Assertion : If $A=\{x, y, z\}$ and $B=\{3,4\}$, then number of relations from $A$ to $B$ is $2^5$. Reason : Number of relations from $A$ to $B$ is $2^{n(A) \times n(B)}$.

Assertion is correct, Reason is correct; Reason is a correct explanation for assertion

Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion

On the set $N$ of natural numbers, define the relation $R$ by $a$ $R$ $b$ iff the GCD of $a$ and $b$ is 2, then $R$ is

Neither reflexive, nor symmetric and transitive.

$n / m$ means that $n$ is factor of $m$, then the relation $f$ is

reflexive, transitive and symmetric

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JEE • MATHEMATICS • RELATIONS

Relations practice · Easy

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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61 Easy

61 of 180

Analyzing +4 | -1

Which of the following relations is not symmetric?

View answer and explanation

Correct option: D

R_4

on power set of a set

X

defined by

A\, R_4 \,B

iff

A \subseteq B.

The correct answer is D.

R_4

is not symmetric because

A \subseteq B

does not necessarily imply

B \subseteq A

62 Easy

62 of 180

Analyzing +4 | -1

If R be a relation defined as

aRb

iff

\left| {a - b} \right| > 0

, then the relation is

View answer and explanation

Correct option: D

Symmetric and transitive

Option D (Symmetric and transitive) is correct because the relation satisfies the conditions for symmetry and transitivity, but not reflexivity.

63 Easy

63 of 180

Application +4 | -1

Suppose that

A_1, A_2, A_3 \ldots A_{30}

are thirty sets each containing 6 elements and

B_1, B_2, B_3 \ldots B_n

are

n

sets each containing 3 elements. If

\bigcup\limits_{{i=1}}^{30} A_i=S=\bigcup\limits_{{i=1}}^n B_i

and each element of

{S}

belongs to exactly

10 A_i S

and to exactly

9 B_i S

then

{n}=

View answer and explanation

Correct option: A

Option A (54) is correct because it is the number of sets

B_i

calculated based on the given conditions of union and element frequency.

64 Easy

64 of 180

Understanding +4 | -1

A=\{a, b, c\},

then the relation

R=\{(b, c)\}

A

View answer and explanation

Correct option: C

Transitive only

Option C (Transitive only) is correct because the relation R = {(b, c)} satisfies the transitive property, but not reflexive or symmetric.

65 Easy

65 of 180

Understanding +4 | -1

Let

{A}=\{{a}, {b},{c}, {d}, {e}, {f}, {g}, {h}\}

and

{R}=\{({a}, {a}),({b}, {b}),({a}, {g}), (b, a), (b, g), (g, a), (g, b), (g, g), (b, b)\}

Consider the following statements:

Assertion :

{R} \subset {A} \times {A}

.

Reason :

{R}

is not a relation on

A.

View answer and explanation

Correct option: C

Assertion is correct, Reason is incorrect

Assertion is correct, Reason is incorrect because R is a subset of A x A, meaning it *is* a relation on A.

66 Easy

66 of 180

Application +4 | -1

If set

A=\{a, b, c\}, R: A \rightarrow A, R=\{(a, b),(b, c)\}

. How many elements should be added for making it symmetric and transitive?

View answer and explanation

Correct option: D

Option D (7) is correct because 7 elements need to be added to the relation R to make it symmetric and transitive.

67 Easy

67 of 180

Understanding +4 | -1

The void relation on a set

A

View answer and explanation

Correct option: B

Symmetric and transitive

Option B (Symmetric and transitive) is correct because the void relation is symmetric and transitive by definition, as the conditions for symmetry and transitivity are vacuously true.

68 Easy

68 of 180

Understanding +4 | -1

Assertion : If

A=\{x, y, z\}

and

B=\{3,4\}

, then number of relations from

A

B

2^5

.

Reason : Number of relations from

A

B

2^{n(A) \times n(B)}

View answer and explanation

Correct option: D

Assertion is incorrect, Reason is correct

Assertion is incorrect, Reason is correct because the number of relations from A to B is 2^(n(A) * n(B)), and in this case, it should be 2^(3*2) = 2⁶, not 2⁵.

69 Easy

69 of 180

Analyzing +4 | -1

On the set

N

of natural numbers, define the relation

R

a

R

b

iff the GCD of

a

and

b

is 2, then

R

View answer and explanation

Correct option: B

Symmetric only

The correct answer is B. The relation

R

is symmetric only because if the GCD of

a

and

b

is 2, then the GCD of

b

and

a

is also 2, but it's not reflexive or transitive.

70 Easy

70 of 180

Understanding +4 | -1

Consider the following statements.

Let

{A}=\{1,2,3,4\}

and

{B}=\{5,7,9\}

{A} \times {B}={B} \times {A}

II.

{n}({A} \times {B})={n}({B} \times {A})

Choose the correct option.

View answer and explanation

Correct option: B

Statement-II is true

Statement-II is true because the number of elements in the Cartesian product of two sets is the product of the number of elements in each set, regardless of the order of the sets.

71 Easy

71 of 180

Analyzing +4 | -1

A = \left\{ {1,\;2,\;3} \right\},

then the relation

R = \left\{ {\left( {1,1} \right),\left( {2,2} \right),\left( {3,1} \right),\left( {1,3} \right)} \right\}

View answer and explanation

Correct option: B

Symmetric

Option B (Symmetric) is correct because for every (a, b) in R, (b, a) is also in R. In this case, (1,3) and (3,1) are both in R.

72 Easy

72 of 180

Understanding +4 | -1

A relation between two persons is defined as follows:

aRb \Leftrightarrow a

and

b

born in different months. Then,

R

View answer and explanation

Correct option: B

Symmetric

Option B (Symmetric) is correct because if a is born in a different month than b, then b is also born in a different month than a.

73 Easy

73 of 180

Analyzing +4 | -1

n / m

means that

n

is factor of

m

, then the relation

f

View answer and explanation

Correct option: D

reflexive, transitive and not symmetric

Option D (reflexive, transitive and not symmetric) is correct because if n is a factor of m, then the relation is reflexive and transitive, but not symmetric.

74 Easy

74 of 180

Analyzing +4 | -1

The relation

R = \left\{ {\left( {a,b} \right):\gcd \left( {a,b} \right) = 1,\,2a \ne b,\,\,a,b \in \mathbb{Z}} \right\}

View answer and explanation

Correct option: C

Symmetric but not transitive

Option C (Symmetric but not transitive) is correct because the relation satisfies the symmetric property but fails the transitive property.

75 Easy

75 of 180

Understanding +4 | -1

Let

X

be the set of all engineering colleges in a state of Indian Republic and R be a relation on

X

defined as two colleges are related iff they are affiliated to the same university, then

R

View answer and explanation

Correct option: D

Equivalence

Option D (Equivalence) is correct because the relation R, where two colleges are related if they are affiliated to the same university, is reflexive, symmetric, and transitive, thus it is an equivalence relation.

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76 Easy

76 of 180

Remembering +4 | -1

R

is a relation in a set

A

such that

(a, a) \in R

for every

a \in A

, then the relation

R

is called

View answer and explanation

Correct option: B

reflexive

Option B (reflexive) is correct because a relation R is reflexive if (a, a) belongs to R for every a in A.

77 Easy

77 of 180

Analyzing +4 | -1

Let

R

be a relation defined on

S

, the set of squares on a chess board such that

xRy

iff

x

and

y

share a common side. Then, which of the following is false for

R

View answer and explanation

Correct option: C

Transitive

Option A is correct because the relation R is not reflexive, as a square cannot share a common side with itself.

78 Easy

78 of 180

Understanding +4 | -1

Let

R

be a relation on the set

N

of natural numbers defined by

nRm \Leftrightarrow n

is a factor of

m

m{\text{(i}}.{\text{e}}.n\,{\text{|}}\,m).

Then,

R

View answer and explanation

Correct option: D

Reflexive, transitive but not symmetric

Option D (Reflexive, transitive but not symmetric) is correct because n is always a factor of itself, if n is a factor of m and m is a factor of p, then n is a factor of p, but if n is a factor of m, m is not necessarily a factor of n.

79 Easy

79 of 180

Analyzing +4 | -1

Given the relation

R = \left\{ {\left( {1,2} \right),\left( {2,3} \right)} \right\}

on the set

A = \left\{ {1,2,3} \right\},

the minimum number of ordered pairs which when added to

R

make it an equivalence relation is

View answer and explanation

Correct option: C

Option C (7) is correct because 7 ordered pairs must be added to R to make it reflexive, symmetric, and transitive, thus forming an equivalence relation.

80 Easy

80 of 180

Analyzing +4 | -1

Consider the following relations:

R=\{(x, y): x, y

are real numbers and

x=w y

for some rational number

w\}

S=\left\{\frac{m}{n}, \frac{p}{q}: m, n, p, q \in Z \text { such that } n, q \neq 0 \text { and } q m=p n\right\}

Then,

View answer and explanation

Correct option: A

S

is an equivalence relation but

R

is not an equivalence relation

Option A is correct. Relation S is an equivalence relation because it satisfies reflexivity, symmetry, and transitivity, while R fails to be symmetric.

81 Easy

81 of 180

Analyzing +4 | -1

For real numbers

x

and

y

, we write

x\;Ry \Leftrightarrow x - y + \sqrt 2

is an irrational number. Then, the relation

R

View answer and explanation

Correct option: A

Reflexive

Option A (Reflexive) is correct. For any real number x, x - x +

\sqrt{2}

\sqrt{2}

, which is irrational, thus xRx.

82 Easy

82 of 180

Understanding +4 | -1

Let

A = \left\{ {1,2,3,4} \right\},

and let

R = \left\{ {\left( {2,2} \right),\left( {3,3} \right),\left( {4,4} \right),\left( {1,2} \right)} \right\}

be a relation on

A

. Then,

R

View answer and explanation

Correct option: C

Transitive

Option C (Transitive) is correct because for any (a, b) and (b, c) in R, (a, c) is also in R.

83 Easy

83 of 180

Analyzing +4 | -1

For the set

A=\{1,2,3\}

, define a relation

R

in the set

A

as follows

R=\{(1,1),(2,2),(3,3),(1,3)\}

. Then, the ordered pair to be added to

\mathrm{R}

to make it the smallest equivalence relation is

View answer and explanation

Correct option: B

(3,1)

Option B ((3,1)) is correct because adding (3,1) makes the relation symmetric, and since it's already reflexive and transitive, it becomes an equivalence relation.

84 Easy

84 of 180

Understanding +4 | -1

Let

L

be the set of all lines in a plane and

R

be a relation on

L

defined by

l_1 R l_2

if and only if

l_1 \perp l_2

then

R

View answer and explanation

Correct option: B

symmetric

Option B (symmetric) is correct because if line

l_1

is perpendicular to

l_2

, then

l_2

is perpendicular to

l_1

85 Easy

85 of 180

Remembering +4 | -1

If the set

A

has

p

elements,

B

has

q

elements, then the number of elements in

A \times B

is:

View answer and explanation

Correct option: C

p q

Option C (

p q

) is correct because the number of elements in the Cartesian product of two sets is the product of the number of elements in each set.

86 Easy

86 of 180

Application +4 | -1

If two sets

A

and

B

are having 99 elements in common, then the number of elements common to each of the sets

A \times B

and

B \times A

are

View answer and explanation

Correct option: B

{99^2}

Option B (

{99^2}

) is correct because the number of common elements in

A \times B

and

B \times A

is the square of the number of common elements in A and B.

87 Easy

87 of 180

Analyzing +4 | -1

\mathrm{A}=[-2,2]

, then the relation

R=\left\{(x, y) \in A \times A / \frac{x^2}{9}+\frac{y^2}{4} \leq 1, \frac{x^2}{4}+\frac{y^2}{9} \leq 1\right\}

View answer and explanation

Correct option: B

symmetric only

Option B (symmetric only) is correct because if (x, y) is in R, then (y, x) is also in R, but the relation is not reflexive or transitive.

88 Easy

88 of 180

Understanding +4 | -1

A=\{1,2\}

and

B=\{0,1\},

then

A \times B=

View answer and explanation

Correct option: A

\{(1,0),(1,1),(2,0),(2,1)\}

Option A (

\{(1,0),(1,1),(2,0),(2,1)\}

) is correct because it represents the Cartesian product of A and B, where each element of A is paired with each element of B.

89 Easy

89 of 180

Analyzing +4 | -1

Let a relation

R

in the set

N

of natural numbers be defined as

(x, y) \in R

iff

x^2-4 x y+3 y^2=0

for all

x, y \in N.

The relation

R

View answer and explanation

Correct option: A

Reflexive and transitive

Option A is correct. The relation R is reflexive because x=y is a solution, and it's transitive because if x=3y and y=3z then x=9z, but x=y or x=3y, therefore transitive.

90 Easy

90 of 180

Analyzing +4 | -1

Let

{R_1} = \left\{ {\left( {a,b} \right) \in N \times N:\left| {a - b} \right| \leqslant 13} \right\}

and

{R_2} = \left\{ {\left( {a,b} \right) \in N \times N:\left| {a - b} \right| \ne 13} \right\}.

Then on

N:

View answer and explanation

Correct option: B

Neither

{R_1}

nor

{R_2}

is an equivalence relation

Option B (Neither

{R_1}

nor

{R_2}

is an equivalence relation) is correct because neither relation is transitive, and therefore neither can be an equivalence relation.

Revision notes and key takeaways

Review RELATIONS in your textbook or class notes, then use this set to check your understanding.

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