Cardinality and Operations of Sets

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Questions and Answers

What does the union of two sets A and B represent?

The elements both in A and B

The elements in A or B or both (correct)

The elements not in A or B

The elements only in A

Which of the following is NOT a property of set operations?

Commutative Property

Reflexive Property (correct)

Distributive Property

Associative Property

What is the symmetric difference of sets A and B defined as?

A - B ∩ B - A

A ∩ B

A - B + B - A (correct)

A ∪ B

If A = {1, 2} and B = {2, 3}, what is the intersection A ∩ B?

{2} Signup and view all the answers

Which of the following identifies the complement of set A?

Elements that are not in A Signup and view all the answers

What is the cardinality of the set of all integers Z?

Infinity Signup and view all the answers

Which of the following statements is true regarding subsets?

The empty set is a subset of every set. Signup and view all the answers

How many subsets does the set A = {1, 2, 3} have?

8 Signup and view all the answers

What defines two sets as disjoint?

They share no common elements. Signup and view all the answers

In the equality of sets, which of the following must be proved?

All elements of the first set must be in the second set and vice versa. Signup and view all the answers

Which notation represents that B is a subset of A?

B⊆A Signup and view all the answers

What does the notation A⊆A imply?

Every set is a subset of itself. Signup and view all the answers

If set E contains even integers and set F contains sums of two odd integers, which conclusion can be made about the equality of E and F?

Every element in E is also in F, confirming their equality. Signup and view all the answers

Which region in a Venn diagram represents the intersection of sets A and B?

The overlapping area of A and B Signup and view all the answers

In a Venn diagram with sets A, B, and C, which area represents (A ∪ B) ∩ C?

The overlapping area of A and B, along with the area of C Signup and view all the answers

What does the region outside both sets A and B in a Venn diagram represent?

The complement of the union of A and B Signup and view all the answers

In a Venn diagram, how is the complement of set A represented?

All areas outside set A Signup and view all the answers

If A = {2, 3, 4, 5, 6} and B = {2, 7, 8, 9, 6}, which region in a Venn diagram would represent A - (B ∩ C) if C = {1, 2, 3, 5, 7}?

The section unique to A not shared with B Signup and view all the answers

Study Notes

Cardinality of Sets

The size of a set is called its cardinality.
A set is finite if its cardinality is an integer.
A set is infinite otherwise.

Equality of Sets

Two sets are equal if they have precisely the same elements.
For sets E and F, E=F if every element of E is also in F and every element of F is also in E.

Subsets

Set A is a subset of set B (denoted A⊆B) if every element of A is also an element of B.
Every set is a subset of itself (A⊆A).
The empty set (∅) is a subset of every set (∅⊆A).

Power Sets

The power set of a set S (denoted 2^S) is the set of all possible subsets of S.
The number of subsets of a set with n elements is 2ⁿ.

Disjoint Sets

Two sets are disjoint if they have no elements in common.
If A & B are disjoint, then the intersection of A and B is empty (AnB=∅).

Set Operations: Union and Intersection

The union of sets A and B (A∪B) is the set of all elements that are in A or B or both.
The intersection of sets A and B (A∩B) is the set of all elements that are in both A and B.

Set Operations: Comparability

Two sets, A and B, are comparable if A⊆B or B⊆A.
This means that one set is a subset of the other.

Set Operations: Venn Diagrams

Venn diagrams visually represent relationships between sets.
They show the overlapping regions that represent the various set intersections.

Set Operations: Three Sets

Set operations and relationships can be extended to three or more sets.
Venn diagrams for three sets show more complex overlapping regions
The intersection of all three sets is the region shared by all three sets.
The union of all three sets includes all elements from any of the sets.

Set Operations: Set Differences

A - B represents elements in A but not in B.
The symmetric difference of A and B (AΔB) consists of elements in either A or B, but not both
De Morgan's Laws apply to set differences and complements (A-(B∪C) = (A-B)∩(A-C), A-(B∩C) = (A-B)∪(A-C)).

Set Operations: Complements

The complement of a set A, denoted A’, is the set of all elements in the universal set that are not in A.
A∪A’ = U (the universal set)
A∩A’=∅ (the empty set)

Set Operations: Counting Integers

This discusses finding the number of integers in a specific range divisible by a certain number (or numbers).

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Description

Explore the fundamentals of set theory, including cardinality, equality, subsets, power sets, disjoint sets, and set operations like union and intersection. This quiz will test your understanding of these essential concepts in mathematics and their applications.