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 maths set concepts intervals subsets subsets and supersets

If U = {1, 2, 3, .......}; A = {2, 4, 6, 8, .......}; B = {1, 3, 5, .......}, then find (A $\cup$ B)'.

  1. A'

  2. B

  3. A
  4. $\phi$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Given,
$U=(1, 2, 3,....)$
$A=\{2, 4, 6, 8,...\}$
$B=\{1, 3, 5, 7,....\}$

We know,
$(A\cup B)'=U-(A\cup B)$

here
$A\cup B=\{1, 2, 3,....\}=U$

so $(A\cup B)'=\phi$.
Multiple choice maths set concepts intervals subsets subsets and supersets

The number of subsets with two elements, of the set $S+{1,2,3,4,.....,10}$ such that minimum of the two numbers is less than $6$ is 

  1. $35$
  2. $38$
  3. $30$
  4. $40$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We need subsets {a, b} from {1, ..., 10} where min(a, b) < 6. If min is 1, there are 9 choices for the other number. If min is 2, there are 8 choices. If min is 3, there are 7. If min is 4, there are 6. If min is 5, there are 5. Sum = 9+8+7+6+5 = 35.

Multiple choice maths set concepts intervals subsets subsets and supersets

Which of the following sets is a universal set for the other four sets? 

(a) The set of even natural numbers 

(b) The set of odd natural numbers

(c) The set of natural numbers 

(d) The set of negative numbers 

(e) The set of integers 

  1. $(e)$
  2. $(a)$
  3. $(b)$
  4. $(c)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We know that $\mathbb{N} \subset \mathbb{Z}$ where $\mathbb{N}$ represents set of natural numbers and $\mathbb{Z}$ represents set of all positive and negative integers.

Since $\mathbb{N} \subset \mathbb{Z}$,

                                 $(a),(b),(c)$ are subsets of $(e)$        $...(1)$

Since $\mathbb{Z}$ represents set of all positive and negative integers.

                                      $(d)$ is a subset of $(e)$                   $...(2)$

From $(1)$ and $(2)$ we get

$(a),(b),(c),(d)$ are subsets of $(e)$.

Hence $(e)$ is the universal set for the other four sets.

Multiple choice

What is the range of the relation (R = {(x, y) | x + y = 5)?

  1. {x | x \in \mathbb{R}}

  2. {y | y \in \mathbb{R}}

  3. {(x, y) | x \in \mathbb{R}, y \in \mathbb{R}}

  4. {(x, y) | x + y = 5}
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The range of a relation is the set of all possible values of the dependent variable. In this case, the dependent variable is (y), so the range of the relation (R = {(x, y) | x + y = 5) is the set of all real numbers, (\mathbb{R}).

Multiple choice

In a metric space (X, d), a set E is open if:

  1. For each x in E, there exists an r > 0 such that B_r(x) ⊆ E

  2. For each x in E, there exists an r > 0 such that B_r(x) ∩ E = ∅

  3. For each x in E, there exists an r > 0 such that B_r(x) ⊆ X - E
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A set E in a metric space (X, d) is open if for each x in E, there exists an open ball B_r(x) centered at x with radius r such that B_r(x) is entirely contained within E.

Multiple choice

In a metric space (X, d), a set E is closed if:

  1. For each x in E, there exists an r > 0 such that B_r(x) ⊆ E

  2. For each x in E, there exists an r > 0 such that B_r(x) ∩ E = ∅

  3. For each x in X - E, there exists an r > 0 such that B_r(x) ∩ E = ∅
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A set E in a metric space (X, d) is closed if for each x in the complement of E (X - E), there exists an open ball B_r(x) centered at x with radius r such that B_r(x) does not intersect E.

Multiple choice

Which mathematical concept is used to represent the collection of all subsets of a given set?

  1. Power Set

  2. Union

  3. Intersection

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

The power set of a set S is the set of all subsets of S, denoted as P(S). It is a fundamental concept in set theory and has applications in various areas of mathematics and computer science.

Multiple choice

Which set theory concept is used to represent the collection of all elements that are common to two or more sets?

  1. Union

  2. Intersection

  3. Complement

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

The intersection of two sets A and B, denoted as A ∩ B, is the set of all elements that are common to both A and B. It is a fundamental operation in set theory and is used to find the common elements between two sets.

Multiple choice

Which set theory concept is used to represent the collection of all elements that are in one set but not in another?

  1. Union

  2. Intersection

  3. Complement

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

The complement of a set A with respect to a universal set U, denoted as A', is the set of all elements in U that are not in A. It is a fundamental operation in set theory and is used to find the elements that are missing from a set.

Multiple choice

Which set theory concept is used to represent the collection of all elements that are in either one set or the other, but not in both?

  1. Union

  2. Intersection

  3. Complement

  4. Symmetric Difference
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The symmetric difference of two sets A and B, denoted as A Δ B, is the set of all elements that are in either A or B, but not in both. It is a fundamental operation in set theory and is used to find the elements that are unique to each set.

Multiple choice

Which set theory concept is used to represent the collection of all subsets of a set that have a certain cardinality?

  1. Power Set

  2. Cardinality

  3. Ordered Set

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

The cardinality of a set is the number of elements in the set. It is a fundamental property of a set and is used to compare the sizes of different sets. The cardinality of a set can be finite or infinite.

Multiple choice

Which set theory concept is used to represent the collection of all subsets of a set that have a certain property?

  1. Power Set

  2. Cardinality

  3. Ordered Set

  4. Partition
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A partition of a set S is a collection of non-empty subsets of S such that every element of S is in exactly one of these subsets. Partitions are used to divide a set into disjoint subsets and are useful for studying the structure of sets.

Multiple choice

What is the power set of the set {1, 2, 3}?

  1. {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}, ∅}

  2. {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

  3. {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}}

  4. {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}, ∅, {∅}}
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 that set.

Multiple choice

What is the cardinality of the set of all subsets of the set {1, 2, 3}?

  1. 2^3

  2. 3^2

  3. 2^4

  4. 3^3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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

Multiple choice

What is the complement of the set {x ∈ R | x^2 < 4}?

  1. {x ∈ R | x^2 ≥ 4}

  2. {x ∈ R | x^2 > 4}

  3. {x ∈ R | x^2 ≤ 4}

  4. {x ∈ R | x^2 ≠ 4}
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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