Methods

Proofs

Equation form expr-0610c330021c16da

A⊈BA \nsubseteq B

Read as: set A is not a subset of set B

Means: set A is not a subset of set B

4 occurrences in this chapter

Equation form expr-0928828231e90719

a∈Aa \in A

Read as: object a is an element of set A

Means: object a is an element of set A

5 occurrences in this chapter

Equation form expr-0a32f6fa082ebbbb

∉A\notin A

Read as: is not an element of set A

Means: is not an element of set A

1 occurrence in this chapter

Equation form expr-0f621dac4808c818

(A∪C)(A \cup C)

Read as: the union of set A and set C

Means: the union of set A and set C

2 occurrences in this chapter

Equation form expr-13d2b4891a10300d

B⊈AB \nsubseteq A

Read as: set B is not a subset of set A

Means: set B is not a subset of set A

3 occurrences in this chapter

Equation form expr-148de9c5a7a44d19

pp

Read as: claim p

Means: claim p

45 occurrences in this chapter

Equation form expr-174d03e08e4f287f

B∩CB \cap C

Read as: the intersection of set B and set C

Means: the intersection of set B and set C

1 occurrence in this chapter

Equation form expr-181dc47425f7c51c

A=∅A = \emptyset

Read as: set A equals the empty set

Means: set A equals the empty set

3 occurrences in this chapter

Equation form expr-1a696505110aab60

x∈Ex \in E

Read as: x is an element of set E

Means: x is an element of set E

7 occurrences in this chapter

Equation form expr-1b16b1df538ba12d

nn

Read as: n

Means: n

1 occurrence in this chapter

Equation form expr-1b1836e594a40c71

A∪B={z:z∈A or z∈B}A \cup B = \Setabs{z}{z \in A \text{ or } z \in B}

Read as: the union of set A and set B equals the set of all z such that z is an element of set A or z is an element of set B

Means: the union of set A and set B equals the set of all z such that z is an element of set A or z is an element of set B

1 occurrence in this chapter

Equation form expr-1c0b47c80ad95a2d

z∈A∪(A∩B)z \in A \cup (A \cap B)

Read as: z is an element of the union of set A with the intersection of set A and set B

Means: z is an element of the union of set A with the intersection of set A and set B

1 occurrence in this chapter

Equation form expr-1db3560c1864169e

B∪C⊆D∪EB \cup C \subseteq D \cup E

Read as: the union of set B and set C is a subset of the union of set D and set E

Means: the union of set B and set C is a subset of the union of set D and set E

2 occurrences in this chapter

Equation form expr-1e05c28e5f477db7

…\dots

Read as: the indicated omitted material

Means: the indicated omitted material

1 occurrence in this chapter

Equation form expr-203ce1cfcfb872f6

z∈(A∪B)∩(A∪C)z \in (A \cup B) \cap (A \cup C)

Read as: z is an element of the intersection of the union of set A and set B with the union of set A and set C

Means: z is an element of the intersection of the union of set A and set B with the union of set A and set C

10 occurrences in this chapter

Equation form expr-20a32376e8cd2e14

z∈(A∪C)z \in (A \cup C)

Read as: z is an element of the union of set A and set C

Means: z is an element of the union of set A and set C

1 occurrence in this chapter

Equation form expr-2225b5a8bdecda32

x∈Ax \in A

Read as: x is an element of set A

Means: x is an element of set A

55 occurrences in this chapter

Equation form expr-22d9305dc102f71a

A≠BA \neq B

Read as: set A is not equal to set B

Means: set A is not equal to set B

4 occurrences in this chapter

Equation form expr-22fead9d1dccca24

∈D\in D

Read as: is an element of set D

Means: is an element of set D

1 occurrence in this chapter

Equation form expr-23a33c3db967907b

y∉A∩By \notin A \cap B

Read as: y is not an element of the intersection of set A and set B

Means: y is not an element of the intersection of set A and set B

1 occurrence in this chapter

Equation form expr-25277529a2746582

z∈(C∖A)z \in (C \setminus A)

Read as: z is an element of the difference of set C and set A

Means: z is an element of the difference of set C and set A

3 occurrences in this chapter

Equation form expr-252f10c83610ebca

ff

Read as: function f

Means: function f

1 occurrence in this chapter

Equation form expr-25d2673cb29ac282

x∈D∪Ex \in D \cup E

Read as: x is an element of the union of set D and set E

Means: x is an element of the union of set D and set E

2 occurrences in this chapter

Equation form expr-273896cb6602ea87

B⊆DB \subseteq D

Read as: set B is a subset of set D

Means: set B is a subset of set D

2 occurrences in this chapter

Equation form expr-29b816495b58c98e

x∉A∩Bx \notin A \cap B

Read as: x is not an element of the intersection of set A and set B

Means: x is not an element of the intersection of set A and set B

2 occurrences in this chapter

Equation form expr-2c894d591039bb4f

…x…\dots x\dots

Read as: the indicated condition involving x

Means: the indicated condition involving x

1 occurrence in this chapter

Equation form expr-2d711642b726b044

xx

Read as: x

Means: x

36 occurrences in this chapter

Equation form expr-2f71310eec649bdf

A∩B⊆AA \cap B \subseteq A

Read as: the intersection of set A and set B is a subset of set A

Means: the intersection of set A and set B is a subset of set A

1 occurrence in this chapter

Equation form expr-346a39b2b468c802

B⊆CB \subseteq C

Read as: set B is a subset of set C

Means: set B is a subset of set C

3 occurrences in this chapter

Equation form expr-36f98a2c2bcb533e

∉A∪B\notin A \cup B

Read as: is not an element of the union of set A and set B

Means: is not an element of the union of set A and set B

1 occurrence in this chapter

Equation form expr-380918b946a52664

==

Read as: set equality

Means: set equality

9 occurrences in this chapter

Equation form expr-39c3d5a5ec4b06af

⊆\subseteq

Read as: the subset relation

Means: the subset relation

9 occurrences in this chapter

Equation form expr-3b2591dd01df745a

x∈Bx \in B

Read as: x is an element of set B

Means: x is an element of set B

27 occurrences in this chapter

Equation form expr-3bf49562bab93a8f

x∉∅x \notin \emptyset

Read as: x is not an element of the empty set

Means: x is not an element of the empty set

2 occurrences in this chapter

Equation form expr-3e8742bab25ce79a

(A∪B)∩(A∪C)(A \cup B) \cap (A \cup C)

Read as: the intersection of the union of set A and set B with the union of set A and set C

Means: the intersection of the union of set A and set B with the union of set A and set C

8 occurrences in this chapter

Equation form expr-3f39d5c348e5b79d

DD

Read as: set D

Means: set D

11 occurrences in this chapter

Equation form expr-43cf364b9f23ce80

B=∅B = \emptyset

Read as: set B equals the empty set

Means: set B equals the empty set

6 occurrences in this chapter

Equation form expr-44a2b8ce148519ee

z∈Bz \in B

Read as: z is an element of set B

Means: z is an element of set B

11 occurrences in this chapter

Equation form expr-44da5aaa33fc4ed0

a∈A∪Ba \in A \cup B

Read as: object a is an element of the union of set A and set B

Means: object a is an element of the union of set A and set B

2 occurrences in this chapter

Equation form expr-450771c906de519d

x∈{x}x \in \{x\}

Read as: x is an element of the singleton set containing x

Means: x is an element of the singleton set containing x

1 occurrence in this chapter

Equation form expr-452e011a23d6de33

A∪(B∩C)=(A∪B)∩(A∪C)A \cup (B \cap C) = (A \cup B) \cap (A \cup C)

Read as: the union of set A with the intersection of set B and set C equals the intersection of the union of set A and set B with the union of set A and set C

Means: the union of set A with the intersection of set B and set C equals the intersection of the union of set A and set B with the union of set A and set C

3 occurrences in this chapter

Equation form expr-454349e422f05297

rr

Read as: claim r

Means: claim r

6 occurrences in this chapter

Equation form expr-46b57457636c794c

A⊆CA \subseteq C

Read as: set A is a subset of set C

Means: set A is a subset of set C

7 occurrences in this chapter

Equation form expr-4794277b410dfc51

¬p\lnot p

Read as: not claim p

Means: not claim p

9 occurrences in this chapter

Equation form expr-4ae81572f06e1b88

QQ

Read as: property Q

Means: property Q

5 occurrences in this chapter

Equation form expr-4aeda343ddc6974e

A⊈A∪BA \nsubseteq A \cup B

Read as: set A is not a subset of the union of set A and set B

Means: set A is not a subset of the union of set A and set B

3 occurrences in this chapter

Equation form expr-50788c6175741372

x∈A∩Bx \in A \cap B

Read as: x is an element of the intersection of set A and set B

Means: x is an element of the intersection of set A and set B

1 occurrence in this chapter

Equation form expr-5112b355a17766ce

C∖AC \setminus A

Read as: the difference of set C and set A

Means: the difference of set C and set A

3 occurrences in this chapter

Equation form expr-515ee3cf005cdd7b

z∈Cz \in C

Read as: z is an element of set C

Means: z is an element of set C

15 occurrences in this chapter

Equation form expr-51cb77c99a583836

z∈A∪Bz \in A \cup B

Read as: z is an element of the union of set A and set B

Means: z is an element of the union of set A and set B

1 occurrence in this chapter

Equation form expr-530efeb0756de186

∈C\in C

Read as: is an element of set C

Means: is an element of set C

1 occurrence in this chapter

Equation form expr-53818f06006fd0f8

A∪(C∖A)=CA \cup (C \setminus A) = C

Read as: the union of set A with the difference of set C and set A equals set C

Means: the union of set A with the difference of set C and set A equals set C

5 occurrences in this chapter

Equation form expr-559aead08264d579

AA

Read as: A

Means: A

51 occurrences in this chapter

Equation form expr-583e90eb7865276c

B⊆AB \subseteq A

Read as: set B is a subset of set A

Means: set B is a subset of set A

1 occurrence in this chapter

Equation form expr-58fd0b552cc34cc3

y∉Ay \notin A

Read as: y is not an element of set A

Means: y is not an element of set A

1 occurrence in this chapter

Equation form expr-594e519ae499312b

zz

Read as: z

Means: z

18 occurrences in this chapter

Equation form expr-5c62e091b8c0565f

PP

Read as: property P

Means: property P

9 occurrences in this chapter

Equation form expr-5d64b261340f62c7

z∈A∪(B∩C)z \in A \cup (B \cap C)

Read as: z is an element of the union of set A with the intersection of set B and set C

Means: z is an element of the union of set A with the intersection of set B and set C

8 occurrences in this chapter

Equation form expr-5ee96a782f0a9fd3

A=BA = B

Read as: set A equals set B

Means: set A equals set B

9 occurrences in this chapter

Equation form expr-606251e5525c947b

A⊈CA \nsubseteq C

Read as: set A is not a subset of set C

Means: set A is not a subset of set C

2 occurrences in this chapter

Equation form expr-61f397eb2c3448c9

{z:z∈A or z∈B}\Setabs{z}{z \in A \text{ or } z \in B}

Read as: the set of all z such that z is an element of set A or z is an element of set B

Means: the set of all z such that z is an element of set A or z is an element of set B

1 occurrence in this chapter

Equation form expr-68c0df64471a7f09

A∪BA \cup B

Read as: the union of set A and set B

Means: the union of set A and set B

7 occurrences in this chapter

Equation form expr-6ad135453b9a4627

z∈B∩Cz \in B \cap C

Read as: z is an element of the intersection of set B and set C

Means: z is an element of the intersection of set B and set C

6 occurrences in this chapter

Equation form expr-6b23c0d5f35d1b11

CC

Read as: set C

Means: set C

16 occurrences in this chapter

Equation form expr-718e1fca6ad235c6

x∉Cx \notin C

Read as: x is not an element of set C

Means: x is not an element of set C

2 occurrences in this chapter

Equation form expr-72cab995fd205b7a

x∈∅x \in \emptyset

Read as: x is an element of the empty set

Means: x is an element of the empty set

4 occurrences in this chapter

Equation form expr-76eada202f49d57f

A∪CA \cup C

Read as: the union of set A and set C

Means: the union of set A and set C

2 occurrences in this chapter

Equation form expr-78541daf3e949c7a

∈∅\in \emptyset

Read as: is an element of the empty set

Means: is an element of the empty set

2 occurrences in this chapter

Equation form expr-795a720997b72208

∩\cap

Read as: intersection

Means: intersection

14 occurrences in this chapter

Equation form expr-7a0ed2d2532fbfb0

A∩B=A∪BA \cap B = A \cup B

Read as: the intersection of set A and set B equals the union of set A and set B

Means: the intersection of set A and set B equals the union of set A and set B

2 occurrences in this chapter

Equation form expr-7c1a695adc01ea28

C⊆EC \subseteq E

Read as: set C is a subset of set E

Means: set C is a subset of set E

2 occurrences in this chapter

Equation form expr-7c53fca3481f18da

x∈Cx \in C

Read as: x is an element of set C

Means: x is an element of set C

10 occurrences in this chapter

Equation form expr-7d2d295cd8fd0e95

y∈By \in B

Read as: y is an element of set B

Means: y is an element of set B

1 occurrence in this chapter

Equation form expr-803d5acbee01ad99

x∈{z:z∈A or z∈B}x \in \Setabs{z}{z \in A \text{ or } z \in B}

Read as: x is an element of the set of all z such that z is an element of set A or z is an element of set B

Means: x is an element of the set of all z such that z is an element of set A or z is an element of set B

1 occurrence in this chapter

Equation form expr-81346791dbd94e64

x∈Dx \in D

Read as: x is an element of set D

Means: x is an element of set D

7 occurrences in this chapter

Equation form expr-81662596ff787b0e

z∈A∪(C∖A)z \in A \cup (C \setminus A)

Read as: z is an element of the union of set A with the difference of set C and set A

Means: z is an element of the union of set A with the difference of set C and set A

6 occurrences in this chapter

Equation form expr-822b488cc527e9ff

¬q\lnot q

Read as: not claim q

Means: not claim q

2 occurrences in this chapter

Equation form expr-8238c028f61fc0f7

A!A

Read as: formula A

Means: formula A

3 occurrences in this chapter

Equation form expr-836a4c57db7fd2c8

A∪B=B∪AA \cup B = B \cup A

Read as: the union of set A and set B equals the union of set B and set A

Means: the union of set A and set B equals the union of set B and set A

3 occurrences in this chapter

Equation form expr-865eaeacbe9bbe8b

x∈A∪Bx \in A \cup B

Read as: x is an element of the union of set A and set B

Means: x is an element of the union of set A and set B

12 occurrences in this chapter

Equation form expr-8ae5fbd5926a724f

∉B\notin B

Read as: is not an element of set B

Means: is not an element of set B

1 occurrence in this chapter

Equation form expr-8d2cacefc75ba038

∅\emptyset

Read as: the empty set

Means: the empty set

4 occurrences in this chapter

Equation form expr-8d346a767c794d52

A⊆A∩(A∪B)A \subseteq A \cap (A \cup B)

Read as: set A is a subset of the intersection of set A with the union of set A and set B

Means: set A is a subset of the intersection of set A with the union of set A and set B

1 occurrence in this chapter

Equation form expr-8dff38d220df5b85

x∈B∪Cx \in B \cup C

Read as: x is an element of the union of set B and set C

Means: x is an element of the union of set B and set C

3 occurrences in this chapter

Equation form expr-8e35c2cd3bf6641b

qq

Read as: claim q

Means: claim q

27 occurrences in this chapter

Equation form expr-8e85f8f2d1232fc4

C⊆A∪(C∖A)C \subseteq A \cup (C \setminus A)

Read as: set C is a subset of the union of set A with the difference of set C and set A

Means: set C is a subset of the union of set A with the difference of set C and set A

2 occurrences in this chapter

Equation form expr-8f0c67897faf0f98

B∪AB \cup A

Read as: the union of set B and set A

Means: the union of set B and set A

3 occurrences in this chapter

Equation form expr-9179efb1240ee176

A∩B≠A∪BA \cap B \neq A \cup B

Read as: the intersection of set A and set B is not equal to the union of set A and set B

Means: the intersection of set A and set B is not equal to the union of set A and set B

3 occurrences in this chapter

Equation form expr-94232b703f817ac6

A∪(C∖A)A \cup (C \setminus A)

Read as: the union of set A with the difference of set C and set A

Means: the union of set A with the difference of set C and set A

3 occurrences in this chapter

Equation form expr-9709cb9a8accca34

∈A∪B\in A \cup B

Read as: is an element of the union of set A and set B

Means: is an element of the union of set A and set B

2 occurrences in this chapter

Equation form expr-9ece59fffb36197d

∉C\notin C

Read as: is not an element of set C

Means: is not an element of set C

2 occurrences in this chapter

Equation form expr-a107d8bed4ed6919

A∪(C∖A)⊆CA \cup (C \setminus A) \subseteq C

Read as: the union of set A with the difference of set C and set A is a subset of set C

Means: the union of set A with the difference of set C and set A is a subset of set C

3 occurrences in this chapter

Equation form expr-a1fce4363854ff88

yy

Read as: y

Means: y

2 occurrences in this chapter

Equation form expr-a5b6b3bcc1013ab6

A∪(B∩C)A \cup (B \cap C)

Read as: the union of set A with the intersection of set B and set C

Means: the union of set A with the intersection of set B and set C

9 occurrences in this chapter

Equation form expr-a7cef6e93d3a33ef

y∈A∪By \in A \cup B

Read as: y is an element of the union of set A and set B

Means: y is an element of the union of set A and set B

1 occurrence in this chapter

Equation form expr-a9b9a05aefaee72b

A≠∅A \neq \emptyset

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

Means: set A is not equal to the empty set

9 occurrences in this chapter

Equation form expr-a9f51566bd6705f7

EE

Read as: set E

Means: set E

11 occurrences in this chapter

Equation form expr-ab8b034ba394de6d

z∈A∩Bz \in A \cap B

Read as: z is an element of the intersection of set A and set B

Means: z is an element of the intersection of set A and set B

2 occurrences in this chapter

Equation form expr-acbf8a92ff994b13

A∪B≠∅A \cup B \neq \emptyset

Read as: the union of set A and set B is not equal to the empty set

Means: the union of set A and set B is not equal to the empty set

4 occurrences in this chapter

Equation form expr-ad753a89977a287e

A∩(A∪B)⊆AA \cap (A \cup B) \subseteq A

Read as: the intersection of set A with the union of set A and set B is a subset of set A

Means: the intersection of set A with the union of set A and set B is a subset of set A

6 occurrences in this chapter

Equation form expr-aea71cbfc818e6c4

{x}\{x\}

Read as: the singleton set containing x

Means: the singleton set containing x

2 occurrences in this chapter

Equation form expr-b277b7f59890a839

A∩BA \cap B

Read as: the intersection of set A and set B

Means: the intersection of set A and set B

1 occurrence in this chapter

Equation form expr-b2fbf7e2d2108677

A∩B≠∅A \cap B \neq \emptyset

Read as: the intersection of set A and set B is not equal to the empty set

Means: the intersection of set A and set B is not equal to the empty set

3 occurrences in this chapter

Equation form expr-ba1c0f0d20ecfc8b

∈E\in E

Read as: is an element of set E

Means: is an element of set E

1 occurrence in this chapter

Equation form expr-ba7f025e4b9d6797

z∈(A∪B)z \in (A \cup B)

Read as: z is an element of the union of set A and set B

Means: z is an element of the union of set A and set B

1 occurrence in this chapter

Equation form expr-bb6b532d3fce7e11

{x:x∈B or x∈C}\Setabs{x}{x \in B \text{ or } x \in C}

Read as: the set of all x such that x is an element of set B or x is an element of set C

Means: the set of all x such that x is an element of set B or x is an element of set C

1 occurrence in this chapter

Equation form expr-bc4c1c1f92cd0fce

C⊆(A∪(C∖A))C \subseteq (A \cup (C \setminus A))

Read as: set C is a subset of the union of set A with the difference of set C and set A

Means: set C is a subset of the union of set A with the difference of set C and set A

1 occurrence in this chapter

Equation form expr-bf32dfb622982985

A∪(B∩C)={z:z∈A or z∈B∩C}A \cup (B \cap C) = \Setabs{z}{z \in A \text{ or } z \in B \cap C}

Read as: the union of set A with the intersection of set B and set C equals the set of all z such that z is an element of set A or z is an element of the intersection of set B and set C

Means: the union of set A with the intersection of set B and set C equals the set of all z such that z is an element of set A or z is an element of the intersection of set B and set C

1 occurrence in this chapter

Equation form expr-c1dd2383ee29783a

A⊆BA \subseteq B

Read as: set A is a subset of set B

Means: set A is a subset of set B

12 occurrences in this chapter

Equation form expr-c37b76b8dd5756ef

A⊆A∪BA \subseteq A \cup B

Read as: set A is a subset of the union of set A and set B

Means: set A is a subset of the union of set A and set B

5 occurrences in this chapter

Equation form expr-c3d66e58e97b248f

A∪B=A∩BA \cup B = A \cap B

Read as: the union of set A and set B equals the intersection of set A and set B

Means: the union of set A and set B equals the intersection of set A and set B

2 occurrences in this chapter

Equation form expr-c898dc65c7570cd6

…x…\dots x \dots

Read as: the indicated condition involving x

Means: the indicated condition involving x

2 occurrences in this chapter

Equation form expr-c8be3a5a4e7422b0

z∈A∪Bz \in A \cup B

Read as: z is an element of the union of set A and set B

Means: z is an element of the union of set A and set B

15 occurrences in this chapter

Equation form expr-ca978112ca1bbdca

aa

Read as: object a

Means: object a

5 occurrences in this chapter

Equation form expr-cbbba670a3f47b53

Γ⊢A\Gamma \Proves !A

Read as: Gamma proves formula A

Means: Gamma proves formula A

1 occurrence in this chapter

Equation form expr-ccb1488d56c77d49

z∈A∩(A∪B)z \in A \cap (A \cup B)

Read as: z is an element of the intersection of set A with the union of set A and set B

Means: z is an element of the intersection of set A with the union of set A and set B

14 occurrences in this chapter

Equation form expr-cfe683e53df50f6f

∈A\in A

Read as: is an element of set A

Means: is an element of set A

3 occurrences in this chapter

Equation form expr-d07113a87d2fa4fa

¬¬p\lnot\lnot p

Read as: not not claim p

Means: not not claim p

3 occurrences in this chapter

Equation form expr-d2ecc6688baa55b4

∪\cup

Read as: union

Means: union

18 occurrences in this chapter

Equation form expr-d3426f0ba71b71db

B≠∅B \neq \emptyset

Read as: set B is not equal to the empty set

Means: set B is not equal to the empty set

2 occurrences in this chapter

Equation form expr-d48c363fa4e9a6a6

A={x}A = \{x\}

Read as: set A equals the singleton set containing x

Means: set A equals the singleton set containing x

1 occurrence in this chapter

Equation form expr-d49351bb36db3c00

A∩(A∪B)=AA \cap (A \cup B) = A

Read as: the intersection of set A with the union of set A and set B equals set A

Means: the intersection of set A with the union of set A and set B equals set A

2 occurrences in this chapter

Equation form expr-d61252d38c5eba2f

B→C!B \lif !C

Read as: formula B implies formula C

Means: formula B implies formula C

3 occurrences in this chapter

Equation form expr-d71c8ecc67edcac1

z∈Az \in A

Read as: z is an element of set A

Means: z is an element of set A

47 occurrences in this chapter

Equation form expr-d7e39bf617e5eabb

z∈{z:z∈A or z∈B∩C}z \in \Setabs{z}{z \in A \text{ or } z \in B \cap C}

Read as: z is an element of the set of all z such that z is an element of set A or z is an element of the intersection of set B and set C

Means: z is an element of the set of all z such that z is an element of set A or z is an element of the intersection of set B and set C

1 occurrence in this chapter

Equation form expr-d9e9e8fe4e8ee5b7

x∈{z:…z…}x \in \Setabs{z}{\dots z\dots}

Read as: x is an element of the set of all z satisfying the indicated condition involving z

Means: x is an element of the set of all z satisfying the indicated condition involving z

1 occurrence in this chapter

Equation form expr-dbcdf113bbc82c27

z∈{z:…z…}z \in \Setabs{z}{\dots z\dots}

Read as: z is an element of the set of all z satisfying the indicated condition involving z

Means: z is an element of the set of all z satisfying the indicated condition involving z

1 occurrence in this chapter

Equation form expr-de9e04577c4e8823

(A∪B)(A \cup B)

Read as: the union of set A and set B

Means: the union of set A and set B

2 occurrences in this chapter

Equation form expr-df7e70e5021544f4

BB

Read as: set B

Means: set B

30 occurrences in this chapter

Equation form expr-e0c61f3ebb6757cf

x∉Bx \notin B

Read as: x is not an element of set B

Means: x is not an element of set B

2 occurrences in this chapter

Equation form expr-e0fd0cc227d96578

D=ED = E

Read as: set D equals set E

Means: set D equals set E

8 occurrences in this chapter

Equation form expr-e332939704d8c108

∈A∪(A∩B)\in A \cup (A \cap B)

Read as: is an element of the union of set A with the intersection of set A and set B

Means: is an element of the union of set A with the intersection of set A and set B

1 occurrence in this chapter

Equation form expr-e50ed6ba575d1aa3

z∈C∖Az \in C \setminus A

Read as: z is an element of the difference of set C and set A

Means: z is an element of the difference of set C and set A

3 occurrences in this chapter

Equation form expr-e54bd061485c646e

B∪CB \cup C

Read as: the union of set B and set C

Means: the union of set B and set C

1 occurrence in this chapter

Equation form expr-e684c4cea1d2fb04

x≠∅x \neq \emptyset

Read as: x is not equal to the empty set

Means: x is not equal to the empty set

1 occurrence in this chapter

Equation form expr-e7574e020a9d1649

A∪(A∩B)=AA \cup (A \cap B) = A

Read as: the union of set A with the intersection of set A and set B equals set A

Means: the union of set A with the intersection of set A and set B equals set A

1 occurrence in this chapter

Equation form expr-eaf1a7a9a0e437c8

z∈A∪Cz \in A \cup C

Read as: z is an element of the union of set A and set C

Means: z is an element of the union of set A and set C

4 occurrences in this chapter

Equation form expr-ef992dde124293fd

z∉Az \notin A

Read as: z is not an element of set A

Means: z is not an element of set A

4 occurrences in this chapter

Equation form expr-f41383ff68321b1e

{z:…z…}\Setabs{z}{\dots z \dots}

Read as: the set of all z satisfying the indicated condition involving z

Means: the set of all z satisfying the indicated condition involving z

1 occurrence in this chapter

Equation form expr-f96891c9879a9014

x∈{x:x∈A∨x∈B}x \in \Setabs{x}{x \in A \lor x \in B}

Read as: x is an element of the set of all x such that x is an element of set A or x is an element of set B

Means: x is an element of the set of all x such that x is an element of set A or x is an element of set B

1 occurrence in this chapter

Equation form expr-fd7d66765960defc

x∉A∪Bx \notin A \cup B

Read as: x is not an element of the union of set A and set B

Means: x is not an element of the union of set A and set B

3 occurrences in this chapter

Equation form expr-feccce16c939a1ff

∉∅\notin \emptyset

Read as: is not an element of the empty set

Means: is not an element of the empty set

1 occurrence in this chapter

Commutativity of set union

For any sets A and B, the union of A with B equals the union of B with A. This proposition is the chapter's first example of unpacking a definition.

Source

Definition of identical sets

Sets A and B are identical if and only if every element of A is an element of B and every element of B is an element of A. The definition explicitly calls the second direction vice versa.

Source

Set-union commutativity with equality unpacked

For any sets A and B, first prove that every x in A or B is in B or A, and second prove the converse direction. This restates the union-commutativity proposition after unpacking set equality and union membership.

Source

Exercise unpacking a nonempty intersection

Restate the claim that the intersection of A and B is not empty without using the intersection symbol, the equality sign, or the empty-set symbol. The source poses the exercise without a solution.

Source

Every set is contained in its union with another set

For all sets A and B, set A is a subset of the union of A and B. The following source proof chooses an arbitrary element of A and applies the definitions of subset and union.

Source

Union preserves two subset assumptions

If set B is a subset of set D and set C is a subset of set E, then the union of B and C is a subset of the union of D and E. The source proves this universal claim by cases.

Source

A nonempty subset containing a given element

If x is an element of set B, then there is a set A that is a subset of B and is not empty. The proof constructs A as the singleton containing x and verifies both required properties.

Source

A union with a nonempty set is nonempty

If set A is not empty, then the union of A and B is not empty. The proof names one element of A, observes that it belongs to the union, and then removes dependence on that chosen name.

Source

Distributivity of union over intersection

For any sets A, B, and C, the union of A with the intersection of B and C equals the intersection of the union of A and B with the union of A and C. The long source proof establishes both inclusions through nested proofs by cases.

Source

A set split into a subset and its relative difference

If set A is a subset of set C, then the union of A with the difference of C and A equals C. The source proves both inclusions and uses excluded middle for whether an arbitrary element of C belongs to A.

Source

A subset of the empty set has no elements

If set A is a subset of set B and B is the empty set, then A has no elements. The source proves the negative conclusion by assuming an element of A and deriving that it belongs to the empty set.

Source

Indirect proof of inclusion in a union

Set A is a subset of the union of A and B. The source gives an indirect proof by assuming a counterexample element and deriving that the same element both belongs and does not belong to the union.

Source

Exercise on an indirect intersection proof

Prove indirectly that the intersection of set A and set B is a subset of set A. The source supplies no proof, so the exercise remains unsolved.

Source

Transitivity of the subset relation

If set A is a subset of set B and set B is a subset of set C, then set A is a subset of set C. The source proves this indirectly from a hypothetical element of A that is not in C.

Source

Equal union and intersection force equal sets

If the union of set A and set B equals their intersection, then A equals B. The source assumes the sets differ, separates the two possible failed inclusions, and derives a contradiction in each case.

Source

Absorption law for intersection and union

For all sets A and B, the intersection of A with the union of A and B equals A. The printed proof is intentionally condensed; the surrounding section explains its two inclusion directions in detail.

Source

Exercise expanding a condensed absorption proof

Expand the supplied short proof that the union of A with the intersection of A and B equals A. Name every inference pattern, justify every step from earlier assumptions or claims, and identify each definition used. The short proof is source material to expand, not a supplied solution.

Source

Cross-reference reference-001646

(Daniel Solow, 2013)

Source occurrence

Cross-reference reference-001647

(Daniel J. Velleman, 2019)

Source occurrence

Cross-reference reference-001648

Book of Proof (external resource; internet required)

Source occurrence

Cross-reference reference-001649

(Richard Hammack, 2013)

Source occurrence

Cross-reference reference-001650

Mathematical Reasoning (external resource; internet required)

Source occurrence

Cross-reference reference-001651

(Ted Sandstrum, 2019)

Source occurrence

Cross-reference reference-001652

(Eric Steinhart, 2018)

Source occurrence

Cross-reference reference-001653

“Introduction to Mathematical Arguments” (external resource; internet required)

Source occurrence

Cross-reference reference-001654

(Michael Hutchings, 2003)

Source occurrence

Cross-reference reference-001655

“How to write proofs” (external resource; internet required)

Source occurrence

Cross-reference reference-001656

(Eugenia Cheng, 2004)

Source occurrence

Cross-reference reference-001657

https://www.youtube.com/watch?v=ZXsQAXx_ao0 (external resource; internet required)

Source occurrence

Cross-reference reference-001658

https://www.youtube.com/watch?v=BQ4yd2W50No (external resource; internet required)

Source occurrence

Cross-reference reference-001659

https://www.youtube.com/watch?v=StTqXEQ2l-Y (external resource; internet required)

Source occurrence

Source disclosures

Source-generated case expression tr074-source-macro-0001

==

Read as: the equality sign

Read in context source