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 cardinality of the set of all functions from the set {1, 2, 3} to the set {a, b, c}?

  1. 3^3

  2. 3^2

  3. 2^3

  4. 2^2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The cardinality of the set of all functions from a set with n elements to a set with m elements is m^n.

Multiple choice

What is the intersection of the sets {1, 2, 3} and {2, 3, 4}?

  1. {1, 2, 3, 4}

  2. {1, 2, 3}

  3. {2, 3}

  4. {1, 4}
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

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

Multiple choice

Which of the following sets has a cardinality of (\aleph_0)?

  1. The set of all natural numbers \(\mathbb{N}\)
  2. The set of all real numbers \(\mathbb{R}\)
  3. The set of all even integers \(2\mathbb{Z}\)
  4. The set of all prime numbers \(\mathbb{P}\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The set of all natural numbers (\mathbb{N}) has a cardinality of (\aleph_0) because it can be put into one-to-one correspondence with the set of all integers (\mathbb{Z}).

Multiple choice

Which of the following sets is uncountable?

  1. The set of all rational numbers \(\mathbb{Q}\)
  2. The set of all algebraic numbers \(\mathbb{A}\)
  3. The set of all transcendental numbers \(\mathbb{T}\)
  4. The set of all constructible numbers \(\mathbb{C}\)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The set of all transcendental numbers (\mathbb{T}) is uncountable, meaning it cannot be put into one-to-one correspondence with the set of all natural numbers (\mathbb{N}).

Multiple choice

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

  1. \(n\)
  2. \(2^n\)
  3. \(n^2\)
  4. \(n!\)
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) because each element of the power set is a subset of the original set, and there are (2^n) possible subsets.

Multiple choice

What is the distributive law of set operations?

  1. \(A\cap(B\cup C) = (A\cap B)\cup(A\cap C)\)
  2. \(A\cup(B\cap C) = (A\cup B)\cap(A\cup C)\)
  3. \(A\Delta(B\cup C) = (A\Delta B)\cup(A\Delta C)\)
  4. \(A\Delta(B\cap C) = (A\Delta B)\cap(A\Delta C)\)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The distributive law of set operations states that (A\cup(B\cap C) = (A\cup B)\cap(A\cup C)).

Multiple choice

What is the cardinality of the set of all subsets of a set with (n) elements?

  1. \(n\)
  2. \(2^n\)
  3. \(n^2\)
  4. \(n!\)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The cardinality of the set of all subsets of a set with (n) elements is (2^n) because each element of the power set is a subset of the original set, and there are (2^n) possible subsets.

Multiple choice

What is the cardinality of the set of all functions from a set with (m) elements to a set with (n) elements?

  1. \(m\)
  2. \(n\)
  3. \(m^n\)
  4. \(n^m\)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The cardinality of the set of all functions from a set with (m) elements to a set with (n) elements is (n^m) because there are (n) choices for the image of each element in the domain.

Multiple choice

What is the cardinality of the set of all real numbers between 0 and 1?

  1. \(\aleph_0\)
  2. \(\aleph_1\)
  3. \(\continuum\)
  4. \(\infty\)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The cardinality of the set of all real numbers between 0 and 1 is (\continuum), which is equal to the cardinality of the set of all real numbers.

Multiple choice

What is the cardinality of the set of all subsets of a set with (\aleph_0) elements?

  1. \(\aleph_0\)
  2. \(\aleph_1\)
  3. \(\continuum\)
  4. \(\infty\)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The cardinality of the set of all subsets of a set with (\aleph_0) elements is (\continuum), which is equal to the cardinality of the set of all real numbers.

Multiple choice

What is the cardinality of the set of all functions from a set with (\aleph_0) elements to a set with (\aleph_1) elements?

  1. \(\aleph_0\)
  2. \(\aleph_1\)
  3. \(\continuum\)
  4. \(\infty\)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The cardinality of the set of all functions from a set with (\aleph_0) elements to a set with (\aleph_1) elements is (\continuum), which is equal to the cardinality of the set of all real numbers.

Multiple choice

Let (\mu) be a measure on a set (X). If (A_1, A_2, \ldots) is a sequence of sets in (X) such that (A_n \subseteq A_{n+1}) for all (n), then what is the limit of (\mu(A_n)) as (n) approaches infinity?

  1. \(\mu(\cup_{n=1}^\infty A_n)\)
  2. \(\mu(\cap_{n=1}^\infty A_n)\)
  3. \(\lim_{n\to\infty} \mu(A_n)\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The correct answer is (\mu(\cup_{n=1}^\infty A_n)). This is known as the monotone convergence theorem.

Multiple choice

Let (\mu) be a measure on a set (X). If (A) and (B) are sets in (X) such that (A \cap B = \emptyset), then what is the relationship between (\mu(A \cup B)) and (\mu(A) + \mu(B))?

  1. \(\mu(A \cup B) = \mu(A) + \mu(B)\)
  2. \(\mu(A \cup B) < \mu(A) + \mu(B)\)
  3. \(\mu(A \cup B) > \mu(A) + \mu(B)\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The correct answer is (\mu(A \cup B) = \mu(A) + \mu(B)). This is known as the additivity property of a measure.

Multiple choice

Let (\mu) be a measure on a set (X). If (A) and (B) are sets in (X) such that (\mu(A) = \mu(B)), then what is the relationship between (\mu(A \cup B)) and (\mu(A \cap B))?

  1. \(\mu(A \cup B) = \mu(A \cap B)\)
  2. \(\mu(A \cup B) < \mu(A \cap B)\)
  3. \(\mu(A \cup B) > \mu(A \cap B)\)
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

The correct answer is (\mu(A \cup B) = \mu(A) + \mu(B) - \mu(A \cap B)). This is known as the inclusion-exclusion principle.

Multiple choice

What is the partition of the set {a, b, c, d, e, f} induced by the equivalence relation of having the same number of sides?

  1. {{a, b, c}, {d, e, f}}

  2. {{a, c, e}, {b, d, f}}

  3. {{a, b, d, e}, {c, f}}

  4. {{a, c, f}, {b, d, e}}
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The partition of the set {a, b, c, d, e, f} induced by the equivalence relation of having the same number of sides is {{a, b, c}, {d, e, f}}.