A Relation ${R}$ is defined in the set of integers ${I}$ as follows $(x, y) \in {R}$ if $x^2+{y}^2=9$ which of the following is false?

${R}=\{(0,3),(0,-3),(3,0),(-3,0)\}$

If $R$ is a relation '$ < $' from $A=\{1,2,3,4\}$ to $B=\{1,3,5\}$, i.e. $(a, b) \in R$ iff $a < b$, then $R \circ R { }^{-1}$ is

$\{(1,3),(1,5),(2,3),(2,5),(3,5),(4,5)\}$

$\{(3,1),(5,1),(3,2),(5,2),(5,3),(5,4)\}$

Let $ \mathrm{N} $ denote the set of all natural numbers. Define two binary relations on $ N $ as $ R_{1}=\{(x, y) \in N \times N: 2 x+y=10\} $ and $ R_{2}=\{(x, y) \in N \times N: x+2 y=10\} $. Then

Range of $ R_{2} $ is $ \{1,2,3,4\} $

Both $ R_{1} $ and $ R_{2} $ are transitive relations

Both $ R_{1} $ and $ R_{2} $ are symmetric relations

Range of $ R_{1} $ is $ \{2,4,8\} $

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JEE • MATHEMATICS • RELATIONS

Relations practice · Medium

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Showing 1-30 of 56 questions

1 Medium

1 of 56

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.

2 Medium

2 of 56

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.

3 Medium

3 of 56

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.

4 Medium

4 of 56

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).

5 Medium

5 of 56

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.

6 Medium

6 of 56

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.

7 Medium

7 of 56

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.

8 Medium

8 of 56

Understanding +4 | -1

\mathrm{R}

be a relation defined as

\mathrm{R}=\{(\mathrm{x}, \mathrm{y}): \mathrm{x}+\mathrm{y}=\mathrm{p}

for some rational

\mathrm{p} ; \mathrm{x}, \mathrm{y}

are real numbers

\}

, then

R

View answer and explanation

Correct option: D

equivalence

Option D (equivalence) is correct because the relation R satisfies reflexive, symmetric, and transitive properties, thus making it an equivalence relation.

9 Medium

9 of 56

Analyzing +4 | -1

Let

A=\{1,2,3,4,5,6,7\}

. Then the relation

R=\{(x, y) \in A \times A: x+y=7\}

View answer and explanation

Correct option: B

Symmetric but neither reflexive nor transitive

Option B is correct because the relation is symmetric (if x+y=7, then y+x=7), but it's neither reflexive (e.g., 1+1≠7) nor transitive (e.g., (1,6) and (6,1) are in R, but (1,1) is not).

10 Medium

10 of 56

Analyzing +4 | -1

Let

\mathrm{R}

be a relation on

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

defined by

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

if and only if

a d(b-c)=b c(a-d)

. Then

R

View answer and explanation

Correct option: C

Symmetric but neither reflexive nor transitive

Option C is correct because the relation R is symmetric but neither reflexive nor transitive, as demonstrated by the provided counterexamples.

11 Medium

11 of 56

Analyzing +4 | -1

S

is a relation over the set

R

of all real numbers and it is given by

(a, b) \in S \Leftrightarrow a b \geq 0.

Then,

S

View answer and explanation

Correct option: D

An equivalence relation

The correct answer is D.

S

is an equivalence relation because it is reflexive, symmetric, and transitive.

12 Medium

12 of 56

Analyzing +4 | -1

Among the relations

\begin{array}{l} S=\left\{(a, b): a, b \in \mathbb{R}-\{0\}, 2+\frac{a}{b} > 0\right\} \text { and } \\ T=\left\{(a, b): a, b \in \mathbb{R}, a^{2}-b^{2} \in \mathbb{Z}\right\},\end{array}

View answer and explanation

Correct option: D

T

is symmetric but

S

is not

Option A (

(-\infty,-\frac{21}{4}] \cup[0, \infty)

) is correct because by finding the range of the function, we get

y \in(-\infty,-\frac{21}{4}] \cup[0, \infty)

13 Medium

13 of 56

Understanding +4 | -1

The number of reflexive relations of a set with four elements is equal to

View answer and explanation

Correct option: B

{2^{12}}

Option B (

{2^{12}}

) is correct because the number of reflexive relations on a set with four elements is

{2^{{4^2} - 4}} = {2^{12}}

14 Medium

14 of 56

Application +4 | -1

The solution of

8 x \equiv 6(\bmod 14)

are

View answer and explanation

Correct option: C

[6],[13]

Option C ([6],[13]) is correct because it provides the two solutions to the congruence equation within the modulo 14 system.

15 Medium

15 of 56

Understanding +4 | -1

The relation

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

on set

A=\{1,2,3\}

is:

View answer and explanation

Correct option: A

Reflexive but not symmetric

Option A (Reflexive but not symmetric) is correct because the relation contains (1,1), (2,2), and (3,3), making it reflexive, but it lacks (2,1) for (1,2) and (3,2) for (2,3), making it not symmetric.

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16 Medium

16 of 56

Understanding +4 | -1

Consider the following statements.

I. Addition, subtraction and multiplication are binary operations on

{R}

.

II. Division is a binary operation on

{R}

but not a binary operation on non-zero real numbers.

View answer and explanation

Correct option: A

Only I is true

Option A (Only I is true) is correct because addition, subtraction, and multiplication are indeed binary operations on the set of real numbers, while division is not.

17 Medium

17 of 56

Application +4 | -1

R

is a relation on the set

A=\{1,2,3,4,5,6,7,8,9\}

given by

x R y \Leftrightarrow y=3 x

, then

R \circ R^{-1}

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it represents the composition of the relation

R

with its inverse

R^{-1}

, where

y = 3x

18 Medium

18 of 56

Understanding +4 | -1

Consider the following statements.

I. Let

{A}

and

{B}

are non-empty sets such that

{A} \subseteq {B}

. Then,

{A} \times {C} \subseteq {B} \times {C}

.

II. For any two sets

{A}

and

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

Choose the correct option.

View answer and explanation

Correct option: A

Only I is true

Only I is true because if A is a subset of B, then the Cartesian product of A with any set C will also be a subset of the Cartesian product of B with C.

19 Medium

19 of 56

Application +4 | -1

Suppose that the number of elements in set

\mathrm{A}

\mathrm{p}

, the number of elements in set

\mathrm{B}

\mathrm{q}

and the number of elements in

\mathrm{A} \times \mathrm{B}

is 7 . Then

\mathrm{p}^{2}+\mathrm{q}^{2}=

View answer and explanation

Correct option: C

Option C (50) is correct because it is the sum of the squares of the number of elements in set A and set B, calculated using the given information about the Cartesian product.

20 Medium

20 of 56

Understanding +4 | -1

A relation

R

on set of point of given plane defined as

(A, B) \in R

O A=O B

where

O

is fixed point on same plane. Then relation

\mathrm{R}

is:

View answer and explanation

Correct option: B

equivalence relation

Option B (equivalence relation) is correct because the relation satisfies reflexive, symmetric, and transitive properties, thus making it an equivalence relation.

21 Medium

21 of 56

Understanding +4 | -1

Let

n(A)=m

and

n(B)=n

. Then, the total number of non-empty relations that can be defined from

A

B

View answer and explanation

Correct option: D

2^{\text {min }}-1

Option

2^{mn}-1

is correct, as the total number of relations from A to B is

2^{mn}

, and subtracting 1 excludes the empty relation.

22 Medium

22 of 56

Understanding +4 | -1

A Relation

{R}

is defined in the set of integers

{I}

as follows

(x, y) \in {R}

x^2+{y}^2=9

which of the following is false?

View answer and explanation

Correct option: D

None of the above

Option D is correct because all the given statements about the relation R are true.

23 Medium

23 of 56

Application +4 | -1

R

is a relation '

<

' from

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

B=\{1,3,5\}

, i.e.

(a, b) \in R

iff

a < b

, then

R \circ R { }^{-1}

View answer and explanation

Correct option: C

\{(3,3),(3,5),(5,3),(5,5)\}

Option C ({ (3,3),(3,5),(5,3),(5,5) }) is correct because it represents the composition of the relation R with its inverse.

24 Medium

24 of 56

Application +4 | -1

Let

A=\{1,2,3,4,5\} ; B=\{2,3,6,7\}

. Then the number of elements in

(A \times B) \cap(B \times A)

is :

View answer and explanation

Correct option: D

Option D (4) is correct because there are 4 elements that are present in both A x B and B x A, which are (2,2), (2,3), (3,2), and (3,3).

25 Medium

25 of 56

Analyzing +4 | -1

Let

{R_1}

be a relation defined by

{R_1} = \left\{ {\left( {a,b} \right){\text{|}}a \geqslant b,a,b \in R} \right\}.

Then,

{R_1}

View answer and explanation

Correct option: B

Reflexive, transitive but not symmetric

Option B (Reflexive, transitive but not symmetric) is correct because the relation satisfies the reflexive and transitive properties but fails to be symmetric.

26 Medium

26 of 56

Application +4 | -1

Let

x

denote the total number of one-one functions from a set

A

with 3 elements to a set

B

with 5 elements and

y

denote the total number of one-one functions from the set

A

to the set

A \times B

. Then,

View answer and explanation

Correct option: A

2y = 91x

The correct answer is A (

2y = 91x

) because it correctly calculates the number of one-one functions and establishes the relationship between x and y.

27 Medium

27 of 56

Understanding +4 | -1

A=\{a, b, c, d\}, B=\{1,2,3\}

, which of the following sets of ordered pairs are not relations from

A

B

View answer and explanation

Correct option: C

\{(a, 2),(b, 3),(3, b)\}

Option C is correct because (3, b) is not an element of A x B since 3 is not in A and b is not in B.

28 Medium

28 of 56

Application +4 | -1

R

is the smallest equivalence relation on the set

\{1,2,3,4\}

such that

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

, then the number of elements in

R

View answer and explanation

Correct option: B

The correct answer is B (10) because it represents the minimum number of ordered pairs required to satisfy the properties of an equivalence relation given the initial conditions.

29 Medium

29 of 56

Understanding +4 | -1

Total number of equivalence relations defined in the set

S=\{a, b, c\}

View answer and explanation

Correct option: A

Option A (5) is correct because the number of equivalence relations on a set is equivalent to the number of partitions of the set, which is 5 for a set with 3 elements.

30 Medium

30 of 56

Understanding +4 | -1

Let

\mathrm{N}

denote the set of all natural numbers. Define two binary relations on

N

R_{1}=\{(x, y) \in N \times N: 2 x+y=10\}

and

R_{2}=\{(x, y) \in N \times N: x+2 y=10\}

. Then

View answer and explanation

Correct option: C

Range of

R_{2}

\{1,2,3,4\}

Option C is correct because the range of R2, defined by x + 2y = 10, with x and y being natural numbers, is {1, 2, 3, 4}.

Revision notes and key takeaways

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Key Takeaways

✓ This set contains 56 single-correct-answer MCQs on RELATIONS.
✓ Try each question before opening its answer. Read the explanation where one is available.
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