Sets and Cardinal Numbers Quiz

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Questions and Answers

What is the cardinal number of the union of two disjoint sets A and B?

n(A) - n(B)

n(A) + n(B) - n(A ∩ B)

n(A ∩ B)

n(A) + n(B) (correct)

If set A has 4 elements, how many subsets does it have?

16 (correct)

Which of the following statements about equivalent sets is true?

If sets are equivalent, they are also equal.

Equivalent sets must have at least one element in common.

Equal sets are always equivalent. (correct)

Equivalent sets can have different elements but the same cardinality. (correct)

What does the power set P(A) include when A = {1, 2}?

{ {1, 2}, {1}, {2}, {} } Signup and view all the answers

Which of the following correctly represents De Morgan's Law for sets?

(A ∩ B)' = A' ∪ B' Signup and view all the answers

Given three sets P, Q, and R, how is the cardinal number of their union calculated?

n(P) + n(Q) + n(R) - n(P ∩ Q) - n(Q ∩ R) - n(P ∩ R) Signup and view all the answers

What is the complement of set P within the universal set S?

S - P Signup and view all the answers

If set A = {3, 5, 7} and set B = {2, 4, 6}, what can be inferred?

A and B are disjoint. Signup and view all the answers

What is the definition of a singleton set?

A set containing exactly one element. Signup and view all the answers

Which statement correctly describes two sets A and B if A ∩ B = ∅?

A and B are disjoint sets. Signup and view all the answers

What does the complement of a set P denote?

All elements in the universal set that are not in P. Signup and view all the answers

According to De Morgan's laws, which of these equalities holds true?

(P ∩ Q)' = P' ∪ Q' Signup and view all the answers

What is the union of sets A = {2, 3, 6, 10, 15} and B = {3, 6, 15, 18, 21, 24}?

{2, 3, 6, 10, 15, 18, 21, 24} Signup and view all the answers

What do we call a set that contains all elements under consideration in a particular problem?

Universal set Signup and view all the answers

If S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and P = {0, 2, 4, 6, 8}, which of the following is true regarding P'?

{1, 3, 5, 7, 9} Signup and view all the answers

What is a power set?

A set that includes all subsets of a given set. Signup and view all the answers

What is the relationship between a set P and set Q if every element of P is also an element of Q?

P is a subset of Q. Signup and view all the answers

If A = {1, 2, 3} and B = {3, 4, 5}, what is A ∩ B?

{3} Signup and view all the answers

Which of the following sets includes the empty set?

{{} , 1, 2} Signup and view all the answers

Which of the following correctly demonstrates De Morgan's Laws?

(A ∩ B)' = A' ∪ B' Signup and view all the answers

If A and B are disjoint sets, what can be inferred about their intersection?

A ∩ B = ∅ (empty set) Signup and view all the answers

What is the cardinal number of the power set of A if A has 5 elements?

32 Signup and view all the answers

Which of these sets is a proper subset of {2, 4, 6}?

{2, 4} Signup and view all the answers

Study Notes

Sets and Their Properties

A universal set ( S ) is represented by a rectangular area, while a subset ( P ) is depicted as a circular shaded region within it. The complement of ( P ) is the unshaded area in ( S ).
Two sets ( P ) and ( Q ) can be represented with intersecting circles. The union ( P \cup Q ) includes elements from both sets, while the intersection ( P \cap Q ) contains shared elements.

Cardinal Numbers

The cardinal number ( n(A) ) of a finite set ( A ) indicates the count of distinct elements within the set.
For example, given ( R = {2, 3, 5, 7} ), the cardinal number is ( n(R) = 4 ).
The formula for the cardinality of the union of two sets is ( n(A \cup B) = n(A) + n(B) - n(A \cap B) ). If ( A ) and ( B ) are disjoint, ( n(A \cup B) = n(A) + n(B) ).
For three sets ( P, Q, R ), the cardinality formula is ( n(P \cup Q \cup R) = n(P) + n(Q) + n(R) - n(P \cap Q) - n(Q \cap R) - n(P \cap R) + n(P \cap Q \cap R) ).

Equivalent and Equal Sets

Two finite sets ( A ) and ( B ) are equivalent if ( n(A) = n(B) ), although they may not be equal. For instance, ( A = {1, 3, 5} ) and ( B = {2, 4, 6} ) are equivalent but not equal.
Equal sets are when all elements of one set are in another; symbolically ( A = B ) indicates equality.

Power Sets

The power set ( P(A) ) is the collection of all possible subsets of a set ( A ).
For example, if ( A = {1, 2, 3} ), then:
- ( P(A) = { {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, \emptyset } )

Set Operations

A non-empty set contains at least one element. The singleton set contains only one element, e.g., ( {1} ).
The universal set collects all elements relevant to a problem, denoted as ( S ). The complement of a set ( P ), denoted as ( P' ), consists of elements in ( S ) but not in ( P ).
De Morgan’s Laws:
- ( (P \cup Q)' = P' \cap Q' )
- ( (P \cap Q)' = P' \cup Q' )

Set Membership and Subsets

If an element belongs to a set ( A ), it is denoted as ( a \in A ). Conversely, ( 3 \notin B ) means 3 is not an element of set ( B ).
Set ( P ) is a subset of set ( Q ) (denoted ( P \subseteq Q )) if every element in ( P ) is also in ( Q ). A proper subset meets the same condition but is not equal to ( Q ).
The empty set, denoted as ( \emptyset ) or ( {} ), contains no elements. It serves as a subset of any set.

Subsets Enumeration

A set with ( n ) elements has ( 2^n ) total subsets and ( 2^n - 1 ) proper subsets. For example, a set with 3 elements has ( 2^3 = 8 ) subsets and ( 7 ) proper subsets.

Intersection and Union of Sets

The intersection ( A \cap B ) includes elements common to both sets. For example, if ( A = {2, 3, 6, 10, 15} ) and ( B = {3, 6, 15, 18, 21, 24} ), then ( A \cap B = {3, 6, 15} ).
The union ( A \cup B ) combines all elements from both sets without duplication.

Number Representation

Natural numbers ( N = {1, 2, 3, \ldots} ) and whole numbers ( W = {0, 1, 2, 3, \ldots} ) can be defined without listing all members, especially for infinite sets.

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Description

Test your understanding of sets, their properties, and cardinal numbers. This quiz covers concepts such as universal sets, subsets, and formulas for calculating cardinalities. Perfect for students looking to deepen their knowledge of set theory.