Chapter 5

Question 1

Suppose A, B, and C are sets; set containment is… (give 2 answers) [proposition]

Answer 1

1. set containment is reflexive
   A ⊆ A
2. set containment is transitive
   if A ⊆ B and B ⊆ C, then A ⊆ C

Question 2

Suppose A, B, and C are sets; state the 3 characteristics of set equality [proposition]

Answer 2

1. set equality is reflexive
   A = A
2. set eqaulity is symmetric
   if A = B, then B = A
3. set equality is transitive
   ​if A = B and B = C, then A = C

Question 3

the empty set [definition / notation]

Answer 3

The empty set, denoted ∅, is the set with no elements. That is, it is the set such that x ∈ ∅ is never true, no matter what x is.

Question 4

state the proposition that says that there is only one empty set [proposition]

Answer 4

Suppose ∅₁ and ∅₂ have the property that x ∈ ∅₁ is never true
andx ∈ ∅₂ is never true.
Then ∅₁ = ∅₂.

Question 5

intersection of two sets A and B [definition / notation]

Answer 5

A ∩ B = {x : x ∈ A and x ∈ B}

or

(x ∈ A ∩ B) ⇐⇒ (x ∈ A and x ∈ B)

Question 6

If A ∩ B = ∅, we say that A and B are… [terminology]

Answer 6

If A ∩ B = ∅, we say that A and B are disjoint.

Question 7

union of 2 sets A and B [definition / notation]

Answer 7

A ∪ B = {x : x ∈ A or x ∈ B}

or

(x ∈ A ∪ B) ⇐⇒ (x ∈ A or x ∈ B)

Question 8

If A and B are sets, then their set difference is…
[definition / notation]

Answer 8

A − B = {x: x ∈ A and x ∉ B}

or

A \ B = {x: x ∈ A and x ∉ B}

Question 9

If A and B are sets, then their symmetric difference is…
[definition / notation]

Answer 9

A∆B = (A − B) ∪ (B − A)

Question 10

Suppose A, B ⊆ X ; then A ⊆ B ⇔ []^C ⊆ []^C
[proposition]

Answer 10

A ⊆ B ⇔ B^C⊆ A^C

Question 11

De Morgan’s laws (2) [theorem]

Answer 11

1. (A ∩ B)^C = A^C ∪ B^C
2. (A ∪ B)^C = A^C ∩ B^C

Question 12

Suppose A, B, and C are sets; then

C ∩ (A ∪ B) = ?

[proposition]

Answer 12

C ∩ (A ∪ B) = (C ∩ A) ∪ (C ∩ B)

Question 13

Suppose A, B, and C are sets; then

C ∪ (A ∩ B) = ?

[proposition]

Answer 13

C ∪ (A ∩ B) = (C ∪ A) ∩ (C ∪ B)

Question 14

Suppose A and B are sets; the cartesian product of A and B is defined… [definition / notation]

Answer 14

A × B := {(a, b): a ∈ A, b ∈ B}

where (a,b) is an ordered pair, NOT a set

Question 15

Equality of ordered pairs is given by… [definition]

Answer 15

(a, b) = (c, d) ⇐⇒ a = c and b = d

Thus (a, b) = (b, a) if and only if a = b .

Question 16

If A and B are nonempty sets such that A ̸= B, then… [proposition]

Answer 16

If A and B are nonempty sets such that A ≠ B,

then A × B ≠ B × A .

Question 17

Let A, B, and C be sets.

A × (B ∪ C) = ?

A × (B ∩ C) = ?

[proposition]

Answer 17

A × (B ∪ C) = (A × B) ∪ (A × C)

and

A × (B ∩ C) = (A × B) ∩ (A × C)

Question 18

A function consists of… (3 points) and is denoted…
[definition / notation]

Answer 18

1. a set A called the domain of the function
2. a set B called the codomain of the function
3. a “rule” f that “assigns” to each a ∈ A an element f(a) ∈ B.

We denote such a function f : A → B

Question 19

definition of a function [definition]

Answer 19

A function with domain A and codomain B is a subset Γ of A × B such that for each a ∈ A, there is one and only one b ∈ B, such that (a, b) ∈ Γ. If (a, b) ∈ Γ, we write b = f(a).