Mathematics

Set Theory and Relations

355 Questions

Set theory involves the study of collections of objects and includes operations like union, intersection, and finding complements. Questions cover power sets, Cartesian products, and properties of empty sets. This foundational mathematical topic frequently appears in various competitive exams and university entrance tests.

set operationspower setscartesian productsset complementsproperties of empty setsset partitions

Set Theory and Relations Questions

Multiple choice

Which of the following is a proper subset of the set {1, 2, 3, 4, 5}?

  1. {1, 2, 3, 4, 5}

  2. {1, 2, 3}

  3. {1, 3, 5}

  4. {2, 4, 5}
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A proper subset is a subset that is not equal to the original set. {1, 3, 5} is a proper subset of {1, 2, 3, 4, 5} because it is a subset and it is not equal to the original set.

Multiple choice

If A and B are two sets, which of the following is always true?

  1. P(A ∩ B) = P(A) ∩ P(B)

  2. P(A ∪ B) = P(A) ∪ P(B)

  3. P(A - B) = P(A) - P(B)

  4. P(A × B) = P(A) × P(B)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The power set of the intersection of two sets is equal to the intersection of the power sets of the two sets.

Multiple choice

What is the cardinality of the power set of a set with n elements?

  1. n

  2. 2^n

  3. n!

  4. ∞
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The cardinality of the power set of a set with n elements is 2^n.

Multiple choice

Which of the following is a set?

  1. A collection of distinct objects

  2. A list of numbers

  3. A mathematical expression

  4. A geometric figure
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A set is a well-defined collection of distinct objects, which can be any type of entity, such as numbers, letters, shapes, or even other sets.

Multiple choice

What is the symbol used to represent the union of two sets?

  1. ∩

  2. ∪

  3. −

  4. ×
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The union of two sets, denoted by ∪, is the set of all elements that are in either of the two sets.

Multiple choice

What is the symbol used to represent the intersection of two sets?

  1. ∩

  2. ∪

  3. −

  4. ×
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The intersection of two sets, denoted by ∩, is the set of all elements that are in both of the two sets.

Multiple choice

What is the symbol used to represent the complement of a set?

  1. ∩

  2. ∪

  3. −

  4. ×
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The complement of a set, denoted by −, is the set of all elements that are not in the given set.

Multiple choice

What is the power set of a set?

  1. The set of all subsets of the given set

  2. The set of all elements of the given set

  3. The set of all complements of the given set

  4. The set of all unions of the given set
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The power set of a set is the set of all subsets of the given set, including the empty set and the set itself.

Multiple choice

What is the Cartesian product of two sets?

  1. The set of all ordered pairs of elements from the two sets

  2. The set of all unordered pairs of elements from the two sets

  3. The set of all elements that are in both of the two sets

  4. The set of all elements that are not in either of the two sets
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Cartesian product of two sets is the set of all ordered pairs of elements, where the first element is from the first set and the second element is from the second set.

Multiple choice

Which of the following is a property of the union of two sets?

  1. The union of two sets is always a set

  2. The union of two sets is always larger than either of the two sets

  3. The union of two sets is always smaller than either of the two sets

  4. The union of two sets is always equal to the sum of the two sets
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The union of two sets is always a set, regardless of the nature of the elements in the sets.

Multiple choice

Which of the following is a property of the intersection of two sets?

  1. The intersection of two sets is always a set

  2. The intersection of two sets is always larger than either of the two sets

  3. The intersection of two sets is always smaller than either of the two sets

  4. The intersection of two sets is always equal to the difference of the two sets
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The intersection of two sets is always a set, regardless of the nature of the elements in the sets.

Multiple choice

What is the Axiom of Foundation?

  1. Every non-empty set contains a unique element that is not a member of itself.

  2. There exists a set that contains all sets.

  3. The union of two sets is a set.

  4. The intersection of two sets is a set.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Axiom of Foundation is an axiom in set theory that prevents the existence of infinite descending chains of sets, such as (A \in B \in C \in \cdots).

Multiple choice

What is the Axiom of Choice?

  1. Every set can be well-ordered.

  2. Every non-empty set contains a unique element that is not a member of itself.

  3. The union of two sets is a set.

  4. The intersection of two sets is a set.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Axiom of Choice is an axiom in set theory that states that every set can be well-ordered, meaning that it can be put into a linear order such that every non-empty subset has a least element.

Multiple choice

What is the name of the mathematical theorem that states that for any two sets (A) and (B), the cardinality of their union is at most the sum of their cardinalities?

  1. Cantor-Bernstein-Shroeder Theorem

  2. Schroeder-Bernstein Theorem

  3. Cantor's Theorem

  4. Dedekind-MacNeille Completion Theorem
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Cantor's Theorem establishes that the cardinality of the union of two sets is at most the sum of their cardinalities.

Multiple choice

What is a partition of a set?

  1. A collection of non-empty subsets of a set whose union is the original set.

  2. A division of a set into disjoint subsets.

  3. A collection of subsets of a set whose intersection is empty.

  4. A collection of subsets of a set whose union is the original set and whose intersection is empty.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A partition of a set is a collection of non-empty subsets of the set, such that every element of the set belongs to exactly one of the subsets.