Chapter 5 - Set Theory

Question 1

The set of all points or elements under consideration denoted by Ω and a point in Ω is denoted by ω.

Answer 1

Universal Set

Question 2

A well-defined collection of distinct elements in the universal set, denoted by capital letters.

Answer 2

Set

Question 3

If an element ω belongs in A
then this is denoted by _____.

Answer 3

ω∈A

Question 4

Is ω∉A≡~(ω∈A)?

Answer 4

Yes they are equivalent.

Question 5

A method of enumerating or listing of all the elements in the set, enclosed in braces

Answer 5

Roster or Tabular Method

Question 6

A method employing a descriptive phrase that specifies the properties that characterize all the elements in the set

Answer 6

Rule method

Question 7

Set A is a subset of set B or B is a superset of A if and only if

Answer 7

∀ω∈Ω, ω∈A →ω∈B (for all ω in Ω, ω is an element of A, then ω is an element of B)

Question 8

To prove that A is a subset of B (A⊂ B), take an ____ element in the _____ set and show that ω∈A→ω∈B.

Answer 8

arbitrary, universal

Question 9

To disprove that A⊂B, find _____ in the _____ for which _____ and _____

Answer 9

ω, universal set, ω∈A and ω∉B.

Question 10

A property that states that: Any set A is a subset of itself, that is, A⊂A for any A.

Answer 10

Reflexive property for set inclusion

Question 11

A property that states that If A ⊂ B and B ⊂ C then A ⊂ C.

Answer 11

Transitive property of set inclusion

Question 12

Set inclusion does not enjoy which property?

Answer 12

Property of symmetry

Question 13

A is a ____ of B if and only if A ⊂ B and B⊄ A. That is, there is at least one element of B which is not in A.

Answer 13

Proper subset

Question 14

Two sets A and B are equal if and only if

Answer 14

A⊂B and B⊂A. This is written as A = B.

Question 15

The properties of Equality of Two Sets include?

Answer 15

1. Reflexive property. A= A for any set A.
2. Symmetry property. For any sets A and B, if A=B then B=A.
3. Transitive property. For any sets A, B and C, if A = B and B = C then A = C.

Question 16

Let A and B be subsets of Ω. The complement of A can be written as?

Answer 16

ω∈Ac↔~(ω∈A)

Question 17

Let A and B be subsets of Ω. The union of A and B can be written as?

Answer 17

ω∈A ∪B ↔(ω∈A ∨ω∈B)

Question 18

Let A and B be subsets of Ω. The intersection of A and B can be written as?

Answer 18

ω∈A ∩B ↔(ω∈A ∧ω∈B)

Question 19

Let A and B be subsets of Ω. The set difference can be written as?

Answer 19

ω∈A –B↔(ω∈A ∧ω∈Bc) ↔ω∈A ∩Bc

Question 20

Which English logician were Venn diagrams named after?

Answer 20

James Venn

Question 21

Two sets A and B are disjoint if and only if?

Answer 21

A ∩ B = ∅, that is, they do not have any elements in common.

Question 22

Identify the Property of Set Operation

A ∪ A = A or A ∩ A = A

Answer 22

Idempotent Laws

Question 23

Identify the Property of Set Operation

A ∪ B = B ∪ A or A ∩ B = B ∩ A

Answer 23

Commutative Laws

Question 24

Identify the Property of Set Operation

(A ∪ B) ∪ C = A ∪ (B ∪ C) or
(A ∩ B) ∩ C = A ∩ (B ∩ C)

Answer 24

Associative Laws

Question 25

Identify the Property of Set Operation A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) or A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Answer 25

Distributive Laws

Question 26

Identify the Property of Set Operation A ∪ ∅ = A or A ∩ Ω = A

Answer 26

Identity Laws

Question 27

Identify the Property of Set Operation A ∪ Ω = Ω or A ∩ ∅ = ∅

Answer 27

Bound Laws

Question 28

Identify the Property of Set Operation A ∪ Ac = Ω or A ∩ Ac = ∅

Answer 28

Complement Laws

Question 29

Identify the Property of Set Operation Ωc = ∅ or ∅c = Ω

Answer 29

0/1 Laws

Question 30

Identify the Property of Set Operation (Ac) c = A

Answer 30

Involution Law

Question 31

Identify the Property of Set Operation (A ∪ B)c = Ac ∩ Bc or (A ∩ B)c = Ac ∪ Bc

Answer 31

de Morgan’s Laws

Question 32

When A ⊂ B, we can assume that A ∪ B = ___?

Answer 32

B

Question 33

When A ⊂ B, we can assume that A ∩ B = ____?

Answer 33

A ∩ B = A

Question 34

When A ⊂ B, we can assume that Ac ∪ B = ___?

Answer 34

Ω

Question 35

When A ⊂ B, we can assume that A ∩ Bc =

Answer 35

∅

Question 36

When A ⊂ B, we can assume that Bc ⊂ Ac. True or False?

Answer 36

True

Question 37

This is a collection of subsets of Ω and is denoted by script letters.

Answer 37

Class of sets

Question 38

The elements of a class are ____ and the subset of a particular class must also be a _____.

Answer 38

sets, class

Question 39

This assigns a set Aλ to each λ∈Λ and is denoted by {Aλ: λ∈Λ}.

Answer 39

Indexed class of sets

Question 40

ω∈Uλ∈Λ(Aλ) ↔∃λ∈Λ, ω∈Aλ

Answer 40

Generalized Union

Question 41

ω∈(intersection)λ∈Λ(Aλ) ↔∃λ∈Λ, ω∈Aλ

Answer 41

Generalized Intersection

Question 42

If the index set is a finite collection, then the generalized union and intersection are called?

Answer 42

finite union and finite intersection

Question 43

If the index set is a countably infinite collection, then the generalized union and intersection are called?

Answer 43

countable union and countable intersection

Question 44

The properties of Generalized Operations include:

Answer 44

1. Distributive Laws 2. de Morgan's Laws

Question 45

This is an indexed class of sets where the index set is the set of positive integers. It is also denoted by An

Answer 45

Sequence of sets

Question 46

An is said to be ______ if and only if Ai ⊂ Aj for any i < j.

Answer 46

monotone nondecreasing (expanding)

Question 47

An is said to be _____ if and only if Ai ⊃ Aj for any i < j.

Answer 47

monotone nonincreasing (contracting)

Question 48

Nondecreasing and nonincreasing sequences are together called ____

Answer 48

monotone sequences

Question 49

The class consisting of all subsets of a set S is called the ______ and is denoted by 2S. That is, 2S={A: A ⊂S}.

Answer 49

power set of S

Question 50

If the set S has no elements, then 2S={___} which is not empty.

Answer 50

∅

Question 51

If S is a finite set containing n elements, then 2S will have ___ elements.

Answer 51

2n

Question 52

A class is ____ if and only if Ai∩Aj = ∅ for all i≠j.

Answer 52

pairwise disjoint

Question 53

A class is pairwise disjoint if and only if ___ = ___ for all i≠j.

Answer 53

Ai ∩ Aj = ∅

Question 54

A class is not pairwise disjoint if ______ of the pairwise intersections is not equal to the ____.

Answer 54

at least one, empty set

Question 55

A class is said to be a partition of the set A if and only if:

Answer 55

1. The class is pairwise disjoint; and, 2. The union of all Ai = A, or the set