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JEE • MATHEMATICS • RELATIONS

Relations practice · Mixed difficulty

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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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Showing 1-30 of 238 questions

1 Easy

1 of 238

Analyzing +4 | -1

In the set

\mathrm{Z}

of all integers, which of the following relation

R

is not an equivalence relation?

View answer and explanation

Correct option: A

x \,R \,y

: if

x \leq y

Option A is correct because the relation

xRy

x \leq y

is not symmetric. If

x \leq y

, it is not necessarily true that

y \leq x

2 Medium

2 of 238

Analyzing +4 | -1

Let

\mathrm{N}

denote the set of all natural numbers and

\mathrm{R}

be the relation on

\mathrm{N} \times \mathrm{N}

defined by

(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{c}, \mathrm{d})

a d(b+c)=b c(a+d)

, then

\mathrm{R}

View answer and explanation

Correct option: D

An equivalence relation

Option D (An equivalence relation) is correct because the relation R is reflexive, symmetric, and transitive, thus satisfying the conditions for an equivalence relation.

3 Easy

3 of 238

Analyzing +4 | -1

Let

W

denote the words in the English dictionary. Define the relation

R

by:

R=\{(x, y) \in W \times W \mid

the words

x

and

y

have at least one letter in common

\}

View answer and explanation

Correct option: B

Reflexive, symmetric and not transitive

Option B (Reflexive, symmetric and not transitive) is correct because a word always shares a letter with itself (reflexive), if word x shares a letter with word y, then word y shares a letter with word x (symmetric), but it's possible for x to share a letter with y, and y to share a letter with z, without x and z sharing any letters (not transitive).

4 Easy

4 of 238

Analyzing +4 | -1

Let

S

be the set of all real numbers. Then, the relation

R = \{ \left( {a,\;b} \right):1 + ab > 0\}

S

View answer and explanation

Correct option: A

Reflexive and symmetric but not transitive

The correct answer is A (Reflexive and symmetric but not transitive) because the relation satisfies the reflexive and symmetric properties but fails the transitive property.

5 Easy

5 of 238

Understanding +4 | -1

Let

X

be a family of sets and

R

be a relation on

X

defined by '

A

is disjoint from

B

'. Then

R

is:

View answer and explanation

Correct option: B

Symmetric

Option B (Symmetric) is correct because if set A is disjoint from set B, then set B is also disjoint from set A.

6 Medium

6 of 238

Application +4 | -1

R

is relation on

N

given by

R = \left\{ {\left( {x,\;y} \right):4x + 3y = 20} \right\}

. Which of the following belongs to

R

View answer and explanation

Correct option: D

\left( {2,\mathrm{ }4} \right)

The correct answer is D ((2, 4)) because it's the only pair of natural numbers that satisfies the equation 4x + 3y = 20.

7 Easy

7 of 238

Understanding +4 | -1

A \Delta B=A \cup B

then

View answer and explanation

Correct option: B

A \cap B=\phi

Option B (

A \cap B=\phi

) is correct because if

A \Delta B = A \cup B

, it means that the intersection of A and B is empty.

8 Easy

8 of 238

Application +4 | -1

A=\left\{x: x^2-5 x+6=0\right\}, B=\{2,4\}

and

C=\{4,5\}

, then

A \times(B \cap C)

View answer and explanation

Correct option: A

\{(2,4),(3,4)\}

Option A ({ (2,4), (3,4) }) is correct because it represents the Cartesian product of set A and the intersection of sets B and C.

9 Easy

9 of 238

Understanding +4 | -1

For

{n}, {m} \in {N}, {n} / {m}

means that

{n}

is a factor of

{m}

, the relation

/

View answer and explanation

Correct option: D

Reflexive transitive and not symmetric

Option D is correct because the divisibility relation is reflexive (n/n), transitive (if n/m and m/k, then n/k), but not symmetric (n/m does not imply m/n).

10 Easy

10 of 238

Understanding +4 | -1

Let

A = \left\{ {x,\;y,\;z} \right\}\;and\;B = \left\{ {a,\;b,\;c,\;d} \right\}

. Which one of the following is not a relation from

A\;to\;B

View answer and explanation

Correct option: D

\left\{ {\left( {z,\;b} \right),\;\left( {y,\;b} \right),\;\left( {a,\;d} \right)} \right\}

Option D is correct because it contains an element (a, d) where 'a' is not in set A, thus violating the definition of a relation from A to B.

11 Medium

11 of 238

Analyzing +4 | -1

Let

R=\{(3,3)(5,5),(9,9),(12,12),(5,12),(3,9),(3,12),(3,5)\}

be a relation on the

\operatorname{set} A=\{3,5,9,12\}

. Then,

R

is :

View answer and explanation

Correct option: D

reflexive, transitive but not symmetric.

Option D (reflexive, transitive but not symmetric) is correct because the given relation satisfies the reflexive and transitive properties but fails to satisfy the symmetric property.

12 Easy

12 of 238

Analyzing +4 | -1

Consider the following two binary relations on the set

A=\{a, b, c\}: R_{1}=\{(\mathrm{c}, a)(b, b),(\mathrm{a}, c),(c, c),(b, c),(a, a)\}

and

R_{2}=\{(a, b),(b, a),(c, c),(c, a),(a, a),(b, b),(a, c)

. Then

View answer and explanation

Correct option: A

R_{2}

is symmetric but it is not transitive

Option A (

R_2

is symmetric but it is not transitive) is correct because

R_2

satisfies the symmetric property but not the transitive property.

13 Easy

13 of 238

Understanding +4 | -1

In order that a relation

R

defined on a non-empty set

A

is an equivalence relation. It is sufficient, if

R

View answer and explanation

Correct option: D

possesses all the above three properties

Option D (possesses all the above three properties) is correct because an equivalence relation must be reflexive, symmetric, and transitive.

14 Easy

14 of 238

Analyzing +4 | -1

Let

R_1

and

R_2

be two relations defined as follows:

R_1=\left\{(a, b) \in R^2: a^2+b^2 \in Q\right\}

and

R_2=\left\{(a, b) \in R^2: a^2+b^2 \notin Q\right\}.

Then,

View answer and explanation

Correct option: C

neither

R_1

nor

R_2

is transitive

The correct answer is C. Neither

R_1

nor

R_2

is transitive because counterexamples can be found for both relations.

15 Medium

15 of 238

Analyzing +4 | -1

The minimum number of elements that must be added to the relation

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

on the set

\{a, b, c\}

so that it becomes symmetric and transitive is

View answer and explanation

Correct option: A

The minimum number of elements that must be added is 7. To make the relation symmetric and transitive, we need to add (b, a), (c, b), (a, c), (c, a), (a, a), (b, b), and (c, c).

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16 Easy

16 of 238

Application +4 | -1

f(x)=4 x-x^2, x \in R

, then

f(b+1)-f(b-1)

is equal to

{m}(2-{b})

. The value of '

{m}

' is

View answer and explanation

Correct option: A

Option A (4) is correct because expanding f(b+1) - f(b-1) and comparing it to m(2-b) gives m = 4.

17 Medium

17 of 238

Understanding +4 | -1

Let

X

be a set containing

n

elements. The number of reflexive relations that can be defined on

X

View answer and explanation

Correct option: B

2^{n^2-n}

Option B (

2^{n^2-n}

) is correct because a reflexive relation must contain n elements, leaving

n^2 - n

elements that can either be in the relation or not, resulting in

2^{n^2-n}

possible relations.

18 Easy

18 of 238

Understanding +4 | -1

Let

\mathrm{L}

denote the set of all straight lines in a plane. Let a relation

R

be defined by

\alpha R \beta \Leftrightarrow \alpha \perp \beta, \alpha, \beta \in L

. Then,

\mathrm{R}

View answer and explanation

Correct option: B

Symmetric

Option B (Symmetric) is correct because if line

\alpha

is perpendicular to line

\beta

, then line

\beta

is also perpendicular to line

\alpha

19 Easy

19 of 238

Analyzing +4 | -1

a R b \Leftrightarrow|a| \leq b

. Then

R

View answer and explanation

Correct option: C

Transitive

Option C (Transitive) is correct because if |a| ≤ b and |b| ≤ c, then |a| ≤ c, satisfying the transitive property.

20 Easy

20 of 238

Understanding +4 | -1

Let

R

be a relation defined by

R=\{(a, b): a \geq b\}

, where

a

and

b

are real numbers, then

R

View answer and explanation

Correct option: B

reflexive, transitive but not symmetric

Option B (reflexive, transitive but not symmetric) is correct because the relation

a \geq b

holds for

a=b

(reflexive), if

a \geq b

and

b \geq c

then

a \geq c

(transitive), but

a \geq b

does not imply

b \geq a

(not symmetric).

21 Easy

21 of 238

Analyzing +4 | -1

Let

A=\{2,3,4\}

and

B=\{8,9,12\}

. Then the number of elements in the relation

R=\left\{\left(\left(a_1, b_1\right),\left(a_2, b_2\right)\right) \in(A \times B, A \times B): a_1\right.

divides

b_2

and

a_2

divides

\left.b_1\right\}

View answer and explanation

Correct option: A

Option A (36) is correct because the number of elements in the relation R is calculated by considering all possible pairs (a1, b2) and (a2, b1) where a1 divides b2 and a2 divides b1.

22 Easy

22 of 238

Application +4 | -1

Let

R

be the relation in the set

\mathrm{N}

given by

R=\{(a, b): a=b-2, b>6\}

. Choose the correct answer.

View answer and explanation

Correct option: C

(6,8) \in R

Option C ((6,8) ∈ R) is correct because it satisfies the condition a = b - 2 and b > 6.

23 Easy

23 of 238

Analyzing +4 | -1

On the set

N

of all natural numbers define the relation

R

a R b

if and only if the G.C.D. of

a

and

b

is 2 . then

R

View answer and explanation

Correct option: B

Symmetric only

Option B (Symmetric only) is correct because if the GCD of a and b is 2, then the GCD of b and a is also 2, but a number's GCD with itself is itself, and transitivity doesn't hold.

24 Easy

24 of 238

Analyzing +4 | -1

R=\{(a, b): 3 a-3 b+\sqrt{7}

is irrational

\}

. Then which among the following options are correct

View answer and explanation

Correct option: C

R

is reflexive but not symmetric

Option C (

R

is reflexive but not symmetric) is correct because the relation satisfies the reflexive property but fails to satisfy the symmetric property.

25 Easy

25 of 238

Understanding +4 | -1

In a set of teachers of a school, two teachers are said to be related if they “teach the same subject”, then the relation is

View answer and explanation

Correct option: D

Equivalence

Option D (Equivalence) is correct because the relation "teach the same subject" is reflexive, symmetric, and transitive, thus it is an equivalence relation.

26 Easy

26 of 238

Application +4 | -1

Given two finite sets

A

and

B

such that

n(A)=2, n(B)=3

. Then total number of relations from

A

B

is?

View answer and explanation

Correct option: C

The correct answer is C (64) because the total number of relations from A to B is 2 raised to the power of the cardinality of A cross B.

27 Medium

27 of 238

Analyzing +4 | -1

Let

P

be the relation defined on the set of all real numbers such that

P=\left\{(a, b): \sec ^2 a-\tan ^2 b=1\right\}

. Then

P

View answer and explanation

Correct option: C

Symmetric and transitive but not reflexive.

Option C (Symmetric and transitive but not reflexive) is correct because the relation satisfies symmetry and transitivity but fails the reflexive property.

28 Easy

28 of 238

Understanding +4 | -1

Let

R

be an equivalence relation on a finite set A having

n

elements. Then, the number of ordered pairs in

R

View answer and explanation

Correct option: B

Greater than or equal to

n

Option B (Greater than or equal to

n

) is correct because an equivalence relation must be reflexive, meaning (a, a) is in R for all a in A, resulting in at least n ordered pairs.

29 Easy

29 of 238

Application +4 | -1

The solution set of

8 x \equiv 6(\bmod 14), x \in Z

, are:

View answer and explanation

Correct option: C

[6] \cup[13]

The correct answer is C ([6] ∪ [13]) because the solution set consists of equivalence classes that satisfy the given congruence relation.

30 Medium

30 of 238

Understanding +4 | -1

Consider the following statements.

I. If the set

{A}

has 3 elements and set

{B}=\{3,4,5\}

, then the number of elements in

{A} \times {B}=9

.

II. The domain of the relation

{R}

defined by

{R}=\{({x}, {x}+5): {x} \in(0,1,2,3,4,5)\} \text { is }\{5,6,7,8,9,10\} .

Choose the correct option.

View answer and explanation

Correct option: A

Only I is true

Only I is true because the number of elements in the Cartesian product is the product of the number of elements in each set, while the domain in statement II is incorrectly calculated.

Revision notes and key takeaways

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Key Takeaways

✓ This set contains 238 single-correct-answer MCQs on RELATIONS.
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