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Flashcards in Chapter 5 Deck (19)
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1

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

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

2

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

  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

3

the empty set [definition / notation]

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.

4

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

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

5

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

A ∩ B = {x : x ∈ A and x ∈ B}

or

(x ∈ A ∩ B) ⇐⇒ (x ∈ A and x ∈ B)

 

6

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

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

7

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

A ∪ B = {x : x ∈ A or x ∈ B}

or

(x ∈ A ∪ B) ⇐⇒ (x ∈ A or x ∈ B)

8

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

A − B = {x: x ∈ A and x ∉ B}

or 

A \ B = {x: x ∈ A and x ∉ B}

9

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

A∆B = (A − B) ∪ (B − A)

10

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

 

A ⊆ B ⇔ BC⊆ AC

11

De Morgan's laws (2) [theorem]

  1. (A ∩ B)C = AC ∪ BC
     
  2. (A ∪ B)C = AC ∩ BC

12

Suppose A, B, and C are sets; then

C ∩ (A ∪ B) = ?

[proposition]

 

C ∩ (A ∪ B) = (C ∩ A) ∪ (C ∩ B)

13

Suppose A, B, and C are sets; then

C ∪ (A ∩ B) = ?

[proposition]

 

C ∪ (A ∩ B) = (C ∪ A) ∩ (C ∪ B)

14

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

A × B := {(a, b): a ∈ A, b ∈ B}

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

15

Equality of ordered pairs is given by... [definition]

(a, b) = (c, d) ⇐⇒ a = c and b = d

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

16

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

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

then A × B ≠ B × A .

17

Let A, B, and C be sets.

A × (B ∪ C) = ?

A × (B ∩ C) = ?

[proposition]

A × (B ∪ C) = (A × B) ∪ (A × C)

and

A × (B ∩ C) = (A × B) ∩ (A × C)

18

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

  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

19

definition of a function [definition]

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).