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

Q 21 :

Let R be a relation on the set N of natural numbers defined by $n R m$ if $n$ divides $m$ . Then R is:

Reflexive and symmetric
Transitive and symmetric
Equivalence
Reflexive, transitive but not symmetric

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Ans. Reflexive, transitive but not symmetric

Explanation:

Since $n$ divides $n, \forall n \in N, R$ is reflexive. $R$ is not symmetric since for $3, 6 \in N, 3 R 6 \neq 6 R 3$ .

$R$ is transitive since for $n, m, r$ , whenever $\frac{n}{m}$ and $\frac{m}{r}$ $\Rightarrow \frac{n}{r},$ i.e., $n$ divides $m$ and $m$ divides $r$ , then $n$ will divide $r$ .

Q 22 :

For real numbers $x$ and $y$ , define $x R y$ if and only if $x - y + \sqrt{2}$ is an irrational number. Then the relation $R$ is:

reflexive
symmetric
transitive
equivalence

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(1)

Ans. reflexive

Explanation:

$R_{x} = x - x + \sqrt{2} = \sqrt{2}$ is an irrational number.

Thus, $R$ is reflexive.

Also, $(\sqrt{2}, 1) \in R$ as $\sqrt{2} - 1 + \sqrt{2} = 2 \sqrt{2} - 1$ is an irrational number, but $(1, \sqrt{2}) \notin R$ as $1 - \sqrt{2} + \sqrt{2} = 1$ is a rational number. So $R$ is not symmetric. Since, ${}_{1}R_{2}$ and ${}^{\sqrt{2}}R_{\sqrt{2}}$ but 1 is not related to $\sqrt{2}$ . So, $R$ is not transitive.

Q 23 :

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 R to make it the smallest equivalence relation is:

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

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(2)

Ans. (3, 1)

Explanation:

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

$\Rightarrow R is reflexive.$

$(1, 3) \in R$

$(1, 3) \in R but (3, 1) \notin R$

$\Rightarrow R is not symmetric.$

To make $R$ an equivalence relation, we can add (3, 1).

Clearly, $R$ is reflexive and transitive. For $R$ to be symmetric, we should add (3, 1) in R.

Q 24 :

Which one of the following is an identity relation?

(1, 2), (2, 3), (1, 3)
(5, 5), (4, 4), (2, 2)
(1, 3), (3, 1), (2, 3)
None of the above

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(2)

Ans. (5, 5), (4, 4), (2, 2)

Explanation:

A relation is called an identity relation if

$A = {(a, a) : a \in A}$

So, (5, 5), (4, 4), (2, 2) is an identity relation.

Q 25 :

If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is:

reflexive
transitive
symmetric
none of these

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Ans. none of these

Explanation:

R on the set {1, 2, 3} be defined by R = {(1, 2)}

It is clear that R is not reflexive, transitive and symmetric.

Q 26 :

Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}

Then R is:

reflexive but not symmetric
reflexive but not transitive
symmetric and transitive
neither symmetric nor transitive

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(1)

Ans. reflexive but not symmetric

Explanation:

Given that,

A = {1, 2, 3}

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

$∵$ (1, 1), (2, 2), (3, 3) $\in$ R

Hence, R is reflexive.

(1, 2) $\in$ R but (2, 1) $\notin$ R

Hence, R is not symmetric.

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

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

Hence, R is transitive.

Q 27 :

Relation R on real numbers is defined as $R = {(a, b) : a \leq b} .$

Reflexive and symmetric but not transitive
Symmetric and transitive but not reflexive
Reflexive and transitive but not symmetric
Equivalence relation

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Ans. Reflexive and transitive but not symmetric

Explanation:

$R = {(a, b) : a \leq b}$

$(a, a) \in R$

$∵$ R is reflexive.

$(a, b), (b, c) \in R$

$a \leq b and b \leq c$

$a \leq c$

$(a, c) \in R$

Therefore, $R$ is transitive.

$(2, 4) \in R$

$(4, 2) \notin R$

Therefore $R$ is not symmetric.

Q 28 :

If $A = {a, b, c},$ $B = {b, c, d}$ and $C = {a, d, c}$ , then $(A - B) \times (B \cap C) =$

${(a, c), (a, d)}$
${(a, b), (c, d)}$
${(c, a), (d, a)}$
${(a, c), (a, d), (b, d)}$

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(1)

If $A = {a, b, c},$ $B = {b, c, d}$ and $C = {a, d, c}$ ,

$A - B = {a},$ $B \cap C = {c, d}$

then $(A - B) \times (B \cap C) = {a} \times {c, d} = {(a, c), (a, d)}$

Q 29 :

Let Y = {1, 2, 3, 4, 5}, A = {1, 2}, B = {3, 4, 5} and $ϕ$ denotes null set. If (A × B) denotes Cartesian product of the sets A and B, then (Y × A) ∩ (Y × B) is

Y
A
B
$ϕ$

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$Y \times A = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (3, 2), (4, 1), (4, 2), (5, 1), (5, 2)}$

$Y \times B = {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 3), (3, 4), (3, 5), (4, 3), (4, 4), (4, 5), (5, 3), (5, 4), (5, 5)}$

$∴$ $(Y \times A) \cap (Y \times B) = ϕ$

Q 30 :

Let $A = {x : x^{2} - 5 x + 6 = 0}$ , B = {2, 4}, C = {4, 5} then $A \times (B \cap C)$

{(2, 4), (3, 4)}
{(4, 2), (4, 3)}
{(2, 4), (3, 4), (4, 4)}
{(2, 2), (3, 3), (4, 4), (5, 5)}

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$A = {x : x^{2} - 5 x + 6 = 0}$

$x^{2} - 5 x + 6 = 0$

$\Rightarrow (x - 3) (x - 2) = 0 \Rightarrow x = 3, 2$

$A = {3, 2}, B = {2, 4}, C = {4, 5}, B \cap C = {4}$

$A \times (B \cap C) = {3, 2} \times {4} = {(3, 4), (2, 4)}$

Q 31 :

The cartesian product $A \times A$ has 9 elements among which two elements are found (−1, 0) and (0, 1), then set A = ?

{1, 0}
{1, −1, 0}
{0, −1}
{1, −1}

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$∵$ $A \times A has 9 elements$

$∴$ $Set A has exactly 3 elements.$

$Also (- 1, 0), (0, 1) \in A \times A \Rightarrow - 1, 0, 1 \in A$

$∴$ $A = {- 1, 0, 1}$

Q 32 :

If A and B be two sets such that $A \times B$ consists of 6 elements. If three elements of $A \times B$ are (1, 4), (2, 6) and (3, 6), find $B \times A$ .

{(1, 4), (1, 6), (2, 4), (2, 6), (3, 4), (3, 6)}
{(4, 1), (4, 2), (4, 3), (6, 1), (6, 2), (6, 3)}
{(4, 4), (6, 6)}
{(4, 1), (6, 2), (6, 3)}

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Since, (1, 4), (2, 6) and (3, 6) are the elements of A × B, therefore 1, 2, 3 are the elements of A and 4, 6 are the elements of B. Also, A × B has 6 elements.

$∴$ A = {1, 2, 3} and B = {4, 6}

$∴$ B × A = {(4, 1), (4, 2), (4, 3), (6, 1), (6, 2), (6, 3)}

Q 33 :

If A and B have $n$ elements in common, then the number of elements common to A × B and B × A is

$0$
$n$
$2 n$
$n^{2}$

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Q 34 :

A set A has 5 elements. Then the maximum number of relations on A (including empty relation) is

$5$
$2^{5}$
$2^{25}$
$25$

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Q 35 :

The function $f$ satisfies the functional equation $3 f (x) + 2 f (\frac{x + 59}{x - 1}) = 10 x + 30$ for all real $x \neq 1$ . The value of $f (7)$ is

8
4
- 8
11

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$Given functional equation is$

$3 f (x) + 2 f (\frac{x + 59}{x - 1}) = 10 x + 30$ ...(i)

$Put x = \frac{x + 59}{x - 1}, we get$

$3 f (\frac{x + 59}{x - 1}) + 2 f [\frac{\frac{x + 59}{x - 1} + 59}{\frac{x + 59}{x - 1} - 1}] = 10 (\frac{x + 59}{x - 1}) + 30$

$\Rightarrow$ $3 f (\frac{x + 59}{x - 1}) + 2 f (x) = \frac{40 x + 560}{x - 1}$ ...(ii)

$Multiply (i) by 3 and (ii) by 2 and then subtract, we get$

$9 f (x) - 4 f (x) = (30 x + 90) - (\frac{80 x + 1120}{x - 1})$

$\Rightarrow$ $5 f (x) = 10 [(3 x + 9) - (\frac{8 x + 112}{x - 1})]$

$\Rightarrow$ $f (x) = 2 [(3 x + 9) - (\frac{8 x + 112}{x - 1})]$

$Putting x = 7, we get$

$f (7) = 2 [30 - \frac{56 + 112}{6}] = 2 (30 - 28) = 4$

Q 36 :

If $f (x) = \log (\frac{1 + x}{1 - x})$ , $- 1 < x < 1$ , then $f (\frac{3 x + x^{3}}{1 + 3 x^{2}}) - f (\frac{2 x}{1 + x^{2}})$ is

${[f (x)]}^{3}$
${[f (x)]}^{2}$
$- f (x)$
$f (x)$

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$f (x) = \log (\frac{1 + x}{1 - x})$

$f (\frac{3 x + x^{3}}{1 + 3 x^{2}}) - f (\frac{2 x}{1 + x^{2}})$

$= \log (\frac{1 + \frac{3 x + x^{3}}{1 + 3 x^{2}}}{1 - \frac{3 x + x^{3}}{1 + 3 x^{2}}}) - \log (\frac{1 + \frac{2 x}{1 + x^{2}}}{1 - \frac{2 x}{1 + x^{2}}})$

$= \log (\frac{1 + 3 x^{2} + 3 x + x^{3}}{1 + 3 x^{2} - 3 x - x^{3}}) - \log (\frac{1 + x^{2} + 2 x}{1 + x^{2} - 2 x})$

$= \log {(\frac{1 + x}{1 - x})}^{3} - \log {(\frac{1 + x}{1 - x})}^{2}$

$= \log (\frac{{(1 + x)}^{3} \times {(1 - x)}^{2}}{{(1 - x)}^{3} \times {(1 + x)}^{2}}) = \log (\frac{1 + x}{1 - x}) = f (x)$

Q 37 :

Let A = {1, 2, 3} and B = {2, 3, 4}, then which of the following relations is a function from A to B?

{(1, 2), (2, 3), (3, 4), (2, 2)}
{(1, 2), (2, 3), (1, 3)}
{(1, 3), (2, 3), (3, 3)}
{(1, 1), (2, 3), (3, 4)}

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A relation is a function from set A to set B, if every element in A maps to one and only one element of B. So, relation {(1, 3), (2, 3), (3, 3)} is a function from A to B.

Q 38 :

If a relation R is defined from a set A = {2, 3, 4, 5} to a set B = {3, 6, 7, 10} as follows $(x, y) \in R \Leftrightarrow x$ divides $y$ . Expression of $R^{- 1}$ is represented by

{(6, 2), (10, 2), (3, 3), (6, 3)}
{(6, 2), (3, 3), (10, 5), (10, 2)}
{(6, 2), (10, 2), (3, 3), (6, 3), (10, 5)}
None of these

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Recall that $a ∣ b$ stands for $a$ divides $b$ . For the elements of the given sets $A$ and $B$ , we find $2 ∣ 6, 2 ∣ 10, 3 ∣ 3, 3 ∣ 6, 5 ∣ 10$

$\Rightarrow$ $R = {(2, 6), (2, 10), (3, 3), (3, 6), (5, 10)}$

$∴$ $R^{- 1} = {(6, 2), (10, 2), (3, 3), (6, 3), (10, 5)}$

Q 39 :

If $n (A) = 4, n (B) = 3, n (A \times B \times C) = 24$ , then $n (C) =$

2
1
12
17

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$n (A \times B \times C) = n (A) \times n (B) \times n (C)$

$∴$ $n (C) = \frac{n (A \times B \times C)}{n (A) \times n (B)} \Rightarrow n (C) = \frac{24}{4 \times 3} = 2$

$∴$ $n (C) = 2$

Q 40 :

If $n (A)$ denotes the number of elements in set $A$ and if $n (A) = 4$ , $n (B) = 5$ and $n (A \cap B) = 3$ , then $n [(A \times B) \cap (B \times A)] =$

8
9
10
11

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Given $n (A) = 4, n (B) = 5, n (A \cap B) = 3$

$n [(A \times B) \cap (B \times A)] = n [(A \cap B) \times (B \cap A)]$

$= n (A \cap B) \times n (B \cap A) = 3 \times 3 = 9$ $[∵ n (A \times B) = n (A) \times n (B)]$

Q 41 :

Let the number of elements of the sets $A$ and $B$ be $p$ and $q$ respectively. Then the number of the relations from the set $A$ to the set $B$ is

$2^{p + q}$
$2^{p q}$
$p + q$
$p q$

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Number of possible relations from $A$ to $B = 2^{p q}$

Q 42 :

If $R$ is a relation on a finite set having $n$ elements, then the number of relations on $A$ is

$2^{n}$
$2^{n^{2}}$
$n^{2}$
$n^{n}$

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Number of elements in set $A = n$

Number of elements in $A \times A = n \times n = n^{2}$

$∵$ Relations $R$ on $A$ are subsets of $A \times A$ .

$∴$ Number of relations $= 2^{n^{2}}$

Q 43 :

Suppose that the number of elements in set $A$ is $p$ , the number of elements in set $B$ is $q$ and the number of elements in $A \times B$ is 7 then $p^{2} + q^{2} =$ ________.

50
42
51
49

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We have, $n (A) = p, n (B) = q$

$n (A \times B) = p \times q \Rightarrow 7 = p q$

$\Rightarrow Either p = 1, q = 7 or q = 1, p = 7$

$∴$ $p^{2} + q^{2} = 1^{2} + 7^{2} = 50$

Q 44 :

If $n (A) = 5$ and $n (B) = 7$ , then the number of relations on $A \times B$ is $2^{k}$ , where unit digit of $k$ is ________.

(5)

$Total number of relations from A to B is 2^{m n} .$

$Here,$ $m = 5, n = 7$ .

$∴$ $Number of relations = 2^{5 \times 7} = 2^{35}$

Q 45 :

Consider the relation R on the set ${- 2, - 1, 0, 1, 2}$ defined by $(a, b) \in R$ if and only if 1 + ab > 0. Then, among the statements:

I. The number of elements in R is 17

II. R is an equivalence relation [2026]

Only I is true
Only II is true
Both I and II are true
Neither I nor II is true

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R on set ${- 2, - 1, 0, 1, 2}$ , if 1 + ab > 0

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

Number of elements = 17

Clearly, 'R' is reflexive and symmetric but not transitive,

Hence, not an equivalence relation.

$∴$ Only statement I is true.

Q 46 :

Let A = {2, 3, 4, 5, 6}. Let R be a relation on the set A x A given by (x, y) R (z, w) if and only if x divides z and y $\leq$ w. Then the number of elements in R is ________. [2026]

(120)

We have, A = {2, 3, 4, 5, 6}

We have, y $\leq$ w

$∙$ If y = 2, then w $\in$ {2, 3, 4, 5, 6}, Number of pairs = 5

$∙$ If y = 3, then w $\in$ {3, 4, 5, 6}, Number of pairs = 4

$∙$ If y = 5, then w $\in$ {4, 5, 6}, Number of pairs = 3

$∙$ If y = 5, then w $\in$ {5, 6}, Number of pairs = 2

$∙$ If y = 6, then w $\in$ {6}, Number of pairs = 1

Total pairs 5 + 4 + 3 + 2 + 1 = 15

Now, x divides z

$∴$ If x = 2, then z $\in$ {2, 4, 6}, Number of pairs = 3

If x = 3, then z $\in$ {3, 6}, Number of pairs = 2

If x = 4, then z $\in$ {4}, Number of pairs = 1

If x = 5, then z $\in$ {5}, Number of pairs = 1

If x = 6, then z $\in$ {6}, Number of pairs = 1

$∴$ Total pairs = 3 + 2 + 1 + 1 + 1 = 8

Total elements = 15 x 8 = 120.

Q 47 :

Let A = {1,4, 7} and B = {2, 3, 8}. Then the number of elements, in the relation $R = {((a_{1}, b_{1}), (a_{2} b_{2})) \in ((A \times B) \times (A \times B)) : a_{1} + b_{2} divides a_{2} + b_{1}}$ is ________. [2026]

(18)

Given, A = {1, 4, 7} and B = {2, 3,8}

All possible sums:

A/B	2	3	8
1	3	4	9
4	6	7	12
7	9	10	15

$a_{1} + b_{2}$	$a_{2} + b_{1}$	Number of pairs
15	15	1
12	12	1
10	10	1
9	9	4
7	7	1
6	6, 12	2
4	4, 12	2
3	3, 6, 9 12, 15	6

$∴$ Number of relations = 18

Q 48 :

Let $R = {(x, y) \in N \times N : \log_{e} (x + y) \leq 2}$ . Then the minimum number of elements, required to be added in R to make it a transitive relation, is ________. [2026]

(15)

Given : $\log_{e} (x + y) \leq 2$

$\Rightarrow x + y \leq e^{2}$

Since, $e^{2} \approx 7.29, so x + y \leq 7$

$∴$ R = {(1, 1), (1, 2), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (3, 1) (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (6, 1)}

For R to be transitive, we have to add (6, 2), (6, 3), (6, 4), (6, 5), (6, 6), (5, 3), (5, 4), (5, 5), (5, 6), (4, 4), (4,5), (4, 6), (3, 5),(3, 6), (2, 3) i.e., 15 elements

Q 49 :

Let R = ${(a, b) | a - b$ is irrational; $a, b$ are real numbers $}$ , then relation $R$ is

reflexive and symmetric relation
transitive and symmetric
symmetric relation
equivalence relation

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$(a, a) \notin R$

$If (a, b) \in R \Rightarrow (b, a) \in R$

$If (a, b) \in R, (b, c) \in R \Rightarrow (a, c) \notin R$

Q 50 :

Let R be a relation on real numbers given by $R = {(a, b) : 3 a - 3 b + \sqrt{7}$ is an irrational number $}$ . Then R is

Reflexive but neither symmetric nor transitive
Reflexive and transitive but not symmetric
Reflexive and symmetric but not transitive
An equivalence relation

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(1)

$Check for reflexivity: As 3 (a - a) + \sqrt{7} = \sqrt{7} which belongs to relation$

$So relation is reflexive.$

$Check for symmetric$

$Take a = \frac{\sqrt{7}}{3}, b = 0 Now (a, b) \in R but (b, a) \notin R$

$As 3 (b - a) + \sqrt{7} = 0 which is rational so relation is not symmetric$

$Check for Transitivity, Take (a, b) as (\frac{\sqrt{7}}{3}, 1) & (b, c) as (1, \frac{2 \sqrt{7}}{3})$

$So now (a, b) \in R & (b, c) \in R but (a, c) \notin R,$

$Which means relation is not transitive$

Topic Question Set

Q 21 :

Let R be a relation on the set N of natural numbers defined by $n R m$ if $n$ divides $m$ . Then R is:

Q 22 :

For real numbers $x$ and $y$ , define $x R y$ if and only if $x - y + \sqrt{2}$ is an irrational number. Then the relation $R$ is:

Q 23 :

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 R to make it the smallest equivalence relation is:

Q 24 :

Which one of the following is an identity relation?

Q 25 :

If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is:

Q 26 :

Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}

Then R is:

Q 27 :

Relation R on real numbers is defined as $R = {(a, b) : a \leq b} .$

Q 28 :

If $A = {a, b, c},$ $B = {b, c, d}$ and $C = {a, d, c}$ , then $(A - B) \times (B \cap C) =$

Q 29 :

Let Y = {1, 2, 3, 4, 5}, A = {1, 2}, B = {3, 4, 5} and $ϕ$ denotes null set. If (A × B) denotes Cartesian product of the sets A and B, then (Y × A) ∩ (Y × B) is

Q 30 :

Let $A = {x : x^{2} - 5 x + 6 = 0}$ , B = {2, 4}, C = {4, 5} then $A \times (B \cap C)$

Q 31 :

The cartesian product $A \times A$ has 9 elements among which two elements are found (−1, 0) and (0, 1), then set A = ?

Q 32 :

If A and B be two sets such that $A \times B$ consists of 6 elements. If three elements of $A \times B$ are (1, 4), (2, 6) and (3, 6), find $B \times A$ .

Q 33 :

If A and B have $n$ elements in common, then the number of elements common to A × B and B × A is

Q 34 :

A set A has 5 elements. Then the maximum number of relations on A (including empty relation) is

Q 35 :

The function $f$ satisfies the functional equation $3 f (x) + 2 f (\frac{x + 59}{x - 1}) = 10 x + 30$ for all real $x \neq 1$ . The value of $f (7)$ is

Q 36 :

If $f (x) = \log (\frac{1 + x}{1 - x})$ , $- 1 < x < 1$ , then $f (\frac{3 x + x^{3}}{1 + 3 x^{2}}) - f (\frac{2 x}{1 + x^{2}})$ is

Q 37 :

Let A = {1, 2, 3} and B = {2, 3, 4}, then which of the following relations is a function from A to B?

Q 38 :

If a relation R is defined from a set A = {2, 3, 4, 5} to a set B = {3, 6, 7, 10} as follows $(x, y) \in R \Leftrightarrow x$ divides $y$ . Expression of $R^{- 1}$ is represented by

Q 39 :

If $n (A) = 4, n (B) = 3, n (A \times B \times C) = 24$ , then $n (C) =$

Q 40 :

If $n (A)$ denotes the number of elements in set $A$ and if $n (A) = 4$ , $n (B) = 5$ and $n (A \cap B) = 3$ , then $n [(A \times B) \cap (B \times A)] =$

Q 41 :

Let the number of elements of the sets $A$ and $B$ be $p$ and $q$ respectively. Then the number of the relations from the set $A$ to the set $B$ is

Q 42 :

If $R$ is a relation on a finite set having $n$ elements, then the number of relations on $A$ is

Q 43 :

Suppose that the number of elements in set $A$ is $p$ , the number of elements in set $B$ is $q$ and the number of elements in $A \times B$ is 7 then $p^{2} + q^{2} =$ ________.

Q 44 :

If $n (A) = 5$ and $n (B) = 7$ , then the number of relations on $A \times B$ is $2^{k}$ , where unit digit of $k$ is ________.

Q 45 :

Consider the relation R on the set ${- 2, - 1, 0, 1, 2}$ defined by $(a, b) \in R$ if and only if 1 + ab > 0. Then, among the statements:

I. The number of elements in R is 17

II. R is an equivalence relation [2026]

Q 46 :

Let A = {2, 3, 4, 5, 6}. Let R be a relation on the set A x A given by (x, y) R (z, w) if and only if x divides z and y $\leq$ w. Then the number of elements in R is ________. [2026]

Q 47 :

Let A = {1,4, 7} and B = {2, 3, 8}. Then the number of elements, in the relation $R = {((a_{1}, b_{1}), (a_{2} b_{2})) \in ((A \times B) \times (A \times B)) : a_{1} + b_{2} divides a_{2} + b_{1}}$ is ________. [2026]

Q 48 :

Let $R = {(x, y) \in N \times N : \log_{e} (x + y) \leq 2}$ . Then the minimum number of elements, required to be added in R to make it a transitive relation, is ________. [2026]

Q 49 :

Let R = ${(a, b) | a - b$ is irrational; $a, b$ are real numbers $}$ , then relation $R$ is

Q 50 :

Let R be a relation on real numbers given by $R = {(a, b) : 3 a - 3 b + \sqrt{7}$ is an irrational number $}$ . Then R is

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Topic Question Set

Q 21 : Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then R is:

Q 22 : For real numbers x and y, define xRy if and only if x-y+2 is an irrational number. Then the relation R is:

Q 23 : 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 R to make it the smallest equivalence relation is:

Q 24 : Which one of the following is an identity relation?

Q 25 : If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is:

Q 26 : Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} Then R is:

Q 27 : Relation R on real numbers is defined as R={(a,b):a≤b}.

Q 28 : If A={a,b,c}, B={b,c,d} and C={a,d,c}, then (A-B)×(B∩C)=

Q 29 : Let Y = {1, 2, 3, 4, 5}, A = {1, 2}, B = {3, 4, 5} and ϕ denotes null set. If (A × B) denotes Cartesian product of the sets A and B, then (Y × A) ∩ (Y × B) is

Q 30 : Let A={x:x2−5x+6=0}, B = {2, 4}, C = {4, 5} then A×(B∩C)

Q 31 : The cartesian product A×A has 9 elements among which two elements are found (−1, 0) and (0, 1), then set A = ?

Q 32 : If A and B be two sets such that A×B consists of 6 elements. If three elements of A×B are (1, 4), (2, 6) and (3, 6), find B×A.

Q 33 : If A and B have n elements in common, then the number of elements common to A × B and B × A is

Q 34 : A set A has 5 elements. Then the maximum number of relations on A (including empty relation) is

Q 35 : The function f satisfies the functional equation 3f(x)+2f(x+59x-1)=10x+30 for all real x≠1. The value of f(7) is

Q 36 : If f(x)=log(1+x1-x), -1<x<1, then f(3x+x31+3x2)-f(2x1+x2) is

Q 37 : Let A = {1, 2, 3} and B = {2, 3, 4}, then which of the following relations is a function from A to B?

Q 38 : If a relation R is defined from a set A = {2, 3, 4, 5} to a set B = {3, 6, 7, 10} as follows (x,y)∈R⇔x divides y. Expression of R-1 is represented by

Q 39 : If n(A)=4, n(B)=3, n(A×B×C)=24, then n(C)=

Q 40 : If n(A) denotes the number of elements in set A and if n(A)=4, n(B)=5 and n(A∩B)=3, then n[(A×B)∩(B×A)]=

Q 41 : Let the number of elements of the sets A and B be p and q respectively. Then the number of the relations from the set A to the set B is

Q 42 : If R is a relation on a finite set having n elements, then the number of relations on A is

Q 43 : Suppose that the number of elements in set A is p, the number of elements in set B is q and the number of elements in A×B is 7 then p2+q2= ________.

Q 44 : If n(A)=5 and n(B)=7, then the number of relations on A×B is 2k, where unit digit of k is ________.

Q 45 : Consider the relation R on the set {–2,–1,0,1,2} defined by (a,b)∈R if and only if 1 + ab > 0. Then, among the statements: I. The number of elements in R is 17 II. R is an equivalence relation [2026]

Q 46 : Let A = {2, 3, 4, 5, 6}. Let R be a relation on the set A x A given by (x, y) R (z, w) if and only if x divides z and y ≤ w. Then the number of elements in R is ________. [2026]

Q 47 : Let A = {1,4, 7} and B = {2, 3, 8}. Then the number of elements, in the relation R={((a1,b1),(a2b2))∈((A×B)×(A×B)) : a1+b2 divides a2+b1} is ________. [2026]

Q 48 : Let R={(x,y)∈N×N:loge(x+y)≤2}. Then the minimum number of elements, required to be added in R to make it a transitive relation, is ________. [2026]

Q 49 : Let R = {(a,b)|a−b is irrational; a, b are real numbers}, then relation R is

Q 50 : Let R be a relation on real numbers given by R={(a,b):3a-3b+7 is an irrational number}. Then R is

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Q 21 :

Let R be a relation on the set N of natural numbers defined by $n R m$ if $n$ divides $m$ . Then R is:

Q 22 :

For real numbers $x$ and $y$ , define $x R y$ if and only if $x - y + \sqrt{2}$ is an irrational number. Then the relation $R$ is:

Q 23 :

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 R to make it the smallest equivalence relation is:

Q 24 :

Which one of the following is an identity relation?

Q 25 :

If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is:

Q 26 :

Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}

Then R is:

Q 27 :

Relation R on real numbers is defined as $R = {(a, b) : a \leq b} .$

Q 28 :

If $A = {a, b, c},$ $B = {b, c, d}$ and $C = {a, d, c}$ , then $(A - B) \times (B \cap C) =$

Q 29 :

Let Y = {1, 2, 3, 4, 5}, A = {1, 2}, B = {3, 4, 5} and $ϕ$ denotes null set. If (A × B) denotes Cartesian product of the sets A and B, then (Y × A) ∩ (Y × B) is

Q 30 :

Let $A = {x : x^{2} - 5 x + 6 = 0}$ , B = {2, 4}, C = {4, 5} then $A \times (B \cap C)$

Q 31 :

The cartesian product $A \times A$ has 9 elements among which two elements are found (−1, 0) and (0, 1), then set A = ?

Q 32 :

If A and B be two sets such that $A \times B$ consists of 6 elements. If three elements of $A \times B$ are (1, 4), (2, 6) and (3, 6), find $B \times A$ .

Q 33 :

If A and B have $n$ elements in common, then the number of elements common to A × B and B × A is

Q 34 :

A set A has 5 elements. Then the maximum number of relations on A (including empty relation) is

Q 35 :

The function $f$ satisfies the functional equation $3 f (x) + 2 f (\frac{x + 59}{x - 1}) = 10 x + 30$ for all real $x \neq 1$ . The value of $f (7)$ is

Q 36 :

If $f (x) = \log (\frac{1 + x}{1 - x})$ , $- 1 < x < 1$ , then $f (\frac{3 x + x^{3}}{1 + 3 x^{2}}) - f (\frac{2 x}{1 + x^{2}})$ is

Q 37 :

Let A = {1, 2, 3} and B = {2, 3, 4}, then which of the following relations is a function from A to B?

Q 38 :

If a relation R is defined from a set A = {2, 3, 4, 5} to a set B = {3, 6, 7, 10} as follows $(x, y) \in R \Leftrightarrow x$ divides $y$ . Expression of $R^{- 1}$ is represented by

Q 39 :

If $n (A) = 4, n (B) = 3, n (A \times B \times C) = 24$ , then $n (C) =$

Q 40 :

If $n (A)$ denotes the number of elements in set $A$ and if $n (A) = 4$ , $n (B) = 5$ and $n (A \cap B) = 3$ , then $n [(A \times B) \cap (B \times A)] =$

Q 41 :

Let the number of elements of the sets $A$ and $B$ be $p$ and $q$ respectively. Then the number of the relations from the set $A$ to the set $B$ is

Q 42 :

If $R$ is a relation on a finite set having $n$ elements, then the number of relations on $A$ is

Q 43 :

Suppose that the number of elements in set $A$ is $p$ , the number of elements in set $B$ is $q$ and the number of elements in $A \times B$ is 7 then $p^{2} + q^{2} =$ ________.

Q 44 :

If $n (A) = 5$ and $n (B) = 7$ , then the number of relations on $A \times B$ is $2^{k}$ , where unit digit of $k$ is ________.

Q 45 :

Consider the relation R on the set ${- 2, - 1, 0, 1, 2}$ defined by $(a, b) \in R$ if and only if 1 + ab > 0. Then, among the statements:

I. The number of elements in R is 17

II. R is an equivalence relation [2026]

Q 46 :

Let A = {2, 3, 4, 5, 6}. Let R be a relation on the set A x A given by (x, y) R (z, w) if and only if x divides z and y $\leq$ w. Then the number of elements in R is ________. [2026]

Q 47 :

Let A = {1,4, 7} and B = {2, 3, 8}. Then the number of elements, in the relation $R = {((a_{1}, b_{1}), (a_{2} b_{2})) \in ((A \times B) \times (A \times B)) : a_{1} + b_{2} divides a_{2} + b_{1}}$ is ________. [2026]

Q 48 :

Let $R = {(x, y) \in N \times N : \log_{e} (x + y) \leq 2}$ . Then the minimum number of elements, required to be added in R to make it a transitive relation, is ________. [2026]

Q 49 :

Let R = ${(a, b) | a - b$ is irrational; $a, b$ are real numbers $}$ , then relation $R$ is

Q 50 :

Let R be a relation on real numbers given by $R = {(a, b) : 3 a - 3 b + \sqrt{7}$ is an irrational number $}$ . Then R is