sets relations and functions jee mains questions | JEE Main Mathematics Sets Relations And Functions

Q 11 :

Let A = {1, 2, 3, ..., 100} and R be a relation on A such that R = {(a, b) : a = 2b + 1}. Let $(a_{1}, a_{2}), (a_{2}, a_{3}), (a_{3}, a_{4}), . . ., (a_{k}, a_{k + 1})$ be a sequence of k elements of R such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer k, for which such a sequence exists, is equal to : [2025]

8
5
6
7

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(2)

Let the smallest value is $a_{k + 1}$ in A = {1, 2, 3, ..., 100}.

Since, $a_{k} = 2 a_{k + 1} + 1 = (2 \times 1) + 1 = 3$

$∴$ $a_{k - 1} = 2 a_{k} + 1 = (2 \times 3) + 1 = 7$

$a_{k - 2} = 2 a_{k - 1} + 1 = (2 \times 7) + 1 = 15$

$a_{k - 3} = 2 a_{k - 2} + 1 = (2 \times 15) + 1 = 31$

$a_{k - 4} = 2 a_{k - 3} + 1 = (2 \times 31) + 1 = 63$

$a_{k - 5} = 2 a_{k - 4} + 1 = (2 \times 63) + 1 = 127 \notin A$

$∴$ The sequence is {(63, 31), (31, 15), (15, 7), (7, 3), (3, 1)}

$∴$ k = 5

Q 12 :

Let A = {–3, –2, –1, 0, 1, 2, 3}. Let R be a relation on A defined by $x R y$ if and only if $0 < x^{2} + 2 y \leq 4$ . Let $l$ be the number of elements in R and m be the minimum number of elements required to be added in R to make it reflexive relation. Then $l$ + m is equal to [2025]

18
20
17
19

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(1)

A = {–3, –2, –1, 0, 1, 2, 3}

xRy if and only if $- 2 y \leq x^{2} \leq 4 - 2 y$

y = –3 $6 \leq x^{2} \leq 10$ $\Rightarrow x \in {- 3, 3}$

y = –2 $4 \leq x^{2} \leq 8$ $\Rightarrow x \in {- 2, 2}$

y = –1 $2 \leq x^{2} \leq 6$ $\Rightarrow x \in {- 2, 2}$

y = 0 $0 \leq x^{2} \leq 4$ $\Rightarrow x \in {- 2, - 1, 0, 1, 2}$

y = 1 $- 2 \leq x^{2} \leq 2$ $\Rightarrow x \in {- 1, 0, 1}$

y = 2 $- 4 \leq x^{2} \leq 0$ $\Rightarrow x \in {0}$

y = 3 $- 6 \leq x^{2} \leq - 2$ $\Rightarrow Value of x does not exist$

R = {(–3, –3), (3, –3), (–2, –2),(2, –2), (–2, –1), (2, –1), (–2, 0), (–1, 0), (0,0), (1, 0), (2, 0), (–1, 1), (0, 1), (1, 1), (0, 2)}

$∴$ $l$ = 15

To make it reflexive we will add (–1, –1), (2, 2), (3, 3) in R

$∴$ $l$ + m = 15 + 3 = 18.

Q 13 :

Let A = {–2, –1, 0, 1, 2, 3}. Let R be a relation on A defined by xRy if and only if y = max{x, 1}. Let $l$ be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then $l$ + m + n is equal to [2025]

13
12
14
11

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(2)

We have, A = {–2, –1, 0, 1, 2, 3} and

R = {(–2, 1), (—1, 1), (0, 1), (1, 1), (2, 2), (3, 3)}.

Now, number of elements in R i.e., $l$ = 6

For R to be reflexive,

R = {(–2, –2), (–1, –1), (0, 0), (–2, 1), (–1, 1), (0, 1), (1, 1), (2, 2), (3, 3)}

So, we need to add three elements to make it reflexive.

$∴$ m = 3

For R to be symmetric,

R = {(–2, 1), (1, –2), (–1, 1), (1, –1), (0, 1), (1, 0), (1, 1), (2, 2), (3, 3)}

So, we need to add three elements to make it symmetric.

$∴$ n = 3

So, $l$ + m + n = 6 + 3 + 3 = 12.

Q 14 :

Let A = {–3, –2, –1, 0, 1, 2, 3} and R be a relation on A defined by xRy if and only if 2x – y $\in$ {0, 1}. Let $l$ be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then $l$ + m + n is equal to : [2025]

17
16
18
15

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(1)

We have, A = {–3, –2, –1, 0, 1, 2, 3}, R is defined on A as xRy such that 2x – y $\in$ {0, 1}.

i.e., 2x – y = 0 or 2x – y = 1

$∴$ R = {(0, 0), (–1, –2), (1, 2), (0, –1), (2,3), (1, 1), (–1, –3)} i.e., $l$ = 7

For R to be reflexive, i.e., we need 5 more elements {(2, 2), (–1, –1), (3, 3), (–3, –3), (–2, –2)} so m = 5 and for R to be symmetric, we need 5 more elements {(–2, –1), (2, 1), (–1, 0), (3, 2), (–3, –1)}, so n = 5.

$∴$ $l$ + m + n = 7 + 5 + 5 = 17.

Q 15 :

Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) $\in$ R if and only if max {x, y} $\in$ {3, 4}. Then among the statements

( $S_{1}$ ) : The number of elements in R is 18, and

( $S_{2}$ ) : The relation R is symmetric but neither reflexive nor transitive. [2025]

both are true
only ( $S_{2}$ ) is true
both are false
only ( $S_{1}$ ) is true

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(2)

R = {(0, 3), (3, 0), (0, 4), (4, 0), (1, 3), (3, 1), (2, 3), (3, 2), (3, 3), (1, 4), (4, 1), (2, 4), (4, 2), (3, 4), (4, 3), (4, 4)}

$∴$ Number of elements in R = 16

$∴$ $S_{1}$ is false.

Now, (0, 0), (1, 1), (2, 2), (5, 5) $\notin$ R $∴$ R is not reflexive.

Again, let (a, b) $\in$ R then (b, a) $\in$ R

As max {a, b} = max {b, a} $∴$ R is symmetric.

Now, R is not transitive as (0, 3), (3, 1) $\in$ R but (0, 1) $\notin$ R.

$∴$ $S_{2}$ is true.

Q 16 :

The number of non-empty equivalence relations on the set {1, 2, 3} is: [2025]

5
6
7
4

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(1)

Partitions of set {1, 2, 3} is {{1}}, {2}, {3}}, {{1, 2}, {3}}, {{1, 3}, {2}}, {{2, 3}, {1}}, {{1, 2, 3}}

$∴$ Number of non-empty equivalence relations on the set {1, 2, 3} = 5.

Q 17 :

Let R = {(1, 2), (2, 3), (3, 3)} be a relation defined on the set {1, 2, 3, 4}. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is: [2025]

7
10
9
8

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3

4

(1)

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

R = {(1, 2), (2, 3), (3, 3)} [Given]

For an equivalence relation 'R' should be reflexive, symmetric and transitive.

$∴$ Minimum elements that should be added are

(2, 1), (3, 2), (1, 1), (2, 2), (4, 4), (1, 3), (3, 1)

$∴$ Minimum number of elements required = 7.

Q 18 :

Let $A = {(x, y) \in R \times R : | x + y | \geq 3}$ and $B = {(x, y) \in R \times R : | x | + | y | \leq 3}$

If $C = {(x, y) \in A \cap B : x = 0 or y = 0}$ , then $\sum_{(x, y) \in C}$ $| x + y |$ is: [2025]

18
12
24
15

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(2)

We have, $A = {(x, y) \in R \times R : | x + y | \geq 3}$ and $B = {(x, y) \in R \times R : | x | + | y | \leq 3}$

From given figure, we get C = {(3, 0), (–3, 0), (0, 3), (0, –3)}

$\sum_{(x, y) \in C} | x + y | = 12$

Q 19 :

Let X = R $\times$ R. Define a relation R on X as:

$(a_{1}, b_{1}) R (a_{2}, b_{2}) \Leftrightarrow b_{1} = b_{2}$ .

Statement I: R is an equivalence relation.

Statement II: For some (a, b) $\in$ X, the set

S = {(x, y) $\in$ X :(x, y) R (a, b)} represents a line parallel to y = x.

In the light of the above statements, choose the correct answer from the options given below: [2025]

Statement I is true but Statement II is false.
Statement I is false but State II is true.
Both Statement I and Statement II are false.
Both Statement I and Statement II is true.

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(1)

Statement I:

Reflexive : $(a_{1}, b_{1}) R (a_{1}, b_{1}) \Rightarrow b_{1} = b_{1}$

$∴$ R is reflexive.

Symmetric : $(a_{1}, b_{1}) R (a_{2}, b_{2}) \Rightarrow b_{1} = b_{2}$

$(a_{2}, b_{2}) R (a_{1}, b_{1}) \Rightarrow b_{2} = b_{1}$

$∴$ R is symmetric.

Transitive : $(a_{1}, b_{1}) R (a_{2}, b_{2}) \Rightarrow b_{1} = b_{2}$

and $(a_{2}, b_{2}) R (a_{3}, b_{3}) \Rightarrow b_{2} = b_{3} \Rightarrow b_{1} = b_{3}$

$\Rightarrow (a_{1}, b_{1}) R (a_{3}, b_{3})$

Hence, relation R is equivalence relation.

$∴$ Statement I is true.

For Statement II: (x, y) R (a, b), for some $(a, b) \in X \Rightarrow y = b$

So, Statement II is false.

Q 20 :

The relation R = {(x, y) : x, y $\in$ Z and x + y is even} is: [2025]

reflexive and transitive but not symmetric.
reflexive and symmetric but not transitive.
symmetric and transitive but not reflexive.
an equivalence relation.

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(4)

R = {(x, y) : x, y $\in$ Z and x + y is even}

For reflexive : x + x = 2x is even

For symmetric : x + y = y + x is even

For transitive : x + y is even and y + z is even then z + x is also even.

So, the relation is an equivalence.

Q 21 :

Define a relation R on the interval $[0, \frac{π}{2})$ by xRy if and only if $\sec^{2} x - \tan^{2} y = 1$ . Then R is : [2025]

an equivalence relation.
reflexive but neither symmetric not transitive.
both reflexive and symmetric but not transitive.
both reflexive and transitive but not symmetric.

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(1)

$\sec^{2} x - \tan^{2} x = 1, \forall x \in [0, \frac{π}{2})$

$∴$ R is reflexive

Consider, $\sec^{2} x - \tan^{2} y = 1$

$\Rightarrow 1 + \tan^{2} x - (\sec^{2} y - 1) = 1$

$\Rightarrow 1 + \tan^{2} x - s e c^{2} y + 1 = 1 \Rightarrow \sec^{2} y - \tan^{2} x = 1$

$∴$ R is symmetric.

Now, if $\sec^{2} x - \tan^{2} y = 1$ and $\sec^{2} y - \tan^{2} z = 1$

Adding both equation, $\sec^{2} x - \tan^{2} y + \sec^{2} y - \tan^{2} z = 2$

$\Rightarrow \sec^{2} x - \tan^{2} z = 1$ [ $∵ \sec^{2} y - \tan^{2} y = 1$ ]

$∴$ R is transitive

Thus R is an equivalence relation.

Q 22 :

Let $S = N \cup {0}$ . Define a relation R from S to R by:

$R = {(x, y) : \log_{e} y = x \log_{e} (\frac{2}{5}), x \in S, y \in R}$

Then, the sum of all the elements in the range of R is equal to: [2025]

$\frac{5}{2}$
$\frac{5}{3}$
$\frac{3}{2}$
$\frac{10}{9}$

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(2)

We have, $S = N \cup {0} = {0, 1, 2, 3, . . . . . .}$

Also, $\log_{e} y = x \log_{e} (\frac{2}{5}) \Rightarrow y = {(\frac{2}{5})}^{x}$

$∴$ Required sum = ${(\frac{2}{5})}^{0} + {(\frac{2}{5})}^{1} + {(\frac{2}{5})}^{2} + . . . . . . . .$

$= \frac{1}{1 - \frac{2}{5}} = \frac{5}{3}$ .

Q 23 :

The number of relation on the set A = {1, 2, 3}, containing at most 6 elements including (1, 2), which are reflexive and transitive but not symmetric, is __________. [2025]

(5)

Given, A = {1, 2, 3}

Let the relation be R on A, which is reflexive and transitive but not symmetric, then

(1, 1), (2, 2), (3, 3), (1, 2) $\in$ R

Remaining elements are

(2, 1), (2, 3), (1, 3), (3, 1), (3, 2)

Case I : If relation contains exactly 4 elements $\Rightarrow$ 1 way

Case II : If relation contains exactly 5 elements, so we can add (1, 3) or (3, 2) $\Rightarrow$ 2 ways

Case III : If relation contains exactly 6 elements, so we can add (2, 3), (1, 3) or (1, 3), (3, 2) or (3, 1), (3, 2) $\Rightarrow$ 3 ways

$∴$ Total number of relations is 6.

Q 24 :

Let A = {1, 2, 3}. The number of relations on A, containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is __________. [2025]

(3)

For transitive : (1, 2) and (2, 3) $\in$ R $\Rightarrow$ (1, 3) $\in$ R

For reflexive : (1, 1), (2, 2), (3, 3) $\in$ R

Now, for (2, 1), (3, 2), (3, 1); (3, 1) cannot be taken for not symmetric relation.

Case I : (2, 1) taken and (3, 2) not taken

Case II : (3, 2) taken and (2, 1) not taken

Case III : (2, 1) and (3, 2) are not taken

Therefore, 3 relations are possible.

Q 25 :

Let $A = {- 2, - 1, 0, 1, 2, 3, 4}$ . Let R be a relation on A defined by $x R y$ if and only if $2 x + y \leq 2$ . Let $l$ be the number of elements in R. Let $m$ and $n$ be the minimum number of elements required to be added in R to make it reflexive and symmetric relations respectively. Then $l + m + n$ is equal to: [2026]

33
32
35
34

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(1)

R = {(-2, a), (-1, b), (0, c), (1, d), (2, e)}

a = {-2, -1, 0, 1, 2, 3, 4};
b = {-2, -1, 0, 1, 2, 3, 4}
c = {-2, -1, 0, 1, 2};
d = {-2, -1, 0};
e = {-2}

∴ No. of elements in R
= 7 + 7 + 5 + 3 + 1 = 23 = $ℓ$

Minimum number of element to be added to make it reflexive = m = 4
⇒ {(1, 1), (2, 2), (3, 3), (4, 4)}

Minimum number of element to be added to make it symmetric = n = 6

⇒ R = {(3, -2), (4, -2), (2, -1), (2, 0), (3, -1), (4, -1)}

$∴$ $ℓ$ + m + n = 23 + 4 + 6 = 33

Q 26 :

Let $R$ be a relation defined on the set ${1, 2, 3, 4} \times {1, 2, 3, 4}$ by $R = {((a, b), (c, d)) : 2 a + 3 b = 3 c + 4 d} .$

Then the number of elements in $R$ is [2026]

15
6
18
12

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2

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4

(4)

(a, b) (c, d)

(1, 1) x
(1, 2) x
(1, 3) (1, 2)
(1, 4) (2, 2)
(2, 1) (1, 1)
(2, 2) (2, 1)
(2, 3) (3, 1)
(2, 4) (4, 1)
(3, 1) x
(3, 2) x
(3, 3) (1, 3)
(3, 4) (2, 3)
(4, 1) (1, 2)
(4, 2) (2, 2)
(4, 3) (3, 2)
(4, 4) (4, 2)

Q 27 :

Let the relation R on the set $M = {1, 2, 3, \dots, 16}$ be given by $R = {(x, y) : 4 y = 5 x - 3, x, y \in M} .$

Then the minimum number of elements required to be added in R, in order to make the relation symmetric, is equal to [2026]

1
2
4
3

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(2)

R = {(3, 3), (7, 8), (11, 13)}

to make it symmetric (8, 7), (13, 11) must be added.

Q 28 :

Let $A = {0, 1, 2, \dots, 9}$ Let R be a relation on A defined by $(x, y) \in R$ if and only if $∣ x - y ∣$ is a multiple of 3.

Given below are two statements:

Statement I: $n (R) = 36$ .

Statement II: R is an equivalence relation.

In the light of the above statements, choose the correct answer from the options given below. [2026]

Statement I is correct but Statement II is incorrect
Statement I is incorrect but Statement II is correct
Both Statement I and Statement II are incorrect
Both Statement I and Statement II are correct

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(2)

$Number of form 3 K = 4$

$Number of form 3 K + 1 = 3$

$Number of form 3 K + 2 = 4$

$4 \times 4 + 3 \times 3 + 3 \times 3 = 34 relations$

$\Rightarrow x R y \Rightarrow y R x$

$\Rightarrow (x - y) = 3 λ, (y - z) = 3 μ$

$\Rightarrow (x - z) = 3 (λ + μ)$

$R is reflexive, symmetric and transitive S_{2} is true$

$Ans. S_{1} is false but S_{2} is true$

Q 29 :

Let A = {2,3,5,7,9} Let R be the relation on A defined by xRy if and only if $2 x \leq 3 y$ . Let $l$ be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then $l + m$ is equal to : [2026]

25
23
27
21

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(1)

$A = {2, 3, 5, 7, 9}$

$y \geq \frac{2 x}{3}$

$\begin{matrix} x = 2, & y = 2, 3, 5, 7, 9 \\ x = 3, & y = 2, 3, 5, 7, 9 \\ x = 5, & y = 5, 7, 9 \\ x = 7, & y = 5, 7, 9 \\ x = 9, & y = 7, 9 \end{matrix}] \Rightarrow ℓ = 18$

To make it symmetric the elements to be added are ${(5, 2), (7, 2), (9, 2), (5, 3), (7, 3), (9, 3), (9, 5)}$

$m = 7$

$∴$ $ℓ + m = 25$

Q 30 :

The number of elements in the relation $R = {(x, y) : 4 x^{2} + y^{2} < 52, x, y \in ℤ}$ is [2026]

86
89
67
77

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(4)

$4 x^{2} + y^{2} < 52, x, y \in ℤ$

$↓$ $↓$

$0 0, \pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 7 \to 1 \times 15 = 15$

$\pm 1$ $0, \pm 1, \pm 2, \pm 3, \dots . . . . ., \pm 6$ $\to 2 \times 13 = 26$

$\pm 2$ $0, \pm 1, \pm 2, \pm 3, . . . . . \dots, \pm 5$ $\to 2 \times 11 = 22$

$\pm 3$ $0, \pm 1, \pm 2, \pm 3$ $\to 2 \times 7 = 14$

$Number of elements = 15 + 26 + 22 + 14 = 77$

Q 31 :

Let $R_{1}$ and $R_{2}$ be two relations defined on $ℝ$ by $a R_{1} b \Leftrightarrow a b \geq 0$ and $a R_{2} b \Leftrightarrow a \geq b .$ Then

$R_{1}$ is an equivalence relation but not $R_{2}$
$R_{2}$ is an equivalence relation but not $R_{1}$
Both $R_{1}$ and $R_{2}$ are equivalence relations
Neither $R_{1}$ nor $R_{2}$ is an equivalence relation

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(4)

$a = 1, b = 0, c = - 2 then transitive fails for R_{1}$

Q 32 :

Let $R$ be the relation on $N$ defined by $R = {(x, y) : x + 2 y = 8} .$ Then, the domain of $R$ is:

${2, 4, 6, 8}$
${2, 4, 8}$
${2, 4, 6}$
${1, 2, 3, 4}$

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(3)

Ans. {2, 4, 6}

Explanation:

Since $R$ is defined as $x + 2 y = 8 .$

Now, $x$ and $y$ are natural numbers.

$∴$ $x = 2,$ $y = 3$

Because if we put $x = 3$ , then $y$ will not be a natural number.

$x = 4,$ $y = 2$

$x = 6,$ $y = 1$

$∴$ Domain of the given relation is {2, 4, 6}

Q 33 :

The relation $R$ in the set of natural numbers $N$ defined as $R = {(x, y) : y = x + 5 and x < 4}$ is:

reflexive
symmetric
transitive
None of these

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(4)

Ans. None of these

Explanation:

$R = {(1, 6), (2, 7), (3, 8)}$

$∵$ $(6, 6) \notin R$

$\Rightarrow R is not reflexive.$

For $(2, 7) \in R$

$\Rightarrow$ $(2, 7) \in R but (7, 2) \notin R$

$\Rightarrow R is not symmetric.$

Clearly, $R$ is not transitive.

(Since, there is no pair in $R$ such that $(x, y)$ and $(y, z) \in R$ , then $(x, z) \in R$ )

$\Rightarrow$ R is not an equivalence relation.

Q 34 :

If $A = {x \in Z : 0 \leq x \leq 12}$ and $R$ is the relation in $A$ given by $R = {(a, b) : a = b} .$ Then, the set of all elements related to 1 is:

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

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(3)

Ans. {1}

Explanation:

The set of all elements related to 1 is

${a \in A : a = 1}$

Hence, {1}

Q 35 :

The range of the relation R = { $(x, x^{2}) : x$ is a prime number less than 13} is:

{2, 3, 5, 7}
{4, 9, 25, 49, 121}
{2, 3, 5, 7, 11}
{1, 4, 9, 25, 49, 121}

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(2)

Ans. {4, 9, 25, 49, 121}

Explanation:

R = {(2, 4), (3, 9), (5, 25), (7, 49), (11, 121)}

Range = {4, 9, 25, 49, 121}

Q 36 :

R is a relation from {11, 12, 13} to {8, 10, 12} defined by $y = x - 3$ . Then $R^{- 1}$ is:

{(8, 11), (10, 13)}
{(11, 8), (13, 10)}
{(10, 13), (8, 11), (8, 10)}
None of the above

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(1)

Ans. {(8,11), (10,13)}

Explanation:

Let $A = {11, 12, 13},$

$B = {8, 10, 12}$

${}_{A}R_{B'}$ $y = x - 3$

$R = {(11, 8), (13, 10)}$

$R^{- 1} = {(8, 11), (10, 13)}$

Q 37 :

Let $T$ be the set of all triangles in the Euclidean plane and let a relation $R$ on $T$ be defined as $a R b$ if $a$ is congruent to $b$ for all $a, b \in T$ , then $R$ is:

reflexive but not symmetric
transitive but not symmetric
equivalence
neither symmetric nor transitive

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(3)

Ans. Equivalence

Explanation:

Each triangle is congruent to itself.

$∴$ Relation is reflexive.

Suppose $∆ A B C ≅ ∆ P Q R$

then $∆ P Q R ≅ ∆ A B C$

$∴$ Relation is symmetric.

Suppose $∆ A B C ≅ ∆ P Q R,$

and $∆ P Q R ≅ ∆ X Y Z$

then $∆ A B C ≅ ∆ X Y Z$

$∴$ Relation is transitive.

$∴$ Relation is an equivalence relation.

Q 38 :

Let us define a relation $R$ in $R$ as $a R b$ if $a \geq b$ . Then $R$ is:

an equivalence relation
reflexive, transitive but not symmetric
symmetric, transitive but not reflexive
neither transitive nor reflexive but symmetric

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(2)

Ans. Reflexive, transitive but not symmetric

Explanation:

Given that, $a R b if a \geq b$

$\Rightarrow a R a \Rightarrow a \geq a,$ which is true.

Hence, $R$ is reflexive.

Let $a R b$ , i.e., $a \geq b$ .

For symmetry, we require $b R a$ , i.e., $b \geq a,$ which is not true in general.

Hence, $R$ is not symmetric.

But if $a R b$ and $b R c$ ,

$\Rightarrow$ $a \geq b$ and $b \geq c$

$\Rightarrow$ $a \geq c$

Hence, $R$ is transitive.

Q 39 :

Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer:

R is reflexive and symmetric but not transitive
R is reflexive and transitive but not symmetric
R is symmetric and transitive but not reflexive
R is an equivalence relation

1

2

3

4

(2)

Ans. R is reflexive and transitive but not symmetric.

Explanation:

$R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}$

It is seen that $(a, a) \in R$ , for every $a \in {1, 2, 3, 4}$

$∴$ R is reflexive.

It is seen that $(1, 2) \in R,$ but $(2, 1) \notin R$ .

$∴$ R is not symmetric.

Also, it is observed that $(a, b),$ $(b, c) \in R$ $\Rightarrow (a, c) \in R,$ for all $a, b, c \in {1, 2, 3, 4}$ .

$∴$ R is transitive.

Hence, R is reflexive and transitive but not symmetric.

Q 40 :

If $R = {(x, y) : x + 2 y = 8}$ is a relation on $N$ , then the range of $R$ is:

{2, 3, 4}
R
{1, 2, 3}
None of these

1

2

3

4

(3)

Ans. {1, 2, 3}

Explanation:

The given relation on $N$ is $R = {(x, y) : x + 2 y = 8}$

Since both $x, y \in N$ ,

$x$ can take the values (2, 4, 6) for all $x, y \in N$

For $x = 2$ , $2 + 2 y = 8$

$\Rightarrow$ $y = 3$

For $x = 4$ , $4 + 2 y = 8$

$\Rightarrow$ $y = 2$

For $x = 6$ , $6 + 2 y = 8$

$\Rightarrow$ $y = 1$

Hence, $R = {(2, 3), (4, 2), (6, 1)}$

Range of $R$ = Set of second elements $= {1, 2, 3}$

Topic Question Set

Q 15 :

Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) $\in$ R if and only if max {x, y} $\in$ {3, 4}. Then among the statements

( $S_{1}$ ) : The number of elements in R is 18, and

( $S_{2}$ ) : The relation R is symmetric but neither reflexive nor transitive. [2025]

Q 16 :

The number of non-empty equivalence relations on the set {1, 2, 3} is: [2025]

Q 17 :

Let R = {(1, 2), (2, 3), (3, 3)} be a relation defined on the set {1, 2, 3, 4}. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is: [2025]

Q 18 :

Let $A = {(x, y) \in R \times R : | x + y | \geq 3}$ and $B = {(x, y) \in R \times R : | x | + | y | \leq 3}$

If $C = {(x, y) \in A \cap B : x = 0 or y = 0}$ , then $\sum_{(x, y) \in C}$ $| x + y |$ is: [2025]

Q 20 :

The relation R = {(x, y) : x, y $\in$ Z and x + y is even} is: [2025]

Q 21 :

Define a relation R on the interval $[0, \frac{π}{2})$ by xRy if and only if $\sec^{2} x - \tan^{2} y = 1$ . Then R is : [2025]

Q 22 :

Let $S = N \cup {0}$ . Define a relation R from S to R by:

$R = {(x, y) : \log_{e} y = x \log_{e} (\frac{2}{5}), x \in S, y \in R}$

Then, the sum of all the elements in the range of R is equal to: [2025]

Q 23 :

The number of relation on the set A = {1, 2, 3}, containing at most 6 elements including (1, 2), which are reflexive and transitive but not symmetric, is __________. [2025]

Q 24 :

Let A = {1, 2, 3}. The number of relations on A, containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is __________. [2025]

Q 26 :

Let $R$ be a relation defined on the set ${1, 2, 3, 4} \times {1, 2, 3, 4}$ by $R = {((a, b), (c, d)) : 2 a + 3 b = 3 c + 4 d} .$

Then the number of elements in $R$ is [2026]

Q 27 :

Let the relation R on the set $M = {1, 2, 3, \dots, 16}$ be given by $R = {(x, y) : 4 y = 5 x - 3, x, y \in M} .$

Then the minimum number of elements required to be added in R, in order to make the relation symmetric, is equal to [2026]

Q 29 :

Let A = {2,3,5,7,9} Let R be the relation on A defined by xRy if and only if $2 x \leq 3 y$ . Let $l$ be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then $l + m$ is equal to : [2026]

Q 30 :

The number of elements in the relation $R = {(x, y) : 4 x^{2} + y^{2} < 52, x, y \in ℤ}$ is [2026]

Q 31 :

Let $R_{1}$ and $R_{2}$ be two relations defined on $ℝ$ by $a R_{1} b \Leftrightarrow a b \geq 0$ and $a R_{2} b \Leftrightarrow a \geq b .$ Then

Q 32 :

Let $R$ be the relation on $N$ defined by $R = {(x, y) : x + 2 y = 8} .$ Then, the domain of $R$ is:

Q 33 :

The relation $R$ in the set of natural numbers $N$ defined as $R = {(x, y) : y = x + 5 and x < 4}$ is:

Q 34 :

If $A = {x \in Z : 0 \leq x \leq 12}$ and $R$ is the relation in $A$ given by $R = {(a, b) : a = b} .$ Then, the set of all elements related to 1 is:

Q 35 :

The range of the relation R = { $(x, x^{2}) : x$ is a prime number less than 13} is:

Q 36 :

R is a relation from {11, 12, 13} to {8, 10, 12} defined by $y = x - 3$ . Then $R^{- 1}$ is:

Q 37 :

Let $T$ be the set of all triangles in the Euclidean plane and let a relation $R$ on $T$ be defined as $a R b$ if $a$ is congruent to $b$ for all $a, b \in T$ , then $R$ is:

Q 38 :

Let us define a relation $R$ in $R$ as $a R b$ if $a \geq b$ . Then $R$ is:

Q 39 :

Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer:

Q 40 :

If $R = {(x, y) : x + 2 y = 8}$ is a relation on $N$ , then the range of $R$ is:

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Topic Question Set

Q 12 : Let A = {–3, –2, –1, 0, 1, 2, 3}. Let R be a relation on A defined by xRy if and only if 0<x2+2y≤4. Let l be the number of elements in R and m be the minimum number of elements required to be added in R to make it reflexive relation. Then l + m is equal to [2025]

Q 15 : Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) ∈ R if and only if max {x, y} ∈ {3, 4}. Then among the statements (S1) : The number of elements in R is 18, and (S2) : The relation R is symmetric but neither reflexive nor transitive. [2025]

Q 16 : The number of non-empty equivalence relations on the set {1, 2, 3} is: [2025]

Q 17 : Let R = {(1, 2), (2, 3), (3, 3)} be a relation defined on the set {1, 2, 3, 4}. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is: [2025]

Q 18 : Let A={(x,y)∈R×R:|x+y|≥3} and B={(x,y)∈R×R:|x|+|y|≤3} If C={(x,y)∈A∩B:x=0 or y=0}, then ∑(x,y)∈C|x+y| is: [2025]

Q 20 : The relation R = {(x, y) : x, y ∈ Z and x + y is even} is: [2025]

Q 21 : Define a relation R on the interval [0,π2) by xRy if and only if sec2x–tan2y=1. Then R is : [2025]

Q 22 : Let S=N∪{0}. Define a relation R from S to R by: R={(x,y):logey=xloge(25), x∈S, y∈R} Then, the sum of all the elements in the range of R is equal to: [2025]

Q 23 : The number of relation on the set A = {1, 2, 3}, containing at most 6 elements including (1, 2), which are reflexive and transitive but not symmetric, is __________. [2025]

Q 24 : Let A = {1, 2, 3}. The number of relations on A, containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is __________. [2025]

Q 25 : Let A={-2,-1,0,1,2,3,4}. Let R be a relation on A defined by xRy if and only if 2x+y≤2. Let l be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations respectively. Then l+m+n is equal to: [2026]

Q 26 : Let R be a relation defined on the set {1,2,3,4}×{1,2,3,4} by R={((a,b),(c,d)):2a+3b=3c+4d}. Then the number of elements in R is [2026]

Q 27 : Let the relation R on the set M={1,2,3,…,16} be given by R={(x,y):4y=5x-3, x,y∈M}. Then the minimum number of elements required to be added in R, in order to make the relation symmetric, is equal to [2026]

Q 29 : Let A = {2,3,5,7,9} Let R be the relation on A defined by xRy if and only if 2x≤3y. Let l be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then l+m is equal to : [2026]

Q 30 : The number of elements in the relation R={(x,y):4x2+y2<52, x,y∈ℤ} is [2026]

Q 31 : Let R1 and R2 be two relations defined on ℝ by aR1b⇔ab≥0 and aR2b⇔a≥b. Then

Q 32 : Let R be the relation on N defined by R={(x,y):x+2y=8}. Then, the domain of R is:

Q 33 : The relation R in the set of natural numbers N defined as R={(x,y):y=x+5 and x<4} is:

Q 34 : If A={x∈Z:0≤x≤12} and R is the relation in A given by R={(a,b):a=b}. Then, the set of all elements related to 1 is:

Q 35 : The range of the relation R = {(x,x2):x is a prime number less than 13} is:

Q 36 : R is a relation from {11, 12, 13} to {8, 10, 12} defined by y=x−3. Then R-1 is:

Q 37 : Let T be the set of all triangles in the Euclidean plane and let a relation R on T be defined as aRb if a is congruent to b for all a,b∈T, then R is:

Q 38 : Let us define a relation R in R as aRb if a≥b. Then R is:

Q 39 : Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer:

Q 40 : If R={(x,y):x+2y=8} is a relation on N, then the range of R is:

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Q 15 :

Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) $\in$ R if and only if max {x, y} $\in$ {3, 4}. Then among the statements

( $S_{1}$ ) : The number of elements in R is 18, and

( $S_{2}$ ) : The relation R is symmetric but neither reflexive nor transitive. [2025]

Q 16 :

The number of non-empty equivalence relations on the set {1, 2, 3} is: [2025]

Q 17 :

Let R = {(1, 2), (2, 3), (3, 3)} be a relation defined on the set {1, 2, 3, 4}. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is: [2025]

Q 18 :

Let $A = {(x, y) \in R \times R : | x + y | \geq 3}$ and $B = {(x, y) \in R \times R : | x | + | y | \leq 3}$

If $C = {(x, y) \in A \cap B : x = 0 or y = 0}$ , then $\sum_{(x, y) \in C}$ $| x + y |$ is: [2025]

Q 20 :

The relation R = {(x, y) : x, y $\in$ Z and x + y is even} is: [2025]

Q 21 :

Define a relation R on the interval $[0, \frac{π}{2})$ by xRy if and only if $\sec^{2} x - \tan^{2} y = 1$ . Then R is : [2025]

Q 22 :

Let $S = N \cup {0}$ . Define a relation R from S to R by:

$R = {(x, y) : \log_{e} y = x \log_{e} (\frac{2}{5}), x \in S, y \in R}$

Then, the sum of all the elements in the range of R is equal to: [2025]

Q 23 :

The number of relation on the set A = {1, 2, 3}, containing at most 6 elements including (1, 2), which are reflexive and transitive but not symmetric, is __________. [2025]

Q 24 :

Let A = {1, 2, 3}. The number of relations on A, containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is __________. [2025]

Q 26 :

Let $R$ be a relation defined on the set ${1, 2, 3, 4} \times {1, 2, 3, 4}$ by $R = {((a, b), (c, d)) : 2 a + 3 b = 3 c + 4 d} .$

Then the number of elements in $R$ is [2026]

Q 27 :

Let the relation R on the set $M = {1, 2, 3, \dots, 16}$ be given by $R = {(x, y) : 4 y = 5 x - 3, x, y \in M} .$

Then the minimum number of elements required to be added in R, in order to make the relation symmetric, is equal to [2026]

Q 29 :

Let A = {2,3,5,7,9} Let R be the relation on A defined by xRy if and only if $2 x \leq 3 y$ . Let $l$ be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then $l + m$ is equal to : [2026]

Q 30 :

The number of elements in the relation $R = {(x, y) : 4 x^{2} + y^{2} < 52, x, y \in ℤ}$ is [2026]

Q 31 :

Let $R_{1}$ and $R_{2}$ be two relations defined on $ℝ$ by $a R_{1} b \Leftrightarrow a b \geq 0$ and $a R_{2} b \Leftrightarrow a \geq b .$ Then

Q 32 :

Let $R$ be the relation on $N$ defined by $R = {(x, y) : x + 2 y = 8} .$ Then, the domain of $R$ is:

Q 33 :

The relation $R$ in the set of natural numbers $N$ defined as $R = {(x, y) : y = x + 5 and x < 4}$ is:

Q 34 :

If $A = {x \in Z : 0 \leq x \leq 12}$ and $R$ is the relation in $A$ given by $R = {(a, b) : a = b} .$ Then, the set of all elements related to 1 is:

Q 35 :

The range of the relation R = { $(x, x^{2}) : x$ is a prime number less than 13} is:

Q 36 :

R is a relation from {11, 12, 13} to {8, 10, 12} defined by $y = x - 3$ . Then $R^{- 1}$ is:

Q 37 :

Let $T$ be the set of all triangles in the Euclidean plane and let a relation $R$ on $T$ be defined as $a R b$ if $a$ is congruent to $b$ for all $a, b \in T$ , then $R$ is:

Q 38 :

Let us define a relation $R$ in $R$ as $a R b$ if $a \geq b$ . Then $R$ is:

Q 39 :

Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer:

Q 40 :

If $R = {(x, y) : x + 2 y = 8}$ is a relation on $N$ , then the range of $R$ is: