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Functions

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Equation guide

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

133 expressions

Expression 2

Rxy

Conventional reading: R relates x to y

Meaning here: This is a relation or function-graph statement about the displayed ordered pair.

9 occurrences
  1. Occurrence 1: functions-relations.tex, line 45, column 10
  2. Occurrence 2: functions-relations.tex, line 50, column 16
  3. Occurrence 3: functions-relations.tex, line 54, column 34
  4. Occurrence 4: functions-relations.tex, line 56, column 26
  5. Occurrence 5: partial-functions.tex, line 59, column 65
  6. Occurrence 6: partial-functions.tex, line 62, column 1
  7. Occurrence 7: partial-functions.tex, line 64, column 1
  8. Occurrence 8: partial-functions.tex, line 68, column 34
  9. Occurrence 9: partial-functions.tex, line 70, column 46

Expression 9

Rf∣Rg

Conventional reading: R sub f relative product R sub g

Meaning here: The relative product of the graphs records the graph of first applying f and then applying g.

1 occurrence
  1. Occurrence 1: composition.tex, line 64, column 21

Expression 14

h:N→N

Conventional reading: h is a function from the natural numbers to the natural numbers

Meaning here: This declaration specifies the function and its stated domain and codomain.

1 occurrence
  1. Occurrence 1: function-basics.tex, line 125, column 1

Expression 15

h(x)={x2if x is evenx+12if x is odd.

Conventional reading: h of x equals x divided by two if x is even, and the quantity x plus one, divided by two, if x is odd

Meaning here: This is the displayed piecewise definition, with the even and odd cases spoken in source order.

1 occurrence
  1. Occurrence 1: function-basics.tex, line 126, column 1

Expression 18

x∈A

Conventional reading: x is an element of A

Meaning here: This mathematical expression states that x is an element of A.

17 occurrences
  1. Occurrence 1: function-kinds.tex, line 32, column 5
  2. Occurrence 2: function-kinds.tex, line 66, column 37
  3. Occurrence 3: functions-relations.tex, line 46, column 17
  4. Occurrence 4: inverses.tex, line 29, column 60
  5. Occurrence 5: inverses.tex, line 64, column 65
  6. Occurrence 6: inverses.tex, line 70, column 33
  7. Occurrence 7: inverses.tex, line 71, column 46
  8. Occurrence 8: inverses.tex, line 72, column 45
  9. Occurrence 9: inverses.tex, line 82, column 22
  10. Occurrence 10: inverses.tex, line 101, column 13
  11. Occurrence 11: inverses.tex, line 129, column 49
  12. Occurrence 12: inverses.tex, line 141, column 5
  13. Occurrence 13: inverses.tex, line 147, column 57
  14. Occurrence 14: composition.tex, line 28, column 63
  15. Occurrence 15: partial-functions.tex, line 23, column 37
  16. Occurrence 16: partial-functions.tex, line 44, column 48
  17. Occurrence 17: partial-functions.tex, line 63, column 31

Expression 21

f

Conventional reading: f

Meaning here: The mathematical object denoted by f in the surrounding source context.

66 occurrences
  1. Occurrence 1: function-basics.tex, line 32, column 34
  2. Occurrence 2: function-basics.tex, line 33, column 4
  3. Occurrence 3: function-basics.tex, line 34, column 4
  4. Occurrence 4: function-basics.tex, line 35, column 4
  5. Occurrence 5: function-basics.tex, line 35, column 37
  6. Occurrence 6: function-basics.tex, line 38, column 31
  7. Occurrence 7: function-basics.tex, line 39, column 29
  8. Occurrence 8: function-basics.tex, line 94, column 32
  9. Occurrence 9: function-basics.tex, line 96, column 21
  10. Occurrence 10: function-basics.tex, line 97, column 55
  11. Occurrence 11: function-basics.tex, line 99, column 10
  12. Occurrence 12: function-basics.tex, line 111, column 35
  13. Occurrence 13: function-basics.tex, line 114, column 44
  14. Occurrence 14: function-basics.tex, line 120, column 15
  15. Occurrence 15: function-kinds.tex, line 31, column 22
  16. Occurrence 16: function-kinds.tex, line 40, column 26
  17. Occurrence 17: function-kinds.tex, line 41, column 22
  18. Occurrence 18: function-kinds.tex, line 71, column 26
  19. Occurrence 19: function-kinds.tex, line 72, column 37
  20. Occurrence 20: functions-relations.tex, line 19, column 57
  21. Occurrence 21: functions-relations.tex, line 25, column 21
  22. Occurrence 22: functions-relations.tex, line 36, column 12
  23. Occurrence 23: functions-relations.tex, line 56, column 60
  24. Occurrence 24: functions-relations.tex, line 81, column 27
  25. Occurrence 25: functions-relations.tex, line 87, column 27
  26. Occurrence 26: functions-relations.tex, line 89, column 7
  27. Occurrence 27: functions-relations.tex, line 94, column 42
  28. Occurrence 28: inverses.tex, line 16, column 60
  29. Occurrence 29: inverses.tex, line 33, column 4
  30. Occurrence 30: inverses.tex, line 51, column 37
  31. Occurrence 31: inverses.tex, line 56, column 51
  32. Occurrence 32: inverses.tex, line 57, column 33
  33. Occurrence 33: inverses.tex, line 59, column 34
  34. Occurrence 34: inverses.tex, line 59, column 40
  35. Occurrence 35: inverses.tex, line 64, column 31
  36. Occurrence 36: inverses.tex, line 71, column 1
  37. Occurrence 37: inverses.tex, line 87, column 61
  38. Occurrence 38: inverses.tex, line 93, column 47
  39. Occurrence 39: inverses.tex, line 99, column 11
  40. Occurrence 40: inverses.tex, line 101, column 49
  41. Occurrence 41: inverses.tex, line 102, column 61
  42. Occurrence 42: inverses.tex, line 109, column 78
  43. Occurrence 43: inverses.tex, line 110, column 67
  44. Occurrence 44: inverses.tex, line 117, column 62
  45. Occurrence 45: inverses.tex, line 124, column 45
  46. Occurrence 46: inverses.tex, line 140, column 12
  47. Occurrence 47: inverses.tex, line 146, column 38
  48. Occurrence 48: inverses.tex, line 148, column 13
  49. Occurrence 49: inverses.tex, line 151, column 18
  50. Occurrence 50: inverses.tex, line 168, column 16
  51. Occurrence 51: inverses.tex, line 172, column 44
  52. Occurrence 52: inverses.tex, line 173, column 28
  53. Occurrence 53: composition.tex, line 14, column 36
  54. Occurrence 54: composition.tex, line 16, column 38
  55. Occurrence 55: composition.tex, line 17, column 10
  56. Occurrence 56: composition.tex, line 18, column 46
  57. Occurrence 57: composition.tex, line 29, column 30
  58. Occurrence 58: composition.tex, line 34, column 57
  59. Occurrence 59: composition.tex, line 41, column 23
  60. Occurrence 60: partial-functions.tex, line 23, column 4
  61. Occurrence 61: partial-functions.tex, line 26, column 37
  62. Occurrence 62: partial-functions.tex, line 46, column 4
  63. Occurrence 63: partial-functions.tex, line 51, column 67
  64. Occurrence 64: partial-functions.tex, line 64, column 13
  65. Occurrence 65: partial-functions.tex, line 71, column 16
  66. Occurrence 66: partial-functions.tex, line 71, column 63

Expression 23

f(x)=y

Conventional reading: f of x equals y

Meaning here: This mathematical expression states that f of x equals y.

14 occurrences
  1. Occurrence 1: function-kinds.tex, line 32, column 25
  2. Occurrence 2: function-kinds.tex, line 66, column 57
  3. Occurrence 3: functions-relations.tex, line 17, column 37
  4. Occurrence 4: functions-relations.tex, line 50, column 1
  5. Occurrence 5: functions-relations.tex, line 62, column 22
  6. Occurrence 6: inverses.tex, line 45, column 58
  7. Occurrence 7: inverses.tex, line 50, column 18
  8. Occurrence 8: inverses.tex, line 70, column 51
  9. Occurrence 9: inverses.tex, line 72, column 65
  10. Occurrence 10: inverses.tex, line 82, column 42
  11. Occurrence 11: inverses.tex, line 101, column 31
  12. Occurrence 12: inverses.tex, line 113, column 6
  13. Occurrence 13: partial-functions.tex, line 45, column 1
  14. Occurrence 14: partial-functions.tex, line 62, column 13

Expression 27

x

Conventional reading: x

Meaning here: The mathematical object denoted by x in the surrounding source context.

13 occurrences
  1. Occurrence 1: function-basics.tex, line 34, column 68
  2. Occurrence 2: function-basics.tex, line 35, column 55
  3. Occurrence 3: function-basics.tex, line 67, column 60
  4. Occurrence 4: function-basics.tex, line 96, column 16
  5. Occurrence 5: function-basics.tex, line 106, column 54
  6. Occurrence 6: function-kinds.tex, line 42, column 7
  7. Occurrence 7: function-kinds.tex, line 72, column 22
  8. Occurrence 8: functions-relations.tex, line 17, column 21
  9. Occurrence 9: inverses.tex, line 45, column 44
  10. Occurrence 10: inverses.tex, line 49, column 66
  11. Occurrence 11: inverses.tex, line 104, column 66
  12. Occurrence 12: composition.tex, line 29, column 37
  13. Occurrence 13: composition.tex, line 47, column 46

Expression 29

(g∘f):A→C

Conventional reading: g composed with f is a function from A to C

Meaning here: This declaration specifies the function and its stated domain and codomain.

1 occurrence
  1. Occurrence 1: composition.tex, line 26, column 10

Expression 32

dom(f)={x∈A:f(x)↓}

Conventional reading: the domain of f equals the set of x in A for which f of x is defined

Meaning here: The domain of f consists exactly of those elements x of A at which f is defined.

1 occurrence
  1. Occurrence 1: partial-functions.tex, line 27, column 19

Expression 37

f(x)=y∈B

Conventional reading: f of x equals y, and y is an element of B

Meaning here: This mathematical expression states that f of x equals y, and y is an element of B.

1 occurrence
  1. Occurrence 1: composition.tex, line 29, column 65

Expression 38

(∀y∈B)(∃x∈A)f(x)=y.

Conventional reading: for every y in B, there exists an x in A such that f of x equals y

Meaning here: This mathematical expression states that for every y in B, there exists an x in A such that f of x equals y.

1 occurrence
  1. Occurrence 1: function-kinds.tex, line 33, column 1

Expression 41

x1≠x2

Conventional reading: x sub one is not equal to x sub two

Meaning here: This mathematical expression states that x sub one is not equal to x sub two.

1 occurrence
  1. Occurrence 1: inverses.tex, line 48, column 9

Expression 42

f(n)=n+1=n+2−1=g(n)

Conventional reading: f of n equals n plus one, which equals n plus two minus one, which equals g of n

Meaning here: This mathematical expression states that f of n equals n plus one, which equals n plus two minus one, which equals g of n.

1 occurrence
  1. Occurrence 1: function-basics.tex, line 113, column 47

Expression 46

f−1:B→A

Conventional reading: f inverse is a function from B to A

Meaning here: This declaration specifies the function and its stated domain and codomain.

1 occurrence
  1. Occurrence 1: inverses.tex, line 129, column 10

Expression 48

A

Conventional reading: A

Meaning here: The mathematical object denoted by A in the surrounding source context.

14 occurrences
  1. Occurrence 1: function-basics.tex, line 30, column 4
  2. Occurrence 2: function-basics.tex, line 32, column 9
  3. Occurrence 3: function-basics.tex, line 33, column 30
  4. Occurrence 4: function-kinds.tex, line 36, column 46
  5. Occurrence 5: function-kinds.tex, line 67, column 42
  6. Occurrence 6: function-kinds.tex, line 115, column 21
  7. Occurrence 7: function-kinds.tex, line 115, column 44
  8. Occurrence 8: functions-relations.tex, line 15, column 39
  9. Occurrence 9: functions-relations.tex, line 16, column 38
  10. Occurrence 10: composition.tex, line 26, column 67
  11. Occurrence 11: composition.tex, line 28, column 17
  12. Occurrence 12: partial-functions.tex, line 22, column 33
  13. Occurrence 13: partial-functions.tex, line 26, column 58
  14. Occurrence 14: partial-functions.tex, line 32, column 42

Expression 54

(g∘f)(x)=2(x+1)

Conventional reading: g composed with f, evaluated at x, equals two times the quantity x plus one

Meaning here: This expression applies function composition in the source convention: first f, then g.

1 occurrence
  1. Occurrence 1: composition.tex, line 49, column 13

Expression 59

g(y)=the x such that f(x)=y.

Conventional reading: g of y equals the x such that f of x equals y

Meaning here: This mathematical expression states that g of y equals the x such that f of x equals y.

1 occurrence
  1. Occurrence 1: inverses.tex, line 39, column 1

Expression 63

f(x)={x2if x is evenx+12if x is odd.

Conventional reading: f of x equals x divided by two if x is even, and the quantity x plus one, divided by two, if x is odd

Meaning here: This is the displayed piecewise definition, with the even and odd cases spoken in source order.

1 occurrence
  1. Occurrence 1: function-kinds.tex, line 87, column 1

Expression 67

if ∀xf(x)=g(x), then f=g

Conventional reading: if f of x equals g of x for every x, then f equals g

Meaning here: This mathematical expression states that if f of x equals g of x for every x, then f equals g.

1 occurrence
  1. Occurrence 1: function-basics.tex, line 117, column 1

Expression 71

f:A→B

Conventional reading: f is a function from A to B

Meaning here: This declaration specifies the function and its stated domain and codomain.

15 occurrences
  1. Occurrence 1: functions-relations.tex, line 24, column 39
  2. Occurrence 2: functions-relations.tex, line 49, column 39
  3. Occurrence 3: functions-relations.tex, line 61, column 16
  4. Occurrence 4: inverses.tex, line 37, column 19
  5. Occurrence 5: inverses.tex, line 55, column 10
  6. Occurrence 6: inverses.tex, line 63, column 4
  7. Occurrence 7: inverses.tex, line 69, column 14
  8. Occurrence 8: inverses.tex, line 87, column 14
  9. Occurrence 9: inverses.tex, line 92, column 8
  10. Occurrence 10: inverses.tex, line 98, column 14
  11. Occurrence 11: inverses.tex, line 117, column 14
  12. Occurrence 12: inverses.tex, line 128, column 4
  13. Occurrence 13: inverses.tex, line 155, column 16
  14. Occurrence 14: composition.tex, line 40, column 5
  15. Occurrence 15: partial-functions.tex, line 31, column 16

Expression 73

y∈B

Conventional reading: y is an element of B

Meaning here: This mathematical expression states that y is an element of B.

17 occurrences
  1. Occurrence 1: function-kinds.tex, line 31, column 43
  2. Occurrence 2: function-kinds.tex, line 66, column 6
  3. Occurrence 3: functions-relations.tex, line 46, column 41
  4. Occurrence 4: inverses.tex, line 30, column 5
  5. Occurrence 5: inverses.tex, line 47, column 27
  6. Occurrence 6: inverses.tex, line 69, column 61
  7. Occurrence 7: inverses.tex, line 81, column 23
  8. Occurrence 8: inverses.tex, line 94, column 16
  9. Occurrence 9: inverses.tex, line 98, column 62
  10. Occurrence 10: inverses.tex, line 100, column 57
  11. Occurrence 11: inverses.tex, line 103, column 27
  12. Occurrence 12: inverses.tex, line 110, column 98
  13. Occurrence 13: inverses.tex, line 113, column 32
  14. Occurrence 14: inverses.tex, line 130, column 32
  15. Occurrence 15: inverses.tex, line 141, column 19
  16. Occurrence 16: partial-functions.tex, line 44, column 16
  17. Occurrence 17: partial-functions.tex, line 63, column 52

Expression 76

×:N2→N

Conventional reading: multiplication is a function from pairs of natural numbers to the natural numbers

Meaning here: This declaration specifies the function and its stated domain and codomain.

1 occurrence
  1. Occurrence 1: function-basics.tex, line 66, column 44

Expression 77

(f↾C)(x)=f(x)

Conventional reading: the restriction of f to C, evaluated at x, equals f of x

Meaning here: This mathematical expression states that the restriction of f to C, evaluated at x, equals f of x.

1 occurrence
  1. Occurrence 1: functions-relations.tex, line 83, column 1

Expression 80

ran(f)={f(x):x∈A}

Conventional reading: the range of f equals the set of values f of x for x in A

Meaning here: The range of f consists exactly of the values f of x obtained as x varies over A.

1 occurrence
  1. Occurrence 1: function-basics.tex, line 39, column 52

Expression 82

f:A→B

Conventional reading: f is a function from A to B

Meaning here: This declaration specifies the function and its stated domain and codomain.

10 occurrences
  1. Occurrence 1: function-basics.tex, line 29, column 19
  2. Occurrence 2: function-kinds.tex, line 45, column 18
  3. Occurrence 3: function-kinds.tex, line 113, column 12
  4. Occurrence 4: functions-relations.tex, line 64, column 56
  5. Occurrence 5: functions-relations.tex, line 79, column 5
  6. Occurrence 6: inverses.tex, line 28, column 66
  7. Occurrence 7: composition.tex, line 24, column 50
  8. Occurrence 8: composition.tex, line 53, column 14
  9. Occurrence 9: composition.tex, line 58, column 14
  10. Occurrence 10: composition.tex, line 63, column 9

Expression 83

g:N→N

Conventional reading: g is a function from the natural numbers to the natural numbers

Meaning here: This declaration specifies the function and its stated domain and codomain.

1 occurrence
  1. Occurrence 1: function-basics.tex, line 103, column 5

Expression 91

y

Conventional reading: y

Meaning here: The mathematical object denoted by y in the surrounding source context.

10 occurrences
  1. Occurrence 1: function-kinds.tex, line 72, column 30
  2. Occurrence 2: functions-relations.tex, line 17, column 29
  3. Occurrence 3: functions-relations.tex, line 54, column 20
  4. Occurrence 4: functions-relations.tex, line 56, column 12
  5. Occurrence 5: functions-relations.tex, line 56, column 38
  6. Occurrence 6: composition.tex, line 30, column 44
  7. Occurrence 7: partial-functions.tex, line 61, column 55
  8. Occurrence 8: partial-functions.tex, line 68, column 20
  9. Occurrence 9: partial-functions.tex, line 70, column 32
  10. Occurrence 10: partial-functions.tex, line 70, column 58

Expression 92

f[C]={f(x):x∈C}

Conventional reading: the image of C under f equals the set of values f of x for x in C

Meaning here: The image of C under f consists exactly of the values f of x for elements x of C.

1 occurrence
  1. Occurrence 1: functions-relations.tex, line 87, column 41

Expression 93

f↾C={⟨x,y⟩∈Rf:x∈C}

Conventional reading: the restriction of f to C equals the ordered pairs x, y in the graph of f whose first coordinate is in C

Meaning here: This mathematical expression states that the restriction of f to C equals the ordered pairs x, y in the graph of f whose first coordinate is in C.

1 occurrence
  1. Occurrence 1: functions-relations.tex, line 84, column 8

Expression 98

g(0)∉N

Conventional reading: g of zero is not a natural number

Meaning here: This mathematical expression states that g of zero is not a natural number.

1 occurrence
  1. Occurrence 1: inverses.tex, line 20, column 18

Expression 99

N→N

Conventional reading: from the natural numbers to the natural numbers

Meaning here: This function type has the natural numbers as both its domain and its codomain.

1 occurrence
  1. Occurrence 1: inverses.tex, line 19, column 64

Expression 102

g∘f:A→C

Conventional reading: g composed with f is a function from A to C

Meaning here: This declaration specifies the function and its stated domain and codomain.

1 occurrence
  1. Occurrence 1: composition.tex, line 41, column 39

Expression 110

g

Conventional reading: g

Meaning here: The mathematical object denoted by g in the surrounding source context.

22 occurrences
  1. Occurrence 1: function-basics.tex, line 104, column 15
  2. Occurrence 2: function-basics.tex, line 105, column 54
  3. Occurrence 3: function-basics.tex, line 111, column 43
  4. Occurrence 4: function-basics.tex, line 114, column 52
  5. Occurrence 5: function-basics.tex, line 120, column 23
  6. Occurrence 6: functions-relations.tex, line 36, column 20
  7. Occurrence 7: inverses.tex, line 18, column 52
  8. Occurrence 8: inverses.tex, line 33, column 23
  9. Occurrence 9: inverses.tex, line 33, column 63
  10. Occurrence 10: inverses.tex, line 46, column 20
  11. Occurrence 11: inverses.tex, line 51, column 18
  12. Occurrence 12: inverses.tex, line 56, column 31
  13. Occurrence 13: inverses.tex, line 87, column 51
  14. Occurrence 14: inverses.tex, line 155, column 53
  15. Occurrence 15: inverses.tex, line 172, column 11
  16. Occurrence 16: inverses.tex, line 173, column 3
  17. Occurrence 17: composition.tex, line 16, column 46
  18. Occurrence 18: composition.tex, line 17, column 23
  19. Occurrence 19: composition.tex, line 19, column 15
  20. Occurrence 20: composition.tex, line 30, column 37
  21. Occurrence 21: composition.tex, line 34, column 65
  22. Occurrence 22: composition.tex, line 41, column 32

Expression 113

f(x1)=f(x2)

Conventional reading: f of x sub one equals f of x sub two

Meaning here: This mathematical expression states that f of x sub one equals f of x sub two.

1 occurrence
  1. Occurrence 1: inverses.tex, line 48, column 28

Expression 115

X:N→R

Conventional reading: the square-root function is a function from the natural numbers to the real numbers

Meaning here: This declaration specifies the function and its stated domain and codomain.

1 occurrence
  1. Occurrence 1: function-basics.tex, line 70, column 42

Expression 117

y=f(x1)=f(x2)

Conventional reading: y equals f of x sub one, which equals f of x sub two

Meaning here: This mathematical expression states that y equals f of x sub one, which equals f of x sub two.

1 occurrence
  1. Occurrence 1: inverses.tex, line 49, column 24

Expression 120

B

Conventional reading: B

Meaning here: The mathematical object denoted by B in the surrounding source context.

12 occurrences
  1. Occurrence 1: function-basics.tex, line 30, column 29
  2. Occurrence 2: function-basics.tex, line 32, column 42
  3. Occurrence 3: function-basics.tex, line 34, column 32
  4. Occurrence 4: function-kinds.tex, line 30, column 68
  5. Occurrence 5: function-kinds.tex, line 36, column 53
  6. Occurrence 6: function-kinds.tex, line 67, column 49
  7. Occurrence 7: function-kinds.tex, line 115, column 28
  8. Occurrence 8: function-kinds.tex, line 115, column 52
  9. Occurrence 9: functions-relations.tex, line 15, column 62
  10. Occurrence 10: functions-relations.tex, line 16, column 46
  11. Occurrence 11: partial-functions.tex, line 22, column 64
  12. Occurrence 12: partial-functions.tex, line 23, column 30

Expression 126

g(y)={xif f(x)=yaif y∉ran(f).

Conventional reading: g of y equals x when f of x equals y, and equals a when y is not in the range of f

Meaning here: This defines g by choosing a preimage x for values in the range of f and the fixed element a otherwise.

1 occurrence
  1. Occurrence 1: inverses.tex, line 75, column 1

Expression 128

g(f(x))=g(y)=z∈C

Conventional reading: g of f of x equals g of y, which equals z, and z is an element of C

Meaning here: The values g of f of x and g of y both equal z, and z belongs to C.

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

Expression 131

{f(x)∈B:x∈A}

Conventional reading: the set of values f of x in B for x in A

Meaning here: This set-builder expression denotes exactly the objects satisfying the spoken condition.

1 occurrence
  1. Occurrence 1: function-kinds.tex, line 47, column 1

5 diagrams

Function from a three-element domain to a four-element codomain

Three domain elements each have exactly one outgoing arrow. All three arrows end at the same codomain element, leaving three codomain elements unused.

  1. Domain: three elements.
  2. Codomain: four elements.
  3. Mapping: three arrows, one from each domain element; one codomain element receives all three arrows and the other three receive none.

Open in reading context.

Surjective function from three domain elements onto two codomain elements

Each of the three domain elements has one outgoing arrow, and both codomain elements receive at least one arrow.

  1. Domain: three elements.
  2. Codomain: two elements.
  3. Mapping: three arrows; every codomain element is a value, so the function is surjective.

Open in reading context.

Injective function from three domain elements into a seven-element codomain

The three domain elements point to three distinct codomain elements. Four codomain elements receive no arrow.

  1. Domain: three elements.
  2. Codomain: seven elements.
  3. Mapping: three arrows with distinct targets, so no two arguments have the same value.

Open in reading context.

Bijective function pairing three domain elements with three codomain elements

Three arrows pair the three domain elements one-to-one with the three codomain elements; every codomain element is used exactly once.

  1. Domain: three elements.
  2. Codomain: three elements.
  3. Mapping: three one-to-one arrows covering the codomain, so the function is both injective and surjective.

Open in reading context.

Composition of f from A to B and g from B to C

Three elements of A map through two elements of B into two of the four elements of C. The first and third A-elements share one intermediate and final value; the second follows the other branch.

  1. Call the three A-elements the first, second, and third A-elements; call the two B-elements B one and B two; and call the four C-elements C one through C four.
  2. Under f, the first and third A-elements share B one as their middle value; the second A-element maps to B two.
  3. Under g, B one maps to C one and B two maps to C two; C three and C four receive no arrow.
  4. The dashed composite arrows therefore send the first and third A-elements to the same element of C, namely C one, while the second A-element goes to C two.

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45 formal objects

  1. Definition: Function — source line 28.
  2. Figure: Function from a three-element domain to a four-element codomain — source line 49.
  3. Function from a three-element domain to a four-element codomain — source line 50.
  4. Example: Multiplication on natural numbers — source line 57.
  5. Example: Multiplication and square-root mappings — source line 64.
  6. Example: Student grades and parents — source line 74.
  7. Example: Successor function and its range — source line 92.
  8. Example: Extensionally equal successor functions — source line 102.
  9. Example: Function defined by even and odd cases — source line 123.
  10. Figure: Surjective function from three domain elements onto two codomain elements — source line 21.
  11. Surjective function from three domain elements onto two codomain elements — source line 22.
  12. Definition: Surjective function — source line 29.
  13. Figure: Injective function from three domain elements into a seven-element codomain — source line 56.
  14. Injective function from three domain elements into a seven-element codomain — source line 57.
  15. Definition: Injective function — source line 64.
  16. Example: Injective and surjective combinations — source line 76.
  17. Figure: Bijective function pairing three domain elements with three codomain elements — source line 104.
  18. Bijective function pairing three domain elements with three codomain elements — source line 105.
  19. Definition: Bijection — source line 112.
  20. Definition: Graph of a function — source line 24.
  21. Proposition: Relations that are function graphs — source line 42.
  22. Definition: Function restriction and image — source line 78.
  23. Definition: Inverse function — source line 27.
  24. Proposition: Injective functions have left inverses — source line 62.
  25. Exercise: A left inverse implies injectivity — source line 86.
  26. Proposition: Surjective functions have right inverses — source line 91.
  27. Exercise: A right inverse implies surjectivity — source line 116.
  28. Proposition: Bijective functions have inverses — source line 127.
  29. Exercise: Construct the inverse of a bijection — source line 137.
  30. Proposition: Left and right inverses coincide — source line 154.
  31. Exercise: Prove left and right inverses coincide — source line 163.
  32. Proposition: Inverses are unique — source line 167.
  33. Figure: Composition of f from A to B and g from B to C — source line 32.
  34. Composition of f from A to B and g from B to C — source line 33.
  35. Definition: Composition — source line 39.
  36. Example: Composing successor and doubling — source line 45.
  37. Exercise: Composition preserves injectivity — source line 52.
  38. Exercise: Composition preserves surjectivity — source line 57.
  39. Exercise: Graph of a composite function — source line 62.
  40. Definition: Partial function and its domain — source line 20.
  41. Example: Total functions as partial functions — source line 30.
  42. Example: Reciprocal as a partial function — source line 37.
  43. Exercise: Partial inverse of a partial function — source line 42.
  44. Definition: Graph of a partial function — source line 50.
  45. Proposition: Single-valued relations define partial functions — source line 58.

19 source reference records

  1. Figure: Function from a three-element domain to a four-element codomain — function-basics.tex, line 43.
  2. Figure: Surjective function from three domain elements onto two codomain elements — function-kinds.tex, line 19.
  3. Figure: Injective function from three domain elements into a seven-element codomain — function-kinds.tex, line 55.
  4. Figure: Bijective function pairing three domain elements with three codomain elements — function-kinds.tex, line 101.
  5. Proposition: Relations that are function graphs — functions-relations.tex, line 64.
  6. Philosophical Reflections — functions-relations.tex, line 70.
  7. Operations on Relations — functions-relations.tex, line 75.
  8. Operations on Relations — functions-relations.tex, line 96.
  9. Inverses of Functions — functions-relations.tex, line 99.
  10. Composition of Functions — functions-relations.tex, line 100.
  11. the Choice chapter — inverses.tex, line 108.
  12. Proposition: Bijective functions have inverses — inverses.tex, line 138.
  13. Kinds of Functions — inverses.tex, line 146.
  14. Proposition: Bijective functions have inverses — inverses.tex, line 149.
  15. Proposition: Left and right inverses coincide — inverses.tex, line 164.
  16. Proposition: Left and right inverses coincide — inverses.tex, line 174.
  17. Inverses of Functions — composition.tex, line 13.
  18. Operations on Relations — composition.tex, line 21.
  19. Figure: Composition of f from A to B and g from B to C — composition.tex, line 24.