z)=s

Conventional reading: f open parenthesis Z close parenthesis equals s

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: reduction.tex, line 68, column 6

Expression 371

(f∘g)(n)=f(g(n))=f(x)=y

Conventional reading: open parenthesis f composed after g close parenthesis open parenthesis n close parenthesis equals f open parenthesis g open parenthesis n close parenthesis close parenthesis equals f open parenthesis x close parenthesis equals y

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: equinumerous-sets.tex, line 81, column 1

Expression 372

f:Z+→Z+

Conventional reading: f from the positive integers to the positive integers

Meaning here: This expression specifies a function, map, or correspondence between the displayed domain and codomain.

1 occurrence
  1. Occurrence 1: non-enumerability.tex, line 207, column 32

Expression 373

f:℘(Z+)→Bω

Conventional reading: f from the power set of the positive integers to the set of binary digits to the power omega

Meaning here: This declares f as a function from subsets of the positive integers to infinite binary sequences.

1 occurrence
  1. Occurrence 1: reduction.tex, line 39, column 10

Expression 376

Z¯∈℘(Z+)

Conventional reading: Z with an overline is an element of the power set of the positive integers

Meaning here: This asserts that Z with an overline is a subset of the positive integers.

1 occurrence
  1. Occurrence 1: non-enumerability.tex, line 165, column 36

Expression 377

A≼N

Conventional reading: A has cardinality no greater than the natural numbers

Meaning here: This asserts that an injection exists from the first displayed set into the second, so the first is no larger.

1 occurrence
  1. Occurrence 1: comparing-size.tex, line 44, column 46

Expression 382

h(n,m)=2n(2m+1)−1

Conventional reading: h open parenthesis n comma m close parenthesis equals 2 to the power n times open parenthesis 2 times m plus 1 close parenthesis minus 1

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 88, column 1

Expression 383

⟨x,y⟩

Conventional reading: the tuple x comma y

Meaning here: This denotes the displayed ordered tuple, whose coordinate order is significant.

1 occurrence
  1. Occurrence 1: pairing.tex, line 55, column 19

Expression 384

g(j)=g(i)

Conventional reading: g open parenthesis j close parenthesis equals g open parenthesis i close parenthesis

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 247, column 56

Expression 385

22·(2·6+1)−1=51

Conventional reading: 2 to the power 2 times open parenthesis 2 times 6 plus 1 close parenthesis minus 1 equals 51

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: pairing-alt.tex, line 98, column 42

Expression 386

s¯(n)=sk(n)

Conventional reading: s with an overline open parenthesis n close parenthesis equals s sub k open parenthesis n close parenthesis

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: non-enumerability.tex, line 119, column 51

Expression 389

f(n)=(−1)n⌈(n−1)2⌉

Conventional reading: f open parenthesis n close parenthesis equals open parenthesis negative 1 close parenthesis to the power n times the ceiling of the fraction n minus 1 over 2

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 141, column 14

Expression 391

(m−1)

Conventional reading: open parenthesis m minus 1 close parenthesis

Meaning here: This denotes the value of the displayed function or indexed construction at its argument.

1 occurrence
  1. Occurrence 1: pairing.tex, line 31, column 19

Expression 392

N3=(N×N)×N={⟨⟨n,m⟩,k⟩:n,m,k∈N}

Conventional reading: the natural numbers cubed equals open parenthesis the natural numbers cross the natural numbers close parenthesis cross the natural numbers, which equals the set of ordered triples n comma m comma k such that n, m, and k are natural numbers

Meaning here: This is set-builder notation specifying exactly which objects belong to the displayed set.

1 occurrence
  1. Occurrence 1: zig-zag.tex, line 86, column 1

Expression 393

f(n)={0if n=1n/2if n is even−(n−1)/2if n is odd and >1

Conventional reading: f open parenthesis n close parenthesis equals: 0 if n equals 1; n divided by 2 if n is even; negative open parenthesis n minus 1 close parenthesis divided by 2 if n is odd and greater than 1

Meaning here: This piecewise function sends the first input to zero and later even and odd inputs to alternating integers.

1 occurrence
  1. Occurrence 1: enumerability.tex, line 155, column 1

Expression 398

f(n)=(−1)n⌈n2⌉

Conventional reading: f open parenthesis n close parenthesis equals open parenthesis negative 1 close parenthesis to the power n times the ceiling of the fraction n over 2

Meaning here: This asserts that the mathematical expression on the left equals the expression on the right.

1 occurrence
  1. Occurrence 1: enumerability-alt.tex, line 103, column 1

73 formal objects

  1. Definition: Informal enumeration — source line 27.
  2. Proposition: Removing repetitions from an enumeration — source line 68.
  3. Definition: Enumeration as a surjection from the positive integers — source line 87.
  4. Definition: Enumerable sets — source line 113.
  5. Example: Enumerating the positive integers and natural numbers — source line 118.
  6. Example: Enumerating positive even and odd integers — source line 124.
  7. Displayed mathematics: Positive even-and-odd enumeration formulas — source line 127.
  8. Exercise: Enumerating the positive square numbers — source line 136.
  9. Example: Alternating enumeration of the integers from positive inputs — source line 140.
  10. Exercise: Union of two enumerable sets using positive-integer enumerations — source line 164.
  11. Exercise: Enumerable subsets of enumerable sets — source line 172.
  12. Exercise: Finite unions of enumerable sets using induction — source line 179.
  13. Proposition: Shifting enumerations between positive and natural indices — source line 194.
  14. Corollary: Enumerable sets via surjections from the natural numbers — source line 208.
  15. Proposition: Replacing a surjective enumeration by a bijective one — source line 227.
  16. Corollary: Enumerable sets via bijections from natural initial segments — source line 252.
  17. Exercise: Injective and surjective characterizations of enumerability — source line 268.
  18. Proposition: Enumerability of pairs of natural numbers — source line 68.
  19. Proposition: Enumerability of finite powers of the natural numbers — source line 114.
  20. Exercise: Enumerability of finite powers of the positive integers — source line 118.
  21. Exercise: Enumerability of all finite positive-integer sequences — source line 122.
  22. Definition: Arithmetic pairing functions and codes — source line 51.
  23. Exercise: Enumerating the nonnegative rational numbers — source line 66.
  24. Exercise: Enumerability of all rational numbers — source line 70.
  25. Exercise: Enumerating finite binary strings — source line 75.
  26. Exercise: Enumerability of finite-arity truth functions — source line 79.
  27. Exercise: Enumerability of finite subsets of an enumerable set — source line 86.
  28. Exercise: Enumerability of finite and cofinite subsets of the natural numbers — source line 91.
  29. Exercise: Countable unions of enumerable sets — source line 99.
  30. Exercise: Inverting a pairing function to enumerate its product — source line 106.
  31. Exercise: Encoding triples of natural numbers — source line 111.
  32. Example: Pairing natural numbers by powers of two — source line 86.
  33. Example: Injective pair encoding by prime powers — source line 106.
  34. Theorem: Non-enumerability of positive-indexed infinite binary sequences — source line 46.
  35. Theorem: Non-enumerability of the power set of the positive integers — source line 147.
  36. Exercise: Diagonal proof for the power set of the natural numbers — source line 202.
  37. Exercise: Diagonal proof for functions on the positive integers — source line 206.
  38. Exercise: Reduction along an injection with positive-integer enumerations — source line 41.
  39. Exercise: Non-enumerability of sets of positive-integer pairs — source line 103.
  40. Exercise: Functions on the natural numbers by positive-indexed reduction — source line 108.
  41. Exercise: Infinite natural-number sequences by positive-indexed reduction — source line 114.
  42. Exercise: Total and partial zero-valued functions — source line 119.
  43. Exercise: Binary surjections from the positive integers — source line 127.
  44. Exercise: Real numbers by positive-indexed reduction — source line 134.
  45. Definition: Equinumerous sets — source line 29.
  46. Proposition: Equinumerosity as an equivalence relation — source line 34.
  47. Proposition: Preservation of enumerability under equinumerosity — source line 57.
  48. Exercise: Equinumerosity of disjoint unions — source line 101.
  49. Exercise: Infinite enumerable sets and the natural numbers — source line 106.
  50. Definition: Cardinality comparison by injection — source line 26.
  51. Definition: Strict cardinal inequality — source line 36.
  52. Theorem: Cantor's theorem on power sets — source line 56.
  53. Exercise: No injection from a power set into its base set — source line 142.
  54. Theorem: Schröder-Bernstein theorem — source line 23.
  55. Definition: Set-theoretic enumeration — source line 39.
  56. Definition: Enumerable and nonenumerable sets in the set-theoretic presentation — source line 57.
  57. Example: Enumerating natural and positive natural numbers — source line 69.
  58. Exercise: Injective and surjective tests for set-theoretic enumerability — source line 77.
  59. Example: Enumerating even and odd natural numbers — source line 83.
  60. Displayed mathematics: Natural even-and-odd enumeration formulas — source line 86.
  61. Exercise: Enumerating square numbers from natural inputs — source line 95.
  62. Example: Alternating enumeration of the integers from natural inputs — source line 99.
  63. Exercise: Union of two enumerable sets in the set-theoretic presentation — source line 126.
  64. Exercise: Finite unions in the set-theoretic presentation — source line 130.
  65. Theorem: Non-enumerability of zero-indexed infinite binary strings — source line 56.
  66. Theorem: Non-enumerability of the power set of the natural numbers — source line 113.
  67. Exercise: Diagonal proof for functions on the natural numbers — source line 154.
  68. Exercise: Reduction along an injection with natural-number enumerations — source line 42.
  69. Exercise: Functions on the natural numbers by zero-indexed reduction — source line 106.
  70. Exercise: Sets of natural-number pairs by reduction — source line 112.
  71. Exercise: Infinite natural-number sequences by zero-indexed reduction — source line 118.
  72. Exercise: Binary surjections from the natural numbers — source line 130.
  73. Exercise: Real numbers by zero-indexed reduction — source line 136.

42 source reference records

  1. Enumerations and Enumerable Sets — enumerability.tex, line 17.
  2. Proposition: Replacing a surjective enumeration by a bijective one — enumerability.tex, line 260.
  3. Proposition: Shifting enumerations between positive and natural indices — enumerability.tex, line 263.
  4. Definition: Enumerable sets — enumerability.tex, line 269.
  5. Pairs, Tuples, Cartesian Products — zig-zag.tex, line 17.
  6. Exercise: Enumerability of finite powers of the positive integers — zig-zag.tex, line 123.
  7. Enumerations and Enumerable Sets — non-enumerability.tex, line 15.
  8. Enumerations and Enumerable Sets — non-enumerability.tex, line 17.
  9. Nonenumerable Sets — reduction.tex, line 15.
  10. Nonenumerable Sets — reduction.tex, line 16.
  11. Reduction — reduction.tex, line 17.
  12. Theorem: Non-enumerability of the power set of the positive integers — reduction.tex, line 47.
  13. Theorem: Non-enumerability of positive-indexed infinite binary sequences — reduction.tex, line 81.
  14. (Gottlob Frege, 1884, §70) — equinumerous-sets.tex, line 24.
  15. Definition: Enumerable sets — equinumerous-sets.tex, line 63.
  16. Enumerations and Enumerable Sets — equinumerous-sets.tex, line 64.
  17. Definition: Enumerable and nonenumerable sets in the set-theoretic presentation — equinumerous-sets.tex, line 64.
  18. Theorem: Non-enumerability of the power set of the natural numbers — comparing-size.tex, line 48.
  19. Georg Cantor (1892) — comparing-size.tex, line 53.
  20. Nonenumerable Sets — comparing-size.tex, line 66.
  21. Nonenumerable Sets — comparing-size.tex, line 67.
  22. Theorem: Cantor's theorem on power sets — comparing-size.tex, line 109.
  23. Theorem: Non-enumerability of the power set of the positive integers — comparing-size.tex, line 110.
  24. Theorem: Cantor's theorem on power sets — comparing-size.tex, line 127.
  25. Theorem: Non-enumerability of the power set of the natural numbers — comparing-size.tex, line 128.
  26. Russell's Paradox — comparing-size.tex, line 138.
  27. Michael Potter (2004), pp. 165–6 — schroder-bernstein.tex, line 37.
  28. Enumerations and Enumerable Sets — size-of-sets-complete.tex, line 38.
  29. Nonenumerable Sets — size-of-sets-complete.tex, line 39.
  30. Reduction — size-of-sets-complete.tex, line 39.
  31. Enumerations and Enumerable Sets — size-of-sets-complete.tex, line 40.
  32. Nonenumerable Sets — size-of-sets-complete.tex, line 41.
  33. Reduction — size-of-sets-complete.tex, line 41.
  34. Enumerations and Enumerable Sets — enumerability-alt.tex, line 16.
  35. Enumerations and Enumerable Sets — non-enumerability-alt.tex, line 15.
  36. Definition: Enumerable and nonenumerable sets in the set-theoretic presentation — non-enumerability-alt.tex, line 24.
  37. Nonenumerable Sets — reduction-alt.tex, line 15.
  38. Nonenumerable Sets — reduction-alt.tex, line 16.
  39. Reduction — reduction-alt.tex, line 17.
  40. Theorem: Non-enumerability of the power set of the natural numbers — reduction-alt.tex, line 48.
  41. Theorem: Non-enumerability of zero-indexed infinite binary strings — reduction-alt.tex, line 84.

Reference correction

Editorial projection note: the source gives this alternate exercise the same link label as the corresponding exercise in the earlier Reduction section. This edition assigns the alternate copy its own link target; unqualified references lead to the earlier copy. The canonical source is unchanged.