Set Theory

Steps towards Z

Equation form expr-007d3ba7b1c1b0d3

⋃A\bigcup A

Read as: the union of capital A

Means: the union of capital A

1 occurrence in this chapter

Equation form expr-00871f04c48376a8

h(f−1(n))=m≠n=f(f−1(n))h(f^{-1}(n)) = m \neq n = f(f^{-1}(n))

Read as: h of the inverse image of n under f equals m; m is not equal to n; and n equals f of the inverse image of n under f

Means: h of the inverse image of n under f equals m; m is not equal to n; and n equals f of the inverse image of n under f

1 occurrence in this chapter

Equation form expr-00d5331c9f0a44ce

ran(g)=ran(f)∖{f(n)}⊊n=dom(g)\ran{g} = \ran{f} \setminus \{f(n)\} \subsetneq n = \dom{g}

Read as: the range of g equals the range of f with f of n removed; this is a proper subset of n; and n equals the domain of g

Means: the range of g equals the range of f with f of n removed; this is a proper subset of n; and n equals the domain of g

1 occurrence in this chapter

Equation form expr-043a718774c572bd

ss

Read as: s

Means: s

2 occurrences in this chapter

Equation form expr-0449d10f00dd9025

a1,…,ana_1, \ldots, a_n

Read as: a subscript one through a subscript n

Means: a subscript one through a subscript n

1 occurrence in this chapter

Equation form expr-05242e1637ae3072

x↦{x}x \mapsto \{x\}

Read as: x maps to the set containing x

Means: x maps to the set containing x

1 occurrence in this chapter

Equation form expr-08f271887ce94707

MM

Read as: capital M

Means: capital M

4 occurrences in this chapter

Equation form expr-0a8b3670a9af94f0

Z−\Z^-

Read as: set theory Z minus

Means: set theory Z minus

1 occurrence in this chapter

Equation form expr-0d09fe1b50ad2d5a

s(n)s(n)

Read as: s of n

Means: s of n

2 occurrences in this chapter

Equation form expr-10ce0200b450ea95

ϕ\phi

Read as: phi

Means: phi

2 occurrences in this chapter

Equation form expr-11baa595827a4e0f

ω\omega

Read as: omega

Means: omega

3 occurrences in this chapter

Equation form expr-124547e317cd8ebf

c∈Ac \in A

Read as: c belongs to capital A

Means: c belongs to capital A

1 occurrence in this chapter

Equation form expr-13d3ca4e970acf51

⟨a,b⟩\tuple{a, b}

Read as: the ordered pair a, then b

Means: the ordered pair a, then b

1 occurrence in this chapter

Equation form expr-14c5989952c8589e

{∅,{∅},{∅,{∅}},{∅,{∅},{∅,{∅}}}}\{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}, \{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}\}

Read as: the set with four members: the empty set; the singleton containing the empty set; the set containing the empty set and that singleton; and the set containing the empty set, that singleton, and the preceding two-member set

Means: the set with four members: the empty set; the singleton containing the empty set; the set containing the empty set and that singleton; and the set containing the empty set, that singleton, and the preceding two-member set

1 occurrence in this chapter

Equation form expr-1600e169f8795a45

{x:x∉x}\Setabs{x}{x \notin x}

Read as: the set of x such that x does not belong to x

Means: the set of x such that x does not belong to x

1 occurrence in this chapter

Equation form expr-1b16b1df538ba12d

nn

Read as: n

Means: n

12 occurrences in this chapter

Equation form expr-2189d39d0356fbd9

{x},{x,y}∈℘(A∪B)\{x\}, \{x, y\} \in \Pow{A \cup B}

Read as: the singleton set containing x and the unordered pair containing x and y both belong to the power set of capital A union capital B

Means: the singleton set containing x and the unordered pair containing x and y both belong to the power set of capital A union capital B

1 occurrence in this chapter

Equation form expr-2225b5a8bdecda32

x∈Ax \in A

Read as: x belongs to capital A

Means: x belongs to capital A

1 occurrence in this chapter

Equation form expr-22c51c4dfc9d3453

⋂∅\bigcap \emptyset

Read as: the intersection of the empty set

Means: the intersection of the empty set

1 occurrence in this chapter

Equation form expr-23f70165bdcd4454

A,BA, B

Read as: capital A, then capital B

Means: capital A, then capital B

2 occurrences in this chapter

Equation form expr-249140f74b23ce3d

ϕ(S)\phi(S)

Read as: phi of capital S

Means: phi of capital S

1 occurrence in this chapter

Equation form expr-252f10c83610ebca

ff

Read as: f

Means: f

4 occurrences in this chapter

Equation form expr-2d711642b726b044

xx

Read as: x

Means: x

1 occurrence in this chapter

Equation form expr-2e36438ba4cdd12f

m∈dom(f)∖ran(f)m \in \dom{f} \setminus \ran{f}

Read as: m belongs to the set difference between the domain of f and the range of f

Means: m belongs to the set difference between the domain of f and the range of f

1 occurrence in this chapter

Equation form expr-31b1584c7e9e92f8

∀A∃U∀x(x∈U↔(∃b∈A)x∈b)\forall A \exists U \forall x(x \in U \liff (\exists b \in A)x \in b)

Read as: for every capital A there exists a capital U such that, for every x, x belongs to capital U if and only if there exists a b in capital A for which x belongs to b

Means: for every capital A there exists a capital U such that, for every x, x belongs to capital U if and only if there exists a b in capital A for which x belongs to b

1 occurrence in this chapter

Equation form expr-31de442ccf56be33

0=∅0 = \emptyset

Read as: zero equals the empty set

Means: zero equals the empty set

1 occurrence in this chapter

Equation form expr-328cd922a54bde3d

℘(A)\Pow{A}

Read as: the power set of capital A

Means: the power set of capital A

1 occurrence in this chapter

Equation form expr-360b47f07cd28848

⋂A={x:(∀y∈A)x∈y}\bigcap A = \Setabs{x}{(\forall y \in A)x \in y}

Read as: the intersection of capital A equals the set of every x that belongs to every y in capital A

Means: the intersection of capital A equals the set of every x that belongs to every y in capital A

1 occurrence in this chapter

Equation form expr-36473535dabf4171

⟨x,y⟩\tuple{x, y}

Read as: the ordered pair x, then y

Means: the ordered pair x, then y

1 occurrence in this chapter

Equation form expr-3e23e8160039594a

bb

Read as: b

Means: b

3 occurrences in this chapter

Equation form expr-414daf5af4e11e32

x∪{x}∈Ix \cup \{x\} \in I

Read as: x union the set containing x belongs to capital I

Means: x union the set containing x belongs to capital I

1 occurrence in this chapter

Equation form expr-44604b0c677aefab

⋂A\bigcap A

Read as: the intersection of capital A

Means: the intersection of capital A

1 occurrence in this chapter

Equation form expr-47a28732d68d18b8

a,b,ca, b, c

Read as: a, then b, then c

Means: a, then b, then c

1 occurrence in this chapter

Equation form expr-48ad02162b20c83f

C={z∈℘(℘(A∪B)):(∃x∈A)(∃y∈B)z=⟨x,y⟩}.C = \Setabs{z \in \Pow{\Pow{A \cup B}}}{(\exists x \in A)(\exists y \in B) z = \tuple{x, y}}.

Read as: capital C is the set of z in the power set of the power set of capital A union capital B such that there exists an x in capital A and a y in capital B for which z is the ordered pair x, y

Means: capital C is the set of z in the power set of the power set of capital A union capital B such that there exists an x in capital A and a y in capital B for which z is the ordered pair x, y

1 occurrence in this chapter

Equation form expr-4a479db6af79906e

a,ba, b

Read as: a, then b

Means: a, then b

1 occurrence in this chapter

Equation form expr-4b7ce6b75f0ec36f

ϕ(x)\phi(x)

Read as: phi of x

Means: phi of x

3 occurrences in this chapter

Equation form expr-4fffb4aadc592253

A/∼\equivclass{A}{\sim}

Read as: the set of equivalence classes of capital A under the equivalence relation tilde

Means: the set of equivalence classes of capital A under the equivalence relation tilde

1 occurrence in this chapter

Equation form expr-506019087e25ec7b

{a,a}\{a, a\}

Read as: the set containing a, then a

Means: the set containing a, then a

1 occurrence in this chapter

Equation form expr-522df6ad7d1bb5ca

Z−\Zminus

Read as: set theory Z minus

Means: set theory Z minus

5 occurrences in this chapter

Equation form expr-527698b0e1409620

∀x(x∈C↔∀X(ϕ(X)→x∈X))row label *\ollabel{bicondelimarbintersection} \forall x(x \in C \liff \forall X(\phi(X) \lif x \in X))\tag{*}

Read as: for every x, x belongs to capital C if and only if, for every capital X, if phi holds of capital X then x belongs to capital X; equation tagged star

Means: for every x, x belongs to capital C if and only if, for every capital X, if phi holds of capital X then x belongs to capital X; equation tagged star

1 occurrence in this chapter

Equation form expr-559aead08264d579

AA

Read as: capital A

Means: capital A

24 occurrences in this chapter

Equation form expr-5870a4b220f63a99

{∅,{∅},{∅,{∅}}}\{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}

Read as: the set with three members: the empty set; the singleton containing the empty set; and the set containing the empty set and that singleton

Means: the set with three members: the empty set; the singleton containing the empty set; and the set containing the empty set and that singleton

1 occurrence in this chapter

Equation form expr-5984fea10a0e9d05

C={x∈S:∀X(ϕ(X)→x∈X)}.C = \Setabs{x \in S}{\forall X(\phi(X) \lif x \in X)}.

Read as: capital C is the set of x in capital S such that, for every capital X, if phi holds of capital X then x belongs to capital X

Means: capital C is the set of x in capital S such that, for every capital X, if phi holds of capital X then x belongs to capital X

1 occurrence in this chapter

Equation form expr-59c124e2130b5077

⟨x,y⟩∈℘(℘(A∪B))\tuple{x,y} \in \Pow{\Pow{A \cup B}}

Read as: the ordered pair x, y belongs to the power set of the power set of capital A union capital B

Means: the ordered pair x, y belongs to the power set of the power set of capital A union capital B

1 occurrence in this chapter

Equation form expr-5a34e27f06dd5914

n∈ran(f)n \in \ran{f}

Read as: n belongs to the range of f

Means: n belongs to the range of f

1 occurrence in this chapter

Equation form expr-5bc2d5a76e85ce41

A×B=CA \times B = C

Read as: capital A times capital B equals capital C

Means: capital A times capital B equals capital C

1 occurrence in this chapter

Equation form expr-5bfdd628d2c71bcf

g=f↾ng = \funrestrictionto{f}{n}

Read as: g equals the restriction of f to n

Means: g equals the restriction of f to n

1 occurrence in this chapter

Equation form expr-5e0359597496ed87

f(n)∈nf(n) \in n

Read as: f of n belongs to n

Means: f of n belongs to n

1 occurrence in this chapter

Equation form expr-5ea7bade116fc2b4

⋂A={x:(∀y∈A)x∈y}={x∈c:(∀y∈A)x∈y}\bigcap A = \Setabs{x}{(\forall y \in A)x \in y} = \Setabs{x \in c}{(\forall y \in A)x \in y}

Read as: the intersection of capital A equals the set of every x that belongs to every y in capital A, and also equals the set of those x in c that belong to every y in capital A

Means: the intersection of capital A equals the set of every x that belongs to every y in capital A, and also equals the set of those x in c that belong to every y in capital A

1 occurrence in this chapter

Equation form expr-5feceb66ffc86f38

00

Read as: zero

Means: zero

2 occurrences in this chapter

Equation form expr-616522c60d7b122c

∼\sim

Read as: the equivalence relation symbol tilde

Means: the equivalence relation symbol tilde

2 occurrences in this chapter

Equation form expr-62e7b9eff078cb65

∅∈I\emptyset \in I

Read as: the empty set belongs to capital I

Means: the empty set belongs to capital I

1 occurrence in this chapter

Equation form expr-6b23c0d5f35d1b11

CC

Read as: capital C

Means: capital C

3 occurrences in this chapter

Equation form expr-6beb67d4d9e77961

R={x∈V:x∉x}={x:x∉x}R = \Setabs{x \in V}{x \notin x} = \Setabs{x}{x \notin x}

Read as: capital R equals the set of x in capital V that do not belong to themselves, and also equals the unrestricted set of every x that does not belong to itself

Means: capital R equals the set of x in capital V that do not belong to themselves, and also equals the unrestricted set of every x that does not belong to itself

1 occurrence in this chapter

Equation form expr-6ce5d640a102e73b

ran(h)⊊dom(h)=s(n)\ran{h} \subsetneq \dom{h} = s(n)

Read as: the range of h is a proper subset of the domain of h, and the domain of h equals s of n

Means: the range of h is a proper subset of the domain of h, and the domain of h equals s of n

1 occurrence in this chapter

Equation form expr-71c0b2f5b1db0328

{x∈A:ϕ(x)}\Setabs{x \in A}{\phi(x)}

Read as: the set of x in capital A such that phi holds of x

Means: the set of x in capital A such that phi holds of x

2 occurrences in this chapter

Equation form expr-7254671d91aa2c75

a∪b=⋃{a,b}a \cup b = \bigcup \{a, b\}

Read as: a union b equals the union of the set containing a, then b

Means: a union b equals the union of the set containing a, then b

1 occurrence in this chapter

Equation form expr-791fc490b3d1664d

⋂{X:ϕ(X)}\bigcap\Setabs{X}{\phi(X)}

Read as: the intersection of the set of capital X such that phi of capital X

Means: the intersection of the set of capital X such that phi of capital X

1 occurrence in this chapter

Equation form expr-7a045c1ef97d790c

A×BA \times B

Read as: capital A times capital B

Means: capital A times capital B

1 occurrence in this chapter

Equation form expr-7c4f080f17ee2525

n∉ran(h)n \notin \ran{h}

Read as: n does not belong to the range of h

Means: n does not belong to the range of h

1 occurrence in this chapter

Equation form expr-7d2d295cd8fd0e95

y∈By \in B

Read as: y belongs to capital B

Means: y belongs to capital B

1 occurrence in this chapter

Equation form expr-7d7ca52a52f87b72

⋃A={x:(∃b∈A)x∈b}\bigcup A = \Setabs{x}{(\exists b \in A) x \in b}

Read as: the union of capital A equals the set of every x that belongs to at least one b in capital A

Means: the union of capital A equals the set of every x that belongs to at least one b in capital A

1 occurrence in this chapter

Equation form expr-7f2315ed2fa0ec08

Z\Z

Read as: set theory Z

Means: set theory Z

1 occurrence in this chapter

Equation form expr-80dbb42071375608

clof(o)\closureofunder{f}{o}

Read as: the closure of o under f

Means: the closure of o under f

1 occurrence in this chapter

Equation form expr-82c2e741af5090cb

A∖BA \setminus B

Read as: capital A set minus capital B

Means: capital A set minus capital B

1 occurrence in this chapter

Equation form expr-8446ba13909ce9f1

s(x)=x∪{x}s(x) = x \cup \{x\}

Read as: s of x equals x union the set containing x

Means: s of x equals x union the set containing x

1 occurrence in this chapter

Equation form expr-889ce6e841fb4dc7

s(n)=n∪{n}s(n) = n \cup \{n\}

Read as: s of n equals n union the set containing n

Means: s of n equals n union the set containing n

1 occurrence in this chapter

Equation form expr-8d2cacefc75ba038

∅\emptyset

Read as: the empty set

Means: the empty set

6 occurrences in this chapter

Equation form expr-8d34e8532295b0a0

clof(o)={x∈ran(f)∪{o}:(∀X∋o)(X is f-closed→x∈X)}.\closureofunder{f}{o} = \Setabs{x \in \ran{f} \cup \{o\}}{(\forall X \ni o)(X \text{ is $f$-closed} \lif x \in X)}.

Read as: the closure of o under f equals the set of x in the union of the range of f with the singleton containing o such that, for every capital X containing o, if capital X is closed under f then x belongs to capital X

Means: the closure of o under f equals the set of x in the union of the range of f with the singleton containing o such that, for every capital X containing o, if capital X is closed under f then x belongs to capital X

1 occurrence in this chapter

Equation form expr-8de0b3c47f112c59

SS

Read as: capital S

Means: capital S

23 occurrences in this chapter

Equation form expr-9229896a74117a83

{a,b,c}\{a, b, c\}

Read as: the set containing a, then b, then c

Means: the set containing a, then b, then c

1 occurrence in this chapter

Equation form expr-97c1a62ad2cdba93

clof(o)=⋂{X:o∈X and X is f-closed}.The general shape of this is a definition of the form:C=⋂{X:ϕ(X)}.\closureofunder{f}{o} & = \bigcap\Setabs{X}{o \in X \text{ and $X$ is $f$-closed}}. \intertext{The general shape of this is a definition of the form:} C & = \bigcap\Setabs{X}{\phi(X)}.

Read as: Source-ordered display. The closure of o under f equals the intersection of the set of all capital X such that o belongs to capital X and capital X is closed under f. The general shape is this: capital C equals the intersection of the set of all capital X such that phi holds of capital X. End display

Means: Source-ordered display. The closure of o under f equals the intersection of the set of all capital X such that o belongs to capital X and capital X is closed under f. The general shape is this: capital C equals the intersection of the set of all capital X such that phi holds of capital X. End display

1 occurrence in this chapter

Equation form expr-9b833f831f17adcf

{a,b}\{a,b\}

Read as: the set containing a, then b

Means: the set containing a, then b

1 occurrence in this chapter

Equation form expr-9cfac11250fa3c37

(∀x∈A){x}∈A(\forall x \in A)\{x\} \in A

Read as: for every x in capital A, the set containing x belongs to capital A

Means: for every x in capital A, the set containing x belongs to capital A

1 occurrence in this chapter

Equation form expr-9e9f1d6a2199951a

∅∈A\emptyset \in A

Read as: the empty set belongs to capital A

Means: the empty set belongs to capital A

1 occurrence in this chapter

Equation form expr-a14f0673fb84fab2

RA∉RAR_A \notin R_A

Read as: capital R subscript capital A does not belong to capital R subscript capital A

Means: capital R subscript capital A does not belong to capital R subscript capital A

1 occurrence in this chapter

Equation form expr-a1bcb469d4276b70

x∪{x}x \cup \{x\}

Read as: x union the set containing x

Means: x union the set containing x

1 occurrence in this chapter

Equation form expr-a542d1e525457e19

{a1,…,an}\{a_1, \ldots, a_n\}

Read as: the set containing a subscript one through a subscript n

Means: the set containing a subscript one through a subscript n

1 occurrence in this chapter

Equation form expr-a7f78625be6cf92f

A∖B={x∈A:x∉B}A \setminus B = \Setabs{x \in A}{x \notin B}

Read as: capital A set minus capital B equals the set of x in capital A such that x does not belong to capital B

Means: capital A set minus capital B equals the set of x in capital A such that x does not belong to capital B

1 occurrence in this chapter

Equation form expr-a83dd0ccbffe39d0

II

Read as: capital I

Means: capital I

4 occurrences in this chapter

Equation form expr-a866cf8fc5550454

RA={x∈A:x∉x}R_A = \Setabs{x \in A}{x \notin x}

Read as: capital R subscript capital A equals the set of x in capital A such that x does not belong to itself

Means: capital R subscript capital A equals the set of x in capital A such that x does not belong to itself

1 occurrence in this chapter

Equation form expr-a9b9a05aefaee72b

A≠∅A \neq \emptyset

Read as: capital A is not equal to the empty set

Means: capital A is not equal to the empty set

4 occurrences in this chapter

Equation form expr-aaa9402664f1a41f

hh

Read as: h

Means: h

2 occurrences in this chapter

Equation form expr-aba0c52818265ba0

x,y∈A∪Bx, y \in A \cup B

Read as: x and y both belong to capital A union capital B

Means: x and y both belong to capital A union capital B

1 occurrence in this chapter

Equation form expr-ac9158b0318c8e70

h(x)={f(x)if f(x)≠nmif f(x)=nh(x) = \begin{cases} f(x) & \text{if }f(x) \neq n\\ m & \text{if }f(x)=n \end{cases}

Read as: h of x is defined by two cases. If f of x is not equal to n, h of x equals f of x. If f of x equals n, h of x equals m. End cases

Means: h of x is defined by two cases. If f of x is not equal to n, h of x equals f of x. If f of x equals n, h of x equals m. End cases

1 occurrence in this chapter

Equation form expr-aedd0611e173903b

ω=clos(∅)\omega = \closureofunder{s}{\emptyset}

Read as: omega equals the closure of the empty set under s

Means: omega equals the closure of the empty set under s

1 occurrence in this chapter

Equation form expr-b3ca23795ebd1e38

{∅}\{\emptyset\}

Read as: the set containing the empty set

Means: the set containing the empty set

1 occurrence in this chapter

Equation form expr-b5fd9bbfca7dbb57

a∪ba \cup b

Read as: a union b

Means: a union b

1 occurrence in this chapter

Equation form expr-bca6d30e31bfc123

ran(f)⊊dom(f)=s(n)=n∪{n}\ran{f}\subsetneq \dom{f} = s(n) = n \cup \{n\}

Read as: the range of f is a proper subset of the domain of f; the domain of f equals s of n; and s of n equals n union the singleton containing n

Means: the range of f is a proper subset of the domain of f; the domain of f equals s of n; and s of n equals n union the singleton containing n

1 occurrence in this chapter

Equation form expr-c532b1cd10391f04

℘(A)={x:x⊆A}\Pow{A} = \Setabs{x}{x \subseteq A}

Read as: the power set of capital A equals the set of every x that is a subset of capital A

Means: the power set of capital A equals the set of every x that is a subset of capital A

1 occurrence in this chapter

Equation form expr-c6a2fc1c9357db35

n∉ran(f)n \notin \ran{f}

Read as: n does not belong to the range of f

Means: n does not belong to the range of f

1 occurrence in this chapter

Equation form expr-c83045b159648cc1

∃I((∃o∈I)∀xx∉o∧(∀x∈I)(∃s∈I)∀z(z∈s↔(z∈x∨z=x)))\exists I( & (\exists o \in I)\forall x\ x \notin o \land {}\\ & (\forall x \in I)(\exists s \in I)\forall z(z \in s \liff (z \in x \lor z = x)))

Read as: there exists a capital I satisfying two conditions. First, some o belongs to capital I and no x belongs to o. Second, for every x in capital I there exists an s in capital I such that, for every z, z belongs to s if and only if z belongs to x or z equals x. End conditions

Means: there exists a capital I satisfying two conditions. First, some o belongs to capital I and no x belongs to o. Second, for every x in capital I there exists an s in capital I such that, for every z, z belongs to s if and only if z belongs to x or z equals x. End conditions

1 occurrence in this chapter

Equation form expr-ca978112ca1bbdca

aa

Read as: a

Means: a

3 occurrences in this chapter

Equation form expr-cfb45c88261c0fae

{{a},{a,b}}=⟨a,b⟩\{\{a\}, \{a, b\}\} = \tuple{a, b}

Read as: the set whose members are the singleton containing a and the unordered pair containing a and b equals the ordered pair a, b

Means: the set whose members are the singleton containing a and the unordered pair containing a and b equals the ordered pair a, b

1 occurrence in this chapter

Equation form expr-d0e8e45096308ff0

∀a∀b∃P∀x(x∈P↔(x=a∨x=b))\forall a \forall b \exists P \forall x (x \in P \liff (x = a \lor x = b))

Read as: for every a and every b there exists a capital P such that, for every x, x belongs to capital P if and only if x equals a or x equals b

Means: for every a and every b there exists a capital P such that, for every x, x belongs to capital P if and only if x equals a or x equals b

1 occurrence in this chapter

Equation form expr-d10c9934d9bc5dea

ran(f)⊆n\ran{f} \subseteq n

Read as: the range of f is a subset of n

Means: the range of f is a subset of n

1 occurrence in this chapter

Equation form expr-d21b2c2986892de7

{∅,{∅}}\{\emptyset, \{\emptyset\}\}

Read as: the set with two members: the empty set and the singleton containing the empty set

Means: the set with two members: the empty set and the singleton containing the empty set

2 occurrences in this chapter

Equation form expr-d2d4146aa438843f

x↦x∪{x}x \mapsto x \cup \{x\}

Read as: x maps to x union the set containing x

Means: x maps to x union the set containing x

1 occurrence in this chapter

Equation form expr-d4735e3a265e16ee

22

Read as: two

Means: two

2 occurrences in this chapter

Equation form expr-d5f0215bb00e639f

o∈ran(f)∪{o}o \in \ran{f}\cup\{o\}

Read as: o belongs to the range of f union the set containing o

Means: o belongs to the range of f union the set containing o

1 occurrence in this chapter

Equation form expr-d870c8f345891827

{{∅}}\{\{\emptyset\}\}

Read as: the set containing the set containing the empty set

Means: the set containing the set containing the empty set

1 occurrence in this chapter

Equation form expr-d8c8e112dff305a1

s(x)=x∪{x}s(x) = x\cup \{x\}

Read as: s of x equals x union the set containing x

Means: s of x equals x union the set containing x

1 occurrence in this chapter

Equation form expr-d92281c68aed716d

℘(℘(A∪B))\Pow{\Pow{A \cup B}}

Read as: the power set of the power set of capital A union capital B

Means: the power set of the power set of capital A union capital B

1 occurrence in this chapter

Equation form expr-d9fb8fff92326c67

{x:x=x}\Setabs{x}{x = x}

Read as: the set of x such that x equals x

Means: the set of x such that x equals x

1 occurrence in this chapter

Equation form expr-de5a6f78116eca62

VV

Read as: capital V

Means: capital V

1 occurrence in this chapter

Equation form expr-de81a0d4aa4a33d2

∀A∃S∀x(x∈S↔(ϕ(x)∧x∈A)).\forall A \exists S \forall x(x \in S \liff (\phi(x) \land x \in A)).

Read as: for every capital A there exists a capital S such that, for every x, x belongs to capital S if and only if phi holds of x and x belongs to capital A

Means: for every capital A there exists a capital S such that, for every x, x belongs to capital S if and only if phi holds of x and x belongs to capital A

1 occurrence in this chapter

Equation form expr-df7e70e5021544f4

BB

Read as: capital B

Means: capital B

3 occurrences in this chapter

Equation form expr-e325514196af49a9

{a}\{a\}

Read as: the set containing a

Means: the set containing a

3 occurrences in this chapter

Equation form expr-e632b7095b0bf32c

TT

Read as: capital T

Means: capital T

2 occurrences in this chapter

Equation form expr-e8c203d680aa45bd

ran(f)∪{o}\ran{f} \cup \{o\}

Read as: the range of f union the set containing o

Means: the range of f union the set containing o

1 occurrence in this chapter

Equation form expr-e9a6c6d19d6658fc

x∈Ix \in I

Read as: x belongs to capital I

Means: x belongs to capital I

1 occurrence in this chapter

Equation form expr-ea48a33bf9362c6f

{a,b}\{a, b\}

Read as: the set containing a, then b

Means: the set containing a, then b

3 occurrences in this chapter

Equation form expr-f692e7fdbed5a6cd

∅={x∈A:x≠x}\emptyset = \Setabs{x \in A}{x \neq x}

Read as: the empty set equals the set of x in capital A such that x is not equal to itself

Means: the empty set equals the set of x in capital A such that x is not equal to itself

1 occurrence in this chapter

Equation form expr-f6bc3f19366e4a3c

RA∉AR_A \notin A

Read as: capital R subscript capital A does not belong to capital A

Means: capital R subscript capital A does not belong to capital A

1 occurrence in this chapter

Equation form expr-f91a818ffb90b295

{X:ϕ(X)}\Setabs{X}{\phi(X)}

Read as: the set of capital X such that phi of capital X

Means: the set of capital X such that phi of capital X

1 occurrence in this chapter

Equation form expr-fce98a284097e299

∀A∃P∀x(x∈P↔(∀z∈x)z∈A)\forall A \exists P \forall x(x \in P \liff (\forall z \in x)z \in A)

Read as: for every capital A there exists a capital P such that, for every x, x belongs to capital P if and only if every z in x belongs to capital A

Means: for every capital A there exists a capital P such that, for every x, x belongs to capital P if and only if every z in x belongs to capital A

1 occurrence in this chapter

Axiom scheme of Separation

For every formula phi of x and every set capital A, the subset of capital A containing exactly the x for which phi holds exists. This is an axiom scheme, not one axiom.

Source

Formal version of the Separation scheme

For any formula phi of x that does not contain capital S, and for every capital A, there exists a capital S whose members are exactly the x that satisfy phi and belong to capital A.

Source

Theorem that there is no universal set

There is no set containing every object. The source states this as the nonexistence of the set of every x equal to itself.

Source

Proposition establishing the empty set

If any set exists, then the empty set exists. Separation forms it from the members of an arbitrary set that are not equal to themselves.

Source

Proposition establishing set difference

For any capital A and capital B, their set difference exists. Separation selects the members of capital A that do not belong to capital B.

Source

Proposition establishing nonempty intersections

If capital A is not empty, its intersection exists and contains exactly the x that belong to every y in capital A. The nonemptiness condition is essential because the intersection of the empty set would be universal.

Source

Axiom of Union

For every capital A there exists a capital U whose members are exactly the members of members of capital A. Equivalently, the union of capital A exists.

Source

Axiom of Pairs

For any a and b there exists a capital P whose members are exactly a and b. Extensionality permits a and b to be the same set.

Source

Proposition deriving basic pair constructions

For arbitrary a and b, the proposition derives the singleton containing a, the binary union of a and b, and the ordered pair a, b. The proof uses Pairs, Union, and Extensionality in source order.

Source

Exercise constructing a three-member set

Show that the set containing a, b, and c exists for arbitrary sets a, b, and c. The exercise is preserved as stated and no solution is supplied.

Source

Exercise constructing a finite set

Show that the set containing a subscript one through a subscript n exists for arbitrary sets in that finite list. The exercise is preserved as stated and no solution is supplied.

Source

Axiom of Powersets

For every capital A, its power set exists and contains exactly the subsets of capital A. The displayed formula characterizes those subsets by their members.

Source

Proposition establishing Cartesian products

For any capital A and capital B, the Cartesian product of capital A with capital B exists. The proof separates the ordered pairs x, y from the power set of the power set of capital A union capital B.

Source

Exercise constructing relations and functions

For arbitrary capital A and capital B, show first that all relations with domain capital A and range capital B form a set, and second that all functions from capital A to capital B form a set. No solution is supplied.

Source

Exercise constructing a quotient set

Given a set capital A and an equivalence relation tilde on it, prove that the set of equivalence classes of capital A under tilde exists. No solution is supplied.

Source

Axiom of Infinity

There is a set capital I containing an empty object and closed under the operation that sends x to x union its singleton. The source gives both the prose statement and its formal quantified version.

Source

Formal statement of the Axiom of Infinity

There exists a capital I with an element o that has no members, and for every x in capital I there is an s in capital I whose members are exactly the members of x together with x itself.

Source

Definition of the natural numbers and omega

Choose a set supplied by Infinity, define s of x as x union the singleton containing x, and define omega as the closure of the empty set under s. Members of omega are called natural numbers.

Source

Proposition that natural numbers are not Dedekind infinite

No natural number is Dedekind infinite. The source proof proceeds by induction and proves the contrapositive step through two cases for the range of an injection.

Source

Definition of set theory Z minus

Set theory Z minus consists of Extensionality, Union, Pairs, Powersets, Infinity, and every instance of the Separation scheme.

Source

Zermelo alternative Axiom of Infinity

There is a capital A containing the empty set and containing the singleton of every one of its members. The source contrasts this with the von Neumann successor operation.

Source

Closure and intersection display

The display first defines the closure of o under f as the intersection of all f-closed sets containing o. It then gives the general form: capital C is the intersection of all capital X satisfying phi.

Source

Intersection membership condition tagged star

For every x, x belongs to capital C exactly when x belongs to every capital X for which phi holds. This is the condition later obtained by Separation from one witness capital S satisfying phi.

Source

Cross-reference reference-001195

the earlier section on the cumulative iterative approach

Source occurrence

Cross-reference reference-001196

the chapter on ordinals

Source occurrence

Cross-reference reference-001197

Dana Scott (1974)

Source occurrence

Cross-reference reference-001198

Tim Button (2021)

Source occurrence

Cross-reference reference-001199

the Set Theory part

Source occurrence

Cross-reference reference-001200

the natural number induction schema

Source occurrence

Cross-reference reference-001201

the definition of the rank of a set

Source occurrence

Cross-reference reference-001202

the theorem that there is no universal set

Source occurrence

Cross-reference reference-001203

the proposition establishing nonempty intersections

Source occurrence

Cross-reference reference-001204

the section giving the iterative story in more detail

Source occurrence

Cross-reference reference-001205

the singleton item of the basic pair constructions proposition

Source occurrence

Cross-reference reference-001206

the binary union item of the basic pair constructions proposition

Source occurrence

Cross-reference reference-001207

the ordered-pair item of the basic pair constructions proposition

Source occurrence

Cross-reference reference-001208

the singleton item of the basic pair constructions proposition

Source occurrence

Cross-reference reference-001209

the proposition deriving basic pair constructions

Source occurrence

Cross-reference reference-001210

the proposition deriving basic pair constructions

Source occurrence

Cross-reference reference-001211

the proposition establishing the empty set

Source occurrence

Cross-reference reference-001212

the proposition deriving basic pair constructions

Source occurrence

Cross-reference reference-001213

the section on Dedekind's argument for an infinite set

Source occurrence

Cross-reference reference-001214

the theorem extracting a Dedekind algebra from a Dedekind-infinite set

Source occurrence

Cross-reference reference-001215

the section on selecting the natural numbers

Source occurrence

Cross-reference reference-001216

the theorem proving by induction that no natural number is Dedekind infinite

Source occurrence

Cross-reference reference-001217

the section on Dedekind's argument for an infinite set

Source occurrence

Cross-reference reference-001218

1908

Source occurrence

Cross-reference reference-001219

Michael Potter (2004), Appendix A

Source occurrence

Cross-reference reference-001220

the Sets, Functions, and Relations part

Source occurrence

Cross-reference reference-001221

the appendix on closure, comprehension, and intersection

Source occurrence

Cross-reference reference-001222

the definition of omega and the natural numbers

Source occurrence

Cross-reference reference-001223

the reflections section on relations

Source occurrence

Cross-reference reference-001224

the reflections section on arithmetization

Source occurrence

Cross-reference reference-001225

the section on Dedekind's argument for an infinite set

Source occurrence

Cross-reference reference-001226

John von Neumann (1925)

Source occurrence

Cross-reference reference-001227

1908

Source occurrence

Cross-reference reference-001228

the chapter on ordinals

Source occurrence

Cross-reference reference-001229

Paul Benacerraf (1965)

Source occurrence

Cross-reference reference-001230

the section presenting set theory Z minus as a milestone

Source occurrence

Cross-reference reference-001231

the Sets, Functions, and Relations part

Source occurrence

Cross-reference reference-001232

the definition of closure under a function

Source occurrence

Cross-reference reference-001233

the proposition establishing nonempty intersections

Source occurrence

Cross-reference reference-001234

the intersection membership condition tagged star

Source occurrence

Cross-reference reference-001235

the intersection membership condition tagged star

Source occurrence

Cross-reference reference-001236

the result giving the properties of closure under a function

Source occurrence