Chapter 2.2-2.3

Question 1

Universal Set

Answer 1

U, is the set of all elements that are being considered

Question 2

Complement

Answer 2

A’, the set of all elements that are in U and not in A

Question 3

U’

Answer 3

∅

Question 4

∅’

Answer 4

U

Question 5

(A’)’

Answer 5

A

Question 6

Well-defined

Answer 6

a set is well-defined if it is possible to determine whether any given item is or is not an element of the set

Question 7

Set that is not well-defined

Answer 7

{x/x is a nice cat} because nice is ambiguous

Question 8

Empty set or null set

Answer 8

is the set with no elements, denoted ∅

Question 9

Finite

Answer 9

a set is finite if the number of elements in the set is a whole number

Question 10

cardinal number or cardinality

Answer 10

the cardinality of a finite set A, denoted n(A), is the number of elements in A

Question 11

Cardinality of the empty set

Answer 11

0

Question 12

Set A is equal to set B if….

Answer 12

if and only if, A and B have exactly the same elements, denoted A=B

Question 13

Set A is equivalent to set B if…

Answer 13

if and only if the cardinality of A, n(A), equals the cardinality of B, n(B), denoted A~B

Question 14

Subset

Answer 14

Let A and B be two sets. A is a subset of B, denoted A ⊆ B if and only if every element of A is also in B
A= {1, 2, 3}
B= {1, 2, 3, 4, 5}

Question 15

Proper Subset

Answer 15

Let A and B be sets, A is a proper subset of B, denoted A⊂B, if and only if A ⊆ B and A≠B
A= {1, 2, 3}
B= {1, 2, 3, 4, 5}

Question 16

Is this a subset?
A= {5, 10, 44}
B= {5, 10, 44}

Answer 16

Yes

Question 17

Is this a subset?

W⊆ ℕ

Answer 17

No

Question 18

Is this a subset?

A’⊆A

Answer 18

No

Question 19

Is this a proper subset?
A= {5, 10, 44}
B= {5, 10, 44}

Answer 19

No, because A=B

Question 20

How to find number of subsets

Answer 20

If n(A)=K then A has 2^K subsets

Question 21

How to find the number of proper subsets

Answer 21

If n(A)=K then A has 2^K - 1 proper subsets

Question 22

Intersection

Answer 22

Let A and B be two sets, the intersection of A and B, denoted A∩ B, is the set of all elements that belong to both A and B
A∩ B= {x/xEA and xEB}

Question 23

U= {1, 2, 3...10}
A= {1, 4, 5, 7}
B= {2, 3, 5, 10}
C= {3, 8, 10}
Find A∩ B

Answer 23

{5}

Question 24

U= {1, 2, 3...10}
A= {1, 4, 5, 7}
B= {2, 3, 5, 10}
C= {3, 8, 10}
Find B∩ C

Answer 24

{3, 10}

Question 25

``` U= {1, 2, 3...10} A= {1, 4, 5, 7} B= {2, 3, 5, 10} C= {3, 8, 10} Find A∩ C ```

Answer 25

∅, A and C are disjoint

Question 26

Union

Answer 26

let A and B be sets, the union of A and B, denoted AUB, is the set of all elements that belong to either A or B AUB={x/xEA or xEB}

Question 27

``` U= {1, 2, 3...10} A= {1, 4, 5, 7} B= {2, 3, 5, 10} C= {3, 8, 10} Find A U B ```

Answer 27

{1, 2, 3, 4, 5, 7, 10}

Question 28

``` U= {1, 2, 3...10} A= {1, 4, 5, 7} B= {2, 3, 5, 10} C= {3, 8, 10} Find A U C ```

Answer 28

{ 1, 3, 4, 5, 7, 8, 10}

Question 29

``` U= {1, 2, 3...10} A= {1, 4, 5, 7} B= {2, 3, 5, 10} C= {3, 8, 10} Find B U C ```

Answer 29

{2, 3, 5, 8, 10}

Question 30

A U ∅

Answer 30

A

Question 31

``` U= {1, 2, 3, 4...10} A= {2, 4, 6, 8} B= {3, 6, 9} C= {1, 4, 8} Find A U (B ∩ C) ```

Answer 31

{2, 4, 6, 8}

Question 32

``` U= {1, 2, 3, 4...10} A= {2, 4, 6, 8} B= {3, 6, 9} C= {1, 4, 8} Find B ∩ C' ```

Answer 32

{3, 6, 9}

Question 33

``` U= {1, 2, 3, 4...10} A= {2, 4, 6, 8} B= {3, 6, 9} C= {1, 4, 8} Find [A ∩ (B U C)]' ```

Answer 33

B U C= {1, 3, 4, 6, 8, 9} A ∩ (B U C) = {4, 6, 8} [A ∩ (B U C)]' = {1, 2, 3, 5, 7, 9, 10}

Question 34

DeMorgan's Laws

Answer 34

For all sets A and B: (A U B)' = A' ∩ B' (A ∩ B)' = A' U B'

Question 35

Commutative Property

Answer 35

A U B = B U A | A ∩ B = B ∩ A

Question 36

Associative Property

Answer 36

``` A ∩ (B ∩ C) = (A ∩ B) ∩ C A U (B U C) = (A U B) U C ```

Question 37

Distributive Property

Answer 37

``` A ∩ (B U C) = (A ∩ B) U (A ∩ C) A U (B ∩ C) = (A U B) ∩ (A U C) ```