Set Theory

Question 1

Define a set

Answer 1

A set is an unordered collection of objects

Question 2

What is the rooster method

Answer 2

S = {a, b, c, d} denotes the set S containing elements a, b, c, and d.

• Ellipses can be used when pattern is clear: S = {a, b, c, d, . . . , z}

Question 3

Explain this set-builder notation: S = {x E z+| x is odd and x < 10)

Answer 3

For some positive integers, x is odd and x < 10. Contains all the elements from this set that satisfy the predicate

Question 4

What are the 7 important sets

Answer 4

• N = {0, 1, 2, 3, . . . }: set of natural numbers, numbers > 0
• Z = {. . . , −2, −1, 0, 1, 2, . . . }: set of integers
• Z + = {1, 2, 3, . . . } set of positive integers
• Q = { p q | p ∈ Z, q ∈ Z, and q ̸= 0} set of rational numbers
• R: set of real numbers
• R +: set of positive real numbers
• C: set of complex numbers

Question 5

Define a universal set (U)

Answer 5

The universal set is the set containing everything under consideration

Question 6

Define an empty set (∅)

Answer 6

an empty set is a set with no elements

Question 7

How is cardinality shown for A

Answer 7

|A|

Question 8

When is there set equality

Answer 8

when sets have the same elements,
A = B ↔ ∀x. (x ∈ A ↔ x ∈ B)

Question 9

What does this mean: (A ⊆ B)

Answer 9

set A is a subset of B, if and only if every element of A is also an element of B: ∀x. (x ∈ A → x ∈ B)

Question 10

If A ⊆ B, but A ̸= B what do we call this subset

Answer 10

this is called a proper subset denoted by A ⊂ B

Question 11

Provide an example of a subset

Answer 11

• The set of all computer science students is a subset of all students.
• The set of integers with squares less than 100 is not a subset of N.

Question 12

What is the given rule to calculate how many sub-sets any given set has

Answer 12

2 to the power of n (2^n), where n is the cardinality of the set n = |A|

Question 13

What is the cardinality of the set: A = {1 , 2, 3}

Answer 13

2^3 = 8

Question 14

What does this symbol denote (∈)

Answer 14

“is in”, it denotes a set membership

Question 15

What is the unions of set A and B denoted by

Answer 15

A and B, denoted by A ∪ B, is the set:
{x | x ∈ A ∨ x ∈ B}

Question 16

Complete this example of what the union symbol does {1, 2, 3} ∪ {3, 4, 5}

Answer 16

{1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}

Question 17

What is the intersection of set A and B denoted by

Answer 17

The intersection of the sets A and B, denoted by A ∩ B, is the set:
{x | x ∈ A ∧ x ∈ B}

Question 18

Complete these examples of what the intersection symbol does:
- {1, 2, 3} ∩ {3, 4, 5}
* {1, 2, 3} ∩ {4, 5, 6}

Answer 18

1. = {3}
2. = ∅

Question 19

What is the difference of set A and B denoted by

Answer 19

The difference of the sets A and B, denoted by A − B, is the set:
{x | x ∈ A ∧ x ∈/ B}

Question 20

Complete these examples of what the difference symbol does:
- {1, 2, 3} − {3, 4, 5}
* {1, 2, 3} − {4, 5, 6}

Answer 20

1. {1,2}
2. {1,2,3}

Question 21

What is the complement of set A and B denoted by

Answer 21

The complement of the set A (with respect to U), denoted by A, is the set:
U − A = {x | x ∈ U ∧ x ∈/ A}

Question 22

Complete these examples of what the complement symbol does

Question 23

What does the | symbol mean

Answer 23

“such that for all elements of x”

Question 24

What does the line above the symbols mean

Answer 24

The complement of

Question 25

What does this mean A ⊆ B

Answer 25

A is a subset of B

Question 26

How do I know the cardinality

Answer 26

The number of numbers in a set

Question 27

Define a power set

Answer 27

The set of all subsets of a particular set
e.g. A = {1 , 2, 3}
P(A) = {∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}

Question 28

What does this mean [a, b]

Answer 28

{x | a ≤ x ≤ b}

Question 29

What does this mean [a, b)

Answer 29

{x | a ≤ x < b}

Question 30

What does this mean (a, b]

Answer 30

{x | a < x ≤ b}

Question 31

What does this mean (a, b)

Answer 31

{x | a < x < b}

Question 32

How is a union of sets A and B denoted

Answer 32

A U B

Question 33

How is an intersection of sets A and B denoted

Answer 33

A n B

Question 34

How is the difference of sets A and B denoted

Answer 34

A - B

Question 35

How is the complement of sets A and B denoted

Answer 35

A’ (there should be a line above)

Question 36

State the Identity Law

Answer 36

A U ∅ = A, A n U = A

Question 37

State the Domination law

Answer 37

A U U = U, A n ∅ = ∅

Question 38

State the Complementation law

Answer 38

A’’ = A

Question 39

State the Commutative law

Answer 39

A U B = B U A , A n B = B n A (Change the ordering)

Question 40

State the Associative law

Answer 40

A U (B U C) = (A U B) U C, same for n (change the bracketing)

Question 41

State the Distributive law

Answer 41

A n (B U C) = (A n B) U (A n C ), same for A U ( B n C)

Question 42

State De Morgan’s law

Answer 42

(A U B)’ = A’ n B’, (A n B)’ = A’ U B’

Question 43

State the Absorption law

Answer 43

A U ( A n B) = A, A n ( A U B) = A

Question 44

State the Complement law

Answer 44

A. U A’ = U, A n A’ = ∅