Mathematics

Set Theory and Relations

368 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

What is the power set of a set?

  1. The set of all subsets of the set

  2. The set of all elements of the set

  3. The set of all complements of the set

  4. The set of all unions of the 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 set. For example, the power set of the set {1, 2, 3} is {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}.

Multiple choice

Which of the following is an example of a countably infinite set?

  1. The set of all natural numbers

  2. The set of all real numbers

  3. The set of all prime numbers

  4. The set of all even numbers
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A countably infinite set is a set that can be put into one-to-one correspondence with the set of natural numbers. The set of all natural numbers is a countably infinite set because it can be put into one-to-one correspondence with the set of natural numbers.

Multiple choice

What is the union of two sets?

  1. The set of all elements that are in both sets

  2. The set of all elements that are in either set

  3. The set of all elements that are in one set but not the other

  4. The set of all elements that are in neither set
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The union of two sets is the set of all elements that are in either set. For example, the union of the sets {1, 2, 3} and {4, 5, 6} is {1, 2, 3, 4, 5, 6}.

Multiple choice

What is the intersection of two sets?

  1. The set of all elements that are in both sets

  2. The set of all elements that are in either set

  3. The set of all elements that are in one set but not the other

  4. The set of all elements that are in neither set
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The intersection of two sets is the set of all elements that are in both sets. For example, the intersection of the sets {1, 2, 3} and {4, 5, 6} is {}.

Multiple choice

What is the complement of a set?

  1. The set of all elements that are in the set

  2. The set of all elements that are not in the set

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

  4. The set of all elements that are in neither set
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The complement of a set is the set of all elements that are not in the set. For example, the complement of the set {1, 2, 3} is {4, 5, 6, ...}.

Multiple choice

What is the cardinality of a set?

  1. The number of elements in the set

  2. The size of the set

  3. The measure of the set

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

The cardinality of a set is the number of elements in the set. It is denoted by the symbol (|A|).

Multiple choice

What is the axiom of choice?

  1. An axiom that states that for any set of non-empty sets, there exists a function that chooses exactly one element from each set

  2. An axiom that states that for any set of non-empty sets, there exists a function that chooses at least one element from each set

  3. An axiom that states that for any set of non-empty sets, there exists a function that chooses at most one element from each set

  4. An axiom that states that for any set of non-empty sets, there exists a function that chooses no elements from each set
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The axiom of choice is an axiom that states that for any set of non-empty sets, there exists a function that chooses exactly one element from each set.

Multiple choice

What is the number of ways to partition a set of (n) elements into (k) non-empty subsets?

  1. \(S_k\)
  2. \(C_k\)
  3. \(S_k + C_k\)
  4. \(S_k - C_k\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The number of ways to partition a set of (n) elements into (k) non-empty subsets is given by the Schröder number (S_k).

Multiple choice

What is the number of ways to partition a set of (n) elements into (k) non-empty subsets, such that each subset contains at least (2) elements?

  1. \(S_k\)
  2. \(C_k\)
  3. \(S_k + C_k\)
  4. \(S_k - C_k\)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The number of ways to partition a set of (n) elements into (k) non-empty subsets, such that each subset contains at least (2) elements, is given by (S_k - C_k).

Multiple choice

A survey is conducted among 100 people to determine their favorite sport. The results show that 60 people like cricket, 40 people like football, and 20 people like both cricket and football. How many people like only cricket or only football?

  1. 80

  2. 60

  3. 40

  4. 20
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let C be the set of people who like cricket and F be the set of people who like football. Then, n(C) = 60, n(F) = 40, and n(C ∩ F) = 20. We want to find n(C ∪ F), which is the number of people who like either cricket or football. Using the inclusion-exclusion principle, we have: n(C ∪ F) = n(C) + n(F) - n(C ∩ F) = 60 + 40 - 20 = 80.

Multiple choice

In set theory, the empty set is denoted by:

  1. ∅

  2. {}

  3. Ø

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The empty set, also known as the null set, is a set with no elements. It is often denoted by the symbol ∅ or {}.

Multiple choice

Which of the following is an example of a set?

  1. {1, 2, 3}

  2. The set of all even numbers

  3. The set of all prime numbers

  4. All of the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A set is a well-defined collection of distinct objects. The given examples are all sets, as they are collections of distinct objects.

Multiple choice

The union of two sets A and B, denoted as A ∪ B, is:

  1. The set of all elements that are in either A or B

  2. The set of all elements that are in both A and B

  3. The set of all elements that are not in either A or B

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The union of two sets is the set of all elements that are in either of the two sets.

Multiple choice

The intersection of two sets A and B, denoted as A ∩ B, is:

  1. The set of all elements that are in either A or B

  2. The set of all elements that are in both A and B

  3. The set of all elements that are not in either A or B

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

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

Multiple choice

The complement of a set A, denoted as A', is:

  1. The set of all elements that are in A

  2. The set of all elements that are not in A

  3. The set of all elements that are in both A and A'

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The complement of a set is the set of all elements that are not in the given set.