The Open Logic Text — accessible offline edition

Sets — Explore equations

Sets

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Expression explorer

Each of the 163 cards below represents one distinct expression. MathML remains structurally navigable; speech and meaning are separate visible prose.

Accessible diagrams

Union of A and B: elements in A or B

Union of A and B: elements in A or B The highlighted region contains every element in A or B, including elements in both sets. AB

Union of A and B: elements in A or B. The highlighted region contains every element in A or B, including elements in both sets.

Set A
The left boundary.
Set B
The right boundary.
Highlighted region
The highlighted region contains every element in A or B, including elements in both sets.

Original TikZ asset: assets/diagrams/union.tikz, SHA-256 621f4f8fac2f86282bece8f499f5502a97bc40bc13a6fd36bf5d8a179c51f969.

Intersection of A and B: elements common to both

Intersection of A and B: elements common to both The highlighted region is the overlap containing the elements that belong to both A and B. AB

Intersection of A and B: elements common to both. The highlighted region is the overlap containing the elements that belong to both A and B.

Set A
The left boundary.
Set B
The right boundary.
Highlighted region
The highlighted region is the overlap containing the elements that belong to both A and B.

Original TikZ asset: assets/diagrams/intersection.tikz, SHA-256 56bcfa9dd91404e902b92526929b698d67fa635acf956ccca70732b56717a46b.

Difference A minus B: elements in A but not B

Difference A minus B: elements in A but not B The highlighted region is the part of A outside B, containing elements in A that are not in B. AB

Difference A minus B: elements in A but not B. The highlighted region is the part of A outside B, containing elements in A that are not in B.

Set A
The left boundary.
Set B
The right boundary.
Highlighted region
The highlighted region is the part of A outside B, containing elements in A that are not in B.

Original TikZ asset: assets/diagrams/difference.tikz, SHA-256 bdb3bede3e70dd888eff930948db4cd8615868a323f2cd4162b0cc6e954fe5ec.

Formal-object register

All 52 formal-object records are retained, including nested displays and external diagram records.

  1. Extensionality — definition; source line 26.
  2. Example — example; source line 46.
  3. Example — example; source line 65.
  4. Example — example; source line 74.
  5. Exercise — exercise; source line 98.
  6. Subset — definition; source line 19.
  7. Example — example; source line 27.
  8. Example — example; source line 34.
  9. Proposition — proposition; source line 52.
  10. Definition — definition; source line 63.
  11. Power Set — definition; source line 75.
  12. Example — example; source line 83.
  13. Exercise — exercise; source line 94.
  14. Exercise — exercise; source line 98.
  15. Example — example; source line 12.
  16. Display Math — display math; source line 16.
  17. Strings — example; source line 46.
  18. Display Math — display math; source line 51.
  19. Infinite sequences — example; source line 62.
  20. Figure — figure; source line 23.
  21. External Diagram Tikz — external diagram tikz; source line 24.
  22. Union — definition; source line 34.
  23. Example — example; source line 42.
  24. Exercise — exercise; source line 54.
  25. Figure — figure; source line 63.
  26. External Diagram Tikz — external diagram tikz; source line 64.
  27. Intersection — definition; source line 71.
  28. Example — example; source line 81.
  29. Exercise — exercise; source line 95.
  30. Definition — definition; source line 109.
  31. Display Math — display math; source line 112.
  32. Definition — definition; source line 120.
  33. Display Math — display math; source line 123.
  34. Example — example; source line 130.
  35. Exercise — exercise; source line 134.
  36. Display Math — display math; source line 139.
  37. Display Math — display math; source line 147.
  38. Figure — figure; source line 155.
  39. External Diagram Tikz — external diagram tikz; source line 156.
  40. Difference — definition; source line 162.
  41. Exercise — exercise; source line 170.
  42. Ordered pair — definition; source line 29.
  43. Exercise — exercise; source line 33.
  44. Cartesian product — definition; source line 52.
  45. Example — example; source line 60.
  46. Example — example; source line 68.
  47. Display Math — display math; source line 73.
  48. Exercise — exercise; source line 79.
  49. Proposition — proposition; source line 83.
  50. Exercise — exercise; source line 110.
  51. Example — example; source line 115.
  52. Russell's Paradox — theorem; source line 45.

Resolved internal references

  1. the Extensionality section
  2. the union diagram
  3. the intersection diagram
  4. the difference diagram
  5. the ordered-pair definition

Distinct expressions

Expression 00bf4c3b593b4bae

(∀x∈A)φ

Conventional reading: for every x in A, phi

Meaning here: The bounded universal statement that every element x of A has property phi.

Source notation and occurrences

Normalized TeX: (\forall x \in A)\phi.

  1. occurrence in Read; subsets.tex, line 64, column 1

Expression 0449d10f00dd9025

a1,…,an

Conventional reading: a sub 1, through a sub n

Meaning here: A finite indexed list beginning with a sub 1 and ending with a sub n.

Source notation and occurrences

Normalized TeX: a_1, \ldots, a_n.

  1. occurrence in Read; basics.tex, line 34, column 39

Expression 0554f64b7bf4af5c

{a,b,c}∩∅=∅

Conventional reading: the set a, b, c intersect the empty set equals the empty set

Meaning here: Intersecting the set containing a, b, and c with the empty set produces the empty set.

Source notation and occurrences

Normalized TeX: \{a, b, c \} \cap \emptyset = \emptyset.

  1. occurrence in Read; unions-and-intersections.tex, line 91, column 58

Expression 0724fb5c0fd75a28

{a,b,c}∩{a,b,d}={a,b}

Conventional reading: the set a, b, c intersect the set a, b, d equals the set a, b

Meaning here: The two displayed sets have exactly a and b in common.

Source notation and occurrences

Normalized TeX: \{a, b, c \} \cap \{a, b, d \} = \{a, b\}.

  1. occurrence in Read; unions-and-intersections.tex, line 86, column 12

Expression 076635432edbc84a

A1=AAk+1=Ak×A

Conventional reading: A to the first power equals A; A to the k plus 1 power equals A to the k power cross A

Meaning here: The recursive definition of positive Cartesian powers of A.

Source notation and occurrences

Normalized TeX: A^1 & = A\\ A^{k+1} & = A^k \times A.

  1. occurrence in Read; pairs-and-products.tex, line 73, column 1

Expression 07b4e72f6bc4dfc5

℘(A)={B:B⊆A}

Conventional reading: the power set of A equals the set of B such that B is a subset of A

Meaning here: The power set of A consists exactly of the subsets B of A.

Source notation and occurrences

Normalized TeX: \Pow{A} = \Setabs{B}{B \subseteq A}.

  1. occurrence in Read; subsets.tex, line 78, column 3

Expression 091d9f26c2fab901

B∗={Λ,0,1,00,01,10,11,000,001,010,011,100,101,110,111,0000,…}.

Conventional reading: the binary alphabet B star equals the set of all finite binary strings: the empty string; zero; one; zero zero; zero one; one zero; one one; then all longer binary strings

Meaning here: The Kleene star of the binary alphabet is the set of every finite string over 0 and 1, including the empty string.

Source notation and occurrences

Normalized TeX: \Bin^* =\{\Lambda,0,1,00,01,10,11,\\ 000,001,010,011,100,101,110,111,0000,\ldots\}..

  1. occurrence in Read; important-sets.tex, line 51, column 1

Expression 0ff391fe9593e64c

Bx1∩Bx2=∅

Conventional reading: B sub x sub 1 intersect B sub x sub 2 equals the empty set

Meaning here: The two indexed subsets B sub x1 and B sub x2 are disjoint.

Source notation and occurrences

Normalized TeX: B_{x_1} \cap B_{x_2} = \emptyset.

  1. occurrence in Read; pairs-and-products.tex, line 92, column 18

Expression 105c2094f14f8e23

⟨a,b⟩=⟨c,d⟩ iff both a=c and b=d.

Conventional reading: the ordered pair a, b equals the ordered pair c, d if and only if a equals c and b equals d

Meaning here: Equality of ordered pairs holds exactly when their first coordinates agree and their second coordinates agree.

Source notation and occurrences

Normalized TeX: \tuple{a, b}= \tuple{c, d}\text{ iff both }a = c \text{ and }b=d..

  1. occurrence in Read; pairs-and-products.tex, line 22, column 1

Expression 1470f9303b23391a

6=1+2+3

Conventional reading: 6 equals 1 plus 2 plus 3

Meaning here: Six is the sum of its proper positive divisors 1, 2, and 3.

Source notation and occurrences

Normalized TeX: 6 = 1 + 2 + 3.

  1. occurrence in Read; basics.tex, line 78, column 44

Expression 197c5fad5930a922

x∉A

Conventional reading: x is not an element of A

Meaning here: The object x does not belong to the set A.

Source notation and occurrences

Normalized TeX: x \notin A.

  1. occurrence in Read; basics.tex, line 15, column 35

Expression 1b16b1df538ba12d

n

Conventional reading: n

Meaning here: The natural-number or finite-cardinality variable n in the surrounding statement.

Source notation and occurrences

Normalized TeX: n.

  1. occurrence in Read; subsets.tex, line 99, column 22
  2. occurrence in Read; important-sets.tex, line 56, column 60
  3. occurrence in Read; important-sets.tex, line 57, column 66
  4. occurrence in Read; pairs-and-products.tex, line 47, column 15
  5. occurrence in Read; pairs-and-products.tex, line 49, column 49
  6. occurrence in Read; pairs-and-products.tex, line 84, column 12
  7. occurrence in Read; pairs-and-products.tex, line 111, column 63
  8. occurrence in Read; pairs-and-products.tex, line 117, column 57

Expression 251adfab628d5456

⋃i∈IAi=⋃{Ai:i∈I}⋂i∈IAi=⋂{Ai:i∈I}

Conventional reading: the union over i in I of A sub i equals the union of the set of A sub i for i in I; the intersection over i in I of A sub i equals the intersection of the set of A sub i for i in I

Meaning here: Indexed union and intersection are identified with union and intersection applied to the corresponding set of indexed sets.

Source notation and occurrences

Normalized TeX: \bigcup_{i \in I} A_i & = \bigcup \Setabs{A_i }{i \in I}\\ \bigcap_{i \in I} A_i & = \bigcap\Setabs{A_i}{i \in I}.

  1. occurrence in Read; unions-and-intersections.tex, line 147, column 1

Expression 2d711642b726b044

x

Conventional reading: x

Meaning here: The object or element denoted by x in the surrounding statement.

Source notation and occurrences

Normalized TeX: x.

  1. occurrence in Read; basics.tex, line 14, column 31
  2. occurrence in Read; basics.tex, line 61, column 6
  3. occurrence in Read; basics.tex, line 84, column 51
  4. occurrence in Read; basics.tex, line 84, column 67
  5. occurrence in Read; basics.tex, line 88, column 17
  6. occurrence in Read; basics.tex, line 90, column 41
  7. occurrence in Read; pairs-and-products.tex, line 89, column 23
  8. occurrence in Read; russells-paradox.tex, line 14, column 19

Expression 31fbb7c5331fe79c

⟨x,y⟩≠⟨y,x⟩

Conventional reading: the ordered pair x, y is not equal to the ordered pair y, x

Meaning here: Reversing unequal coordinates changes an ordered pair.

Source notation and occurrences

Normalized TeX: \tuple{x, y} \neq \tuple{y, x}.

  1. occurrence in Read; pairs-and-products.tex, line 17, column 15

Expression 320275c0669894d5

a1a2a3a4…

Conventional reading: a sub 1, a sub 2, a sub 3, a sub 4, and so on

Meaning here: A sequence formed by concatenating the indexed symbols a1, a2, a3, a4, and continuing.

Source notation and occurrences

Normalized TeX: a_1a_2a_3a_4\dots.

  1. occurrence in Read; important-sets.tex, line 65, column 1

Expression 33def926ff9f9112

A×B={⟨0,1⟩,⟨0,a⟩,⟨0,b⟩,⟨1,1⟩,⟨1,a⟩,⟨1,b⟩}.

Conventional reading: A cross B equals the set of ordered pairs 0,1; 0,a; 0,b; 1,1; 1,a; and 1,b

Meaning here: This explicitly enumerates the Cartesian product when A is {0,1} and B is {1,a,b}.

Source notation and occurrences

Normalized TeX: A \times B = \{ \tuple{0, 1}, \tuple{0, a}, \tuple{0, b}, \tuple{1, 1}, \tuple{1, a}, \tuple{1, b} \}..

  1. occurrence in Read; pairs-and-products.tex, line 62, column 1

Expression 383dfb0e1886dbb4

A×B=Bx1∪…∪Bxn

Conventional reading: A cross B equals B sub x sub 1 union through B sub x sub n

Meaning here: The Cartesian product is decomposed as the union of the disjoint rows indexed by x1 through xn.

Source notation and occurrences

Normalized TeX: A \times B = B_{x_1} \cup \dots \cup B_{x_n}.

  1. occurrence in Read; pairs-and-products.tex, line 93, column 21

Expression 3899a1383712a029

{a1,…,an}

Conventional reading: the set containing a sub 1 through a sub n

Meaning here: A finite set with the indexed elements a1 through an.

Source notation and occurrences

Normalized TeX: \{a_{1}, \dots, a_{n}\}.

  1. occurrence in Read; basics.tex, line 33, column 39

Expression 39a7bea0822b3f28

A={x1,…,xn}

Conventional reading: A equals the set containing x sub 1 through x sub n

Meaning here: The finite set A is enumerated by x1 through xn.

Source notation and occurrences

Normalized TeX: A = \{x_1, \dots, x_n\}.

  1. occurrence in Read; pairs-and-products.tex, line 92, column 61

Expression 3a3cd8873330d9e4

S={Ruth}

Conventional reading: S equals the singleton set containing Ruth

Meaning here: The set S has Ruth as its only element.

Source notation and occurrences

Normalized TeX: S=\{\textrm{Ruth}\}.

  1. occurrence in Read; basics.tex, line 49, column 47

Expression 407dfdc9b77f0374

⋃A={x:x belongs to an element of A}, i.e.,={x:there is a B∈A so that x∈B}

Conventional reading: the union of A equals the set of x belonging to an element of A; that is, the set of x for which there is a B in A with x in B

Meaning here: Two equivalent set-builder descriptions of the union of a set A of sets.

Source notation and occurrences

Normalized TeX: \bigcup A & = \Setabs{x}{x \text{ belongs to !!a{element} of } A}, \text{ i.e.,}\\ & = \Setabs{x}{\text{there is a } B \in A \text{ so that } x \in B}.

  1. occurrence in Read; unions-and-intersections.tex, line 112, column 1

Expression 40bd480a01ba6c86

Bx={⟨x,y⟩:y∈B}

Conventional reading: B sub x equals the set of ordered pairs x, y such that y is in B

Meaning here: B sub x is the row of the Cartesian product whose first coordinate is the fixed element x.

Source notation and occurrences

Normalized TeX: B_x = \Setabs{\tuple{x, y}}{y \in B}.

  1. occurrence in Read; pairs-and-products.tex, line 90, column 41

Expression 4306cdb1bcbd69e1

{6}={x:x is perfect and 0≤x≤10}

Conventional reading: the singleton set containing 6 equals the set of x such that x is perfect and 0 is less than or equal to x, which is less than or equal to 10

Meaning here: Six is the unique perfect number between 0 and 10, so both set descriptions have the same sole element.

Source notation and occurrences

Normalized TeX: \{6\} = \Setabs{x}{x\text{ is perfect and }0 \leq x \leq 10}.

  1. occurrence in Read; basics.tex, line 81, column 1

Expression 43ba6b47e3507bec

A∪B={x:x∈A∨x∈B}

Conventional reading: A union B equals the set of x such that x is in A or x is in B

Meaning here: The union of A and B is defined by membership in at least one of the two sets.

Source notation and occurrences

Normalized TeX: A \cup B = \Setabs{x}{x \in A \lor x \in B}.

  1. occurrence in Read; unions-and-intersections.tex, line 37, column 1

Expression 4a2308d566befc9a

{a,b,c}∪{a,0,1}={a,b,c,0,1}

Conventional reading: the set a, b, c union the set a, 0, 1 equals the set a, b, c, 0, 1

Meaning here: The displayed union contains each element occurring in either input set, with the repeated a listed once.

Source notation and occurrences

Normalized TeX: \{ a, b, c\} \cup \{ a, 0, 1\} = \{a, b, c, 0, 1\}.

  1. occurrence in Read; unions-and-intersections.tex, line 45, column 7

Expression 4d48640dbde12a85

N⊊Z⊊Q

Conventional reading: the natural numbers are a proper subset of the integers, which are a proper subset of the rational numbers

Meaning here: Each displayed number system is strictly contained in the next.

Source notation and occurrences

Normalized TeX: \Nat \subsetneq \Int \subsetneq \Rat.

  1. occurrence in Read; important-sets.tex, line 33, column 34

Expression 5096f6dc7b4d3427

⋂A={x:x belongs to every element of A}, i.e.,={x:for all B∈A,x∈B}

Conventional reading: the intersection of A equals the set of x belonging to every element of A; that is, the set of x such that for every B in A, x is in B

Meaning here: Two equivalent set-builder descriptions of the intersection of a set A of sets.

Source notation and occurrences

Normalized TeX: \bigcap A & = \Setabs{x}{x \text{ belongs to every !!{element} of } A}, \text{ i.e.,}\\ & = \Setabs{x}{\text{for all } B \in A, x \in B}.

  1. occurrence in Read; unions-and-intersections.tex, line 123, column 1

Expression 559aead08264d579

A

Conventional reading: A

Meaning here: The set denoted by A in the surrounding statement.

Source notation and occurrences

Normalized TeX: A.

  1. occurrence in Read; basics.tex, line 14, column 60
  2. occurrence in Read; basics.tex, line 27, column 6
  3. occurrence in Read; basics.tex, line 28, column 24
  4. occurrence in Read; basics.tex, line 99, column 63
  5. occurrence in Read; subsets.tex, line 20, column 31
  6. occurrence in Read; subsets.tex, line 21, column 10
  7. occurrence in Read; subsets.tex, line 22, column 1
  8. occurrence in Read; subsets.tex, line 24, column 6
  9. occurrence in Read; subsets.tex, line 43, column 22
  10. occurrence in Read; subsets.tex, line 45, column 52
  11. occurrence in Read; subsets.tex, line 47, column 8
  12. occurrence in Read; subsets.tex, line 48, column 32
  13. occurrence in Read; subsets.tex, line 57, column 36
  14. occurrence in Read; subsets.tex, line 58, column 24
  15. occurrence in Read; subsets.tex, line 76, column 44
  16. occurrence in Read; subsets.tex, line 77, column 21
  17. occurrence in Read; subsets.tex, line 99, column 14
  18. occurrence in Read; important-sets.tex, line 48, column 27
  19. occurrence in Read; important-sets.tex, line 48, column 67
  20. occurrence in Read; important-sets.tex, line 49, column 18
  21. occurrence in Read; important-sets.tex, line 50, column 24
  22. occurrence in Read; important-sets.tex, line 50, column 48
  23. occurrence in Read; important-sets.tex, line 57, column 18
  24. occurrence in Read; important-sets.tex, line 63, column 13
  25. occurrence in Read; important-sets.tex, line 64, column 30
  26. occurrence in Read; important-sets.tex, line 66, column 38
  27. occurrence in Read; unions-and-intersections.tex, line 16, column 49
  28. occurrence in Read; unions-and-intersections.tex, line 18, column 42
  29. occurrence in Read; unions-and-intersections.tex, line 19, column 39
  30. occurrence in Read; unions-and-intersections.tex, line 21, column 30
  31. occurrence in Read; unions-and-intersections.tex, line 26, column 4
  32. occurrence in Read; unions-and-intersections.tex, line 35, column 30
  33. occurrence in Read; unions-and-intersections.tex, line 36, column 45
  34. occurrence in Read; unions-and-intersections.tex, line 61, column 4
  35. occurrence in Read; unions-and-intersections.tex, line 72, column 37
  36. occurrence in Read; unions-and-intersections.tex, line 73, column 54
  37. occurrence in Read; unions-and-intersections.tex, line 110, column 4
  38. occurrence in Read; unions-and-intersections.tex, line 111, column 17
  39. occurrence in Read; unions-and-intersections.tex, line 121, column 4
  40. occurrence in Read; unions-and-intersections.tex, line 122, column 17
  41. occurrence in Read; unions-and-intersections.tex, line 135, column 15
  42. occurrence in Read; unions-and-intersections.tex, line 152, column 68
  43. occurrence in Read; unions-and-intersections.tex, line 158, column 27
  44. occurrence in Read; unions-and-intersections.tex, line 164, column 1
  45. occurrence in Read; pairs-and-products.tex, line 53, column 12
  46. occurrence in Read; pairs-and-products.tex, line 69, column 4
  47. occurrence in Read; pairs-and-products.tex, line 69, column 33
  48. occurrence in Read; pairs-and-products.tex, line 84, column 4
  49. occurrence in Read; pairs-and-products.tex, line 89, column 30
  50. occurrence in Read; pairs-and-products.tex, line 111, column 55
  51. occurrence in Read; pairs-and-products.tex, line 116, column 4
  52. occurrence in Read; pairs-and-products.tex, line 116, column 37
  53. occurrence in Read; pairs-and-products.tex, line 117, column 17
  54. occurrence in Read; pairs-and-products.tex, line 118, column 17
  55. occurrence in Read; pairs-and-products.tex, line 121, column 59
  56. occurrence in Read; pairs-and-products.tex, line 123, column 34
  57. occurrence in Read; russells-paradox.tex, line 28, column 10

Expression 5ddb406720d7553f

⟨x,y⟩∈A×B

Conventional reading: the ordered pair x, y is an element of A cross B

Meaning here: The ordered pair with x from A and y from B belongs to the Cartesian product.

Source notation and occurrences

Normalized TeX: \tuple{x, y} \in A \times B.

  1. occurrence in Read; pairs-and-products.tex, line 90, column 6

Expression 5f21a16235075c7b

{a,b}⊆{a,b,c}

Conventional reading: the set a, b is a subset of the set a, b, c

Meaning here: Every element of {a,b} occurs in {a,b,c}.

Source notation and occurrences

Normalized TeX: \{ a, b \} \subseteq \{ a, b, c \}.

  1. occurrence in Read; subsets.tex, line 30, column 16

Expression 607c77fd8a0e6a0e

{0}⊆{0,{0}}

Conventional reading: the singleton set containing 0 is a subset of the set containing 0 and the singleton set containing 0

Meaning here: The outer set on the right contains 0, so {0} is one of its subsets.

Source notation and occurrences

Normalized TeX: \{0\} \subseteq \{0, \{0\}\}.

  1. occurrence in Read; subsets.tex, line 38, column 46

Expression 657633a67ef1eea7

{x,y}={y,x}

Conventional reading: the set x, y equals the set y, x

Meaning here: Changing the written order of elements does not change an unordered set.

Source notation and occurrences

Normalized TeX: \{x, y\} = \{y, x\}.

  1. occurrence in Read; pairs-and-products.tex, line 16, column 13

Expression 67b47ba1f53809ad

{a,b,c}∪∅={a,b,c}

Conventional reading: the set a, b, c union the empty set equals the set a, b, c

Meaning here: Union with the empty set leaves a set unchanged.

Source notation and occurrences

Normalized TeX: \{a, b, c \} \cup \emptyset = \{a, b, c \}.

  1. occurrence in Read; unions-and-intersections.tex, line 50, column 64

Expression 6bdbfa6e4cc0cd79

0≤x≤10

Conventional reading: 0 is less than or equal to x, which is less than or equal to 10

Meaning here: The value x lies in the closed interval from 0 through 10.

Source notation and occurrences

Normalized TeX: 0 \leq x \leq 10.

  1. occurrence in Read; basics.tex, line 85, column 16

Expression 6d603fa9c7e78307

⟨a,b⟩={{a},{a,b}}

Conventional reading: the ordered pair a, b equals the set containing the singleton a and the set a, b

Meaning here: The Wiener-Kuratowski definition represents the ordered pair (a,b) as {{a},{a,b}}.

Source notation and occurrences

Normalized TeX: \tuple{a, b} = \{\{a\}, \{a, b\}\}.

  1. occurrence in Read; pairs-and-products.tex, line 30, column 2

Expression 733dac99c1e14c52

(∀x∈A)x∈B

Conventional reading: for every x in A, x is in B

Meaning here: Every element of A belongs to B; equivalently, A is a subset of B.

Source notation and occurrences

Normalized TeX: (\forall x \in A)x \in B.

  1. occurrence in Read; subsets.tex, line 69, column 58

Expression 7663a874ace4e740

⋃iAi={x:x belongs to one of the Ai}⋂iAi={x:x belongs to every Ai}.

Conventional reading: the union over i of A sub i equals the set of x belonging to at least one A sub i; the intersection over i of A sub i equals the set of x belonging to every A sub i

Meaning here: Set-builder definitions for an indexed union and an indexed intersection.

Source notation and occurrences

Normalized TeX: \bigcup_i A_i & = \Setabs{x}{x \text{ belongs to one of the } A_i}\\ \bigcap_i A_i & = \Setabs{x}{x \text{ belongs to every } A_i}..

  1. occurrence in Read; unions-and-intersections.tex, line 139, column 1

Expression 7925baff038ff640

N={0,1,2,3,…}the set of natural numbersZ={…,−2,−1,0,1,2,…}the set of integersQ={mn:m,n∈Z and n≠0}the set of rationalsR=(−∞,∞)the set of real numbers (the continuum)

Conventional reading: the natural numbers equal 0, 1, 2, 3, and so on; the integers equal the negative whole numbers, zero, and the positive whole numbers; the rational numbers are fractions m over n with integers m and n and nonzero n; the real numbers run from negative infinity to infinity

Meaning here: A display introducing the natural numbers, integers, rational numbers, and real numbers by standard notation and descriptions.

Source notation and occurrences

Normalized TeX: \Nat = \{0, 1, 2, 3, \ldots\} \\ \shoveright{\text{the set of natural numbers}}\\ \shoveleft{\Int = \{\ldots, -2, -1, 0, 1, 2, \ldots\}} \\ \shoveright{\text{the set of integers}}\\ \shoveleft{\Rat = \Setabs{\nicefrac{m}{n}}{m, n \in \Int\text{ and }n \neq 0}}\\ \shoveright{\text{the set of rationals}}\\ \shoveleft{\Real = (-\infty, \infty)}\\ \text{the set of real numbers (the continuum)}.

  1. occurrence in Read; important-sets.tex, line 16, column 1

Expression 7cdc9f457d52cdfd

⋃A={a,b,d,e}

Conventional reading: the union of A equals the set containing a, b, d, and e

Meaning here: For the displayed family A, collecting all elements of its member sets gives {a,b,d,e}.

Source notation and occurrences

Normalized TeX: \bigcup A = \{ a, b, d, e \}.

  1. occurrence in Read; unions-and-intersections.tex, line 132, column 6

Expression 7dde7df2e48238db

{a,b,c}∩{0,1}=∅

Conventional reading: the set a, b, c intersect the set 0, 1 equals the empty set

Meaning here: The two displayed sets are disjoint.

Source notation and occurrences

Normalized TeX: \{ a, b, c\} \cap \{ 0, 1\} = \emptyset.

  1. occurrence in Read; unions-and-intersections.tex, line 83, column 1

Expression 8261830763a82dce

S={x:x is a sibling of Richard}.

Conventional reading: S equals the set of x such that x is a sibling of Richard

Meaning here: S is defined as exactly the people who are siblings of Richard.

Source notation and occurrences

Normalized TeX: S = \Setabs{x}{x \text{ is a sibling of Richard}}..

  1. occurrence in Read; basics.tex, line 67, column 1

Expression 86b5a927b6c103df

A×B={⟨x,y⟩:x∈A and y∈B}.

Conventional reading: A cross B equals the set of ordered pairs x, y such that x is in A and y is in B

Meaning here: The set-builder definition of the Cartesian product of A and B.

Source notation and occurrences

Normalized TeX: A \times B = \Setabs{\tuple{x, y}}{x \in A \text{ and } y \in B}..

  1. occurrence in Read; pairs-and-products.tex, line 55, column 1

Expression 87e5a3ff7d2be647

A⊈B

Conventional reading: A is not a subset of B

Meaning here: At least one element of A is not an element of B.

Source notation and occurrences

Normalized TeX: A \not\subseteq B.

  1. occurrence in Read; subsets.tex, line 22, column 37

Expression 8c2574892063f995

R

Conventional reading: R

Meaning here: The proposed Russell set R of all sets that are not members of themselves.

Source notation and occurrences

Normalized TeX: R.

  1. occurrence in Read; russells-paradox.tex, line 56, column 46
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  3. occurrence in Read; russells-paradox.tex, line 59, column 6
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  7. occurrence in Read; russells-paradox.tex, line 61, column 4
  8. occurrence in Read; russells-paradox.tex, line 61, column 33
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  10. occurrence in Read; russells-paradox.tex, line 66, column 17
  11. occurrence in Read; russells-paradox.tex, line 66, column 38
  12. occurrence in Read; russells-paradox.tex, line 66, column 70
  13. occurrence in Read; russells-paradox.tex, line 67, column 17
  14. occurrence in Read; russells-paradox.tex, line 68, column 56

Expression 8c7e8c877f9d038b

A={{a,b},{a,d,e},{a,d}}

Conventional reading: A equals the set whose elements are the set a, b; the set a, d, e; and the set a, d

Meaning here: A is a set of three sets, used to illustrate union and intersection.

Source notation and occurrences

Normalized TeX: A = \{ \{ a, b \}, \{ a, d, e \}, \{ a, d \} \}.

  1. occurrence in Read; unions-and-intersections.tex, line 131, column 9

Expression 8fca84ca934ef037

⋂A={a}

Conventional reading: the intersection of A equals the singleton set containing a

Meaning here: For the displayed family A, a is the only element common to every member set.

Source notation and occurrences

Normalized TeX: \bigcap A = \{ a \}.

  1. occurrence in Read; unions-and-intersections.tex, line 132, column 41

Expression 8ff8e4c11c35764c

{a,b,c}∪{a}={a,b,c}

Conventional reading: the set a, b, c union the singleton a equals the set a, b, c

Meaning here: Adding an already present element by union does not change the set.

Source notation and occurrences

Normalized TeX: \{a, b, c \} \cup \{a \} = \{a, b, c\}.

  1. occurrence in Read; unions-and-intersections.tex, line 47, column 67

Expression 9229896a74117a83

{a,b,c}

Conventional reading: the set containing a, b, and c

Meaning here: A set with elements a, b, and c.

Source notation and occurrences

Normalized TeX: \{a, b, c\}.

  1. occurrence in Read; subsets.tex, line 86, column 7

Expression 95912c652e718f03

Z+={1,2,3,…}

Conventional reading: the positive integers equal the set 1, 2, 3, and so on

Meaning here: The set of positive integers begins 1, 2, 3 and continues without end.

Source notation and occurrences

Normalized TeX: \PosInt = \{1, 2, 3, \dots\}.

  1. occurrence in Read; important-sets.tex, line 40, column 55

Expression 95b8ba896291ea12

∀x(x∈A→φ)

Conventional reading: for every x, if x is in A then phi

Meaning here: The ordinary quantified form equivalent to the bounded universal notation over A.

Source notation and occurrences

Normalized TeX: \forall x(x \in A \lif \phi).

  1. occurrence in Read; subsets.tex, line 64, column 37

Expression 988f0f64587f9608

x=x1…xn∈A∗

Conventional reading: x equals x sub 1 through x sub n and is an element of A star

Meaning here: The finite string x is the sequence x1 through xn over the alphabet A.

Source notation and occurrences

Normalized TeX: x=x_{1}\ldots x_{n}\in A^{*}.

  1. occurrence in Read; important-sets.tex, line 56, column 4

Expression 994ccdf6939990cd

R={x:x∉x}

Conventional reading: R equals the set of x such that x is not an element of itself

Meaning here: The proposed Russell set is defined as all sets that are not members of themselves.

Source notation and occurrences

Normalized TeX: R = \Setabs{x}{x \notin x}.

  1. occurrence in Read; russells-paradox.tex, line 40, column 1
  2. occurrence in Read; russells-paradox.tex, line 46, column 18
  3. occurrence in Read; russells-paradox.tex, line 50, column 4
  4. occurrence in Read; russells-paradox.tex, line 75, column 12

Expression a53881d54a0cc74c

{a,a,b}={a,b}={b,a}.

Conventional reading: the set a, a, b equals the set a, b equals the set b, a

Meaning here: Repeated elements and written order do not affect the identity of a set.

Source notation and occurrences

Normalized TeX: \{a, a, b\} = \{a, b\} = \{b,a\}..

  1. occurrence in Read; basics.tex, line 38, column 3

Expression a6cee7730bb0455a

⟨⟨x,y⟩,z⟩

Conventional reading: the ordered pair whose first coordinate is the ordered pair x, y and whose second coordinate is z

Meaning here: A nested-pair representation of the ordered triple (x,y,z).

Source notation and occurrences

Normalized TeX: \tuple{\tuple{x, y},z}.

  1. occurrence in Read; pairs-and-products.tex, line 44, column 36

Expression a8bdb87746a1d744

Bx1={⟨x1,y1⟩⟨x1,y2⟩…⟨x1,ym⟩}Bx2={⟨x2,y1⟩⟨x2,y2⟩…⟨x2,ym⟩}⋮⋮Bxn={⟨xn,y1⟩⟨xn,y2⟩…⟨xn,ym⟩}

Conventional reading: array of Cartesian-product rows: B sub x sub 1 equals the set of ordered pairs x sub 1, y sub 1 through x sub 1, y sub m; B sub x sub 2 equals the set of ordered pairs x sub 2, y sub 1 through x sub 2, y sub m; the pattern continues; B sub x sub n equals the set of ordered pairs x sub n, y sub 1 through x sub n, y sub m

Meaning here: A grid arranging A cross B into one row B sub x for each x in A, with one ordered pair for each y in B.

Source notation and occurrences

Normalized TeX: \begin{array}{rcccc} B_{x_1} = & \{\tuple{x_1, y_1} & \tuple{x_1, y_2} & \dots & \tuple{x_1, y_m}\}\\ B_{x_2} = & \{\tuple{x_2, y_1} & \tuple{x_2, y_2} & \dots & \tuple{x_2, y_m}\}\\ \vdots & & \vdots\\ B_{x_n} = & \{\tuple{x_n, y_1} & \tuple{x_n, y_2} & \dots & \tuple{x_n, y_m}\} \end{array}.

  1. occurrence in Read; pairs-and-products.tex, line 97, column 1

Expression ab6502632839669a

℘({a,b,c})

Conventional reading: the power set of the set a, b, c

Meaning here: The set of every subset of {a,b,c}.

Source notation and occurrences

Normalized TeX: \Pow{\{a,b,c\}}.

  1. occurrence in Read; subsets.tex, line 87, column 1

Expression af6b0b2279da0a6e

{x:φ(x)}

Conventional reading: the set of x such that phi of x

Meaning here: The putative set of all x satisfying phi; the surrounding discussion may be asking whether such a set exists.

Source notation and occurrences

Normalized TeX: \Setabs{x}{\phi(x)}.

  1. occurrence in Read; basics.tex, line 60, column 1
  2. occurrence in Read; basics.tex, line 90, column 1
  3. occurrence in Read; unions-and-intersections.tex, line 14, column 44
  4. occurrence in Read; russells-paradox.tex, line 13, column 38
  5. occurrence in Read; russells-paradox.tex, line 18, column 22

Expression b84881531d3aa83a

⟨a,b⟩=⟨c,d⟩

Conventional reading: the ordered pair a, b equals the ordered pair c, d

Meaning here: An asserted equality between two ordered pairs.

Source notation and occurrences

Normalized TeX: \tuple{a, b}= \tuple{c, d}.

  1. occurrence in Read; pairs-and-products.tex, line 34, column 60

Expression c1dd2383ee29783a

A⊆B

Conventional reading: A is a subset of B

Meaning here: Every element of A is also an element of B.

Source notation and occurrences

Normalized TeX: A \subseteq B.

  1. occurrence in Read; subsets.tex, line 21, column 51
  2. occurrence in Read; subsets.tex, line 23, column 4
  3. occurrence in Read; subsets.tex, line 44, column 38
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  5. occurrence in Read; subsets.tex, line 69, column 38
  6. occurrence in Read; unions-and-intersections.tex, line 55, column 15
  7. occurrence in Read; unions-and-intersections.tex, line 96, column 26

Expression c220527e2692262d

Q⊊R

Conventional reading: the rational numbers are a proper subset of the real numbers

Meaning here: Every rational number is real, and some real numbers are not rational.

Source notation and occurrences

Normalized TeX: \Rat \subsetneq \Real.

  1. occurrence in Read; important-sets.tex, line 36, column 6

Expression c8ea1dac1f40d313

⟨x1,…,xn⟩

Conventional reading: the ordered n tuple x sub 1 through x sub n

Meaning here: An ordered tuple of length n with coordinates x1 through xn.

Source notation and occurrences

Normalized TeX: \tuple{x_1, \dots, x_n}.

  1. occurrence in Read; pairs-and-products.tex, line 47, column 27

Expression d34d91a3dcbf080f

{0}∈{0,{0}}

Conventional reading: the singleton set containing 0 is an element of the set containing 0 and the singleton set containing 0

Meaning here: The right-hand set contains two elements: 0 and the set {0}; the left-hand singleton is the second element.

Source notation and occurrences

Normalized TeX: \{0\} \in \{0, \{0\}\}.

  1. occurrence in Read; subsets.tex, line 38, column 12

Expression d4689498ff4d99cd

⟨⟨⟨x,y⟩,z⟩,u⟩

Conventional reading: the ordered pair whose first coordinate is the ordered pair whose first coordinate is the ordered pair x, y and second coordinate z, and whose second coordinate is u

Meaning here: The nested-pair representation of the ordered quadruple (x,y,z,u).

Source notation and occurrences

Normalized TeX: \tuple{\tuple{\tuple{x,y},z},u}.

  1. occurrence in Read; pairs-and-products.tex, line 46, column 1

Expression d4808682e780fca0

A∖B={x:x∈A and x∉B}.

Conventional reading: A set minus B equals the set of x such that x is in A and x is not in B

Meaning here: The set-builder definition of set difference.

Source notation and occurrences

Normalized TeX: A\setminus B = \Setabs{x}{x\in A \text{ and } x \notin B}..

  1. occurrence in Read; unions-and-intersections.tex, line 165, column 1

Expression d6fed58fa6810190

B={0,1}

Conventional reading: the binary alphabet B equals the set containing zero and one

Meaning here: The binary alphabet B has exactly the symbols 0 and 1.

Source notation and occurrences

Normalized TeX: \Bin = \{0, 1\}.

  1. occurrence in Read; important-sets.tex, line 42, column 9

Expression df7e70e5021544f4

B

Conventional reading: B

Meaning here: The set denoted by B in the surrounding statement.

Source notation and occurrences

Normalized TeX: B.

  1. occurrence in Read; basics.tex, line 27, column 14
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  8. occurrence in Read; subsets.tex, line 43, column 50
  9. occurrence in Read; subsets.tex, line 46, column 17
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  14. occurrence in Read; unions-and-intersections.tex, line 16, column 57
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  28. occurrence in Read; pairs-and-products.tex, line 84, column 33

Expression e137c5e656bf2ddf

A∗={∅}∪A∪A2∪A3∪…

Conventional reading: A star equals the union of the singleton set containing the empty sequence, A, A squared, A cubed, and so on.

Meaning here: All finite strings over A are collected by taking the union of every finite Cartesian power, with the empty string at length zero.

Source notation and occurrences

Normalized TeX: A^* = \{\emptyset\} \cup A \cup A^2 \cup A^3 \cup \dots.

  1. occurrence in Read; pairs-and-products.tex, line 124, column 1

Expression e30febe63300fbc5

N⊆Z⊆Q⊆R

Conventional reading: the natural numbers are a subset of the integers, which are a subset of the rational numbers, which are a subset of the real numbers

Meaning here: The standard chain of inclusions among the four number systems.

Source notation and occurrences

Normalized TeX: \Nat \subseteq \Int \subseteq \Rat \subseteq \Real.

  1. occurrence in Read; important-sets.tex, line 30, column 44

Expression e691119e3a8f7bb8

∃x(x∈A∧φ)

Conventional reading: there exists an x such that x is in A and phi

Meaning here: The ordinary quantified form equivalent to the bounded existential notation over A.

Source notation and occurrences

Normalized TeX: \exists x(x \in A \land \phi).

  1. occurrence in Read; subsets.tex, line 65, column 56

Expression e8b405298dfd1494

{a,b,c,d}

Conventional reading: the set containing a, b, c, and d

Meaning here: A set with elements a, b, c, and d.

Source notation and occurrences

Normalized TeX: \{a, b, c, d\}.

  1. occurrence in Read; subsets.tex, line 95, column 21

Expression ebe091a297bc654f

R∈R

Conventional reading: R is an element of R

Meaning here: The Russell set R contains itself, considered as one side of the paradox.

Source notation and occurrences

Normalized TeX: R \in R.

  1. occurrence in Read; russells-paradox.tex, line 51, column 1
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  3. occurrence in Read; russells-paradox.tex, line 57, column 32
  4. occurrence in Read; russells-paradox.tex, line 58, column 54
  5. occurrence in Read; russells-paradox.tex, line 64, column 27

Expression f193a6d57ae512bb

℘({a,b,c})={∅,{a},{b},{c},{a,b},{b,c},{a,c},{a,b,c}}

Conventional reading: the power set of the set a, b, c equals the set containing the empty set; the singleton a; the singleton b; the singleton c; the sets a,b; b,c; a,c; and a,b,c

Meaning here: A complete enumeration of all eight subsets of {a,b,c}.

Source notation and occurrences

Normalized TeX: \Pow{\{ a, b, c \}} = \{\emptyset, \{a \}, \{b\}, \{c\}, \{a, b\}, \{b, c\}, \{a, c\}, \{a, b, c\}\}.

  1. occurrence in Read; subsets.tex, line 88, column 1

Expression f2294c710c2c2d57

(∃x∈A)φ

Conventional reading: there exists an x in A such that phi

Meaning here: The bounded existential statement that some element x of A has property phi.

Source notation and occurrences

Normalized TeX: (\exists x \in A)\phi.

  1. occurrence in Read; subsets.tex, line 65, column 20

Expression fc48f18e891a815f

{a,b,c}∩{a,b}={a,b}

Conventional reading: the set a, b, c intersect the set a, b equals the set a, b

Meaning here: Intersecting a set with the displayed subset {a,b} returns that subset.

Source notation and occurrences

Normalized TeX: \{a, b, c\} \cap \{a, b\} = \{a, b\}.

  1. occurrence in Read; unions-and-intersections.tex, line 89, column 6

Expression fcb7577c000b48cf

A∩B={x:x∈A∧x∈B}

Conventional reading: A intersect B equals the set of x such that x is in A and x is in B

Meaning here: The intersection of A and B is defined by simultaneous membership in both sets.

Source notation and occurrences

Normalized TeX: A \cap B = \Setabs{x}{x \in A \land x \in B}.

  1. occurrence in Read; unions-and-intersections.tex, line 74, column 1

Expression fd9be8490770582e

⟨x1,y⟩≠⟨x2,y⟩

Conventional reading: the ordered pair x sub 1, y is not equal to the ordered pair x sub 2, y

Meaning here: Ordered pairs with distinct first coordinates x1 and x2 are distinct even when their second coordinate y is the same.

Source notation and occurrences

Normalized TeX: \tuple{x_1, y} \neq \tuple{x_2, y}.

  1. occurrence in Read; pairs-and-products.tex, line 91, column 43