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

Q 1 :

Let A = {1, 2, 3, 4, 5, 6, 7}. Then the relation $R = {(x, y) \in A \times A : x + y = 7}$ is [2023]

an equivalence relation
reflexive but neither symmetric nor transitive
transitive but neither symmetric nor reflexive
symmetric but neither reflexive nor transitive

1

2

3

4

(4)

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

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

For Reflexive: Let $y = x$

So, $x + x = 7 \Rightarrow x = \frac{7}{2}$ , which is not possible.

So, the given relation is not reflexive.

For Symmetric: $x R y \Rightarrow x + y = 7$

$\Rightarrow$ $y + x = 7$ $∴$ $x R y \Rightarrow y R x$

Hence, the given relation is symmetric.

For Transitive: Let $x R y$ and $y R z \Rightarrow x + y = 7$ and $y + z = 7$ .

But it does not imply that $x + z = 7$ .

So, $R$ is not transitive.

Q 2 :

Let A = {2, 3, 4} and B = {8, 9, 12}. Then the number of elements in the relation

$R = {((a_{1}, b_{1}), (a_{2}, b_{2})) \in (A \times B, A \times B) : a_{1}$ divides $b_{2}$ and $a_{2}$ divides $b_{1}}$ is [2023]

24
12
36
18

1

2

3

4

(3)

$a_{1}$ divides $b_{2}$

Each element has 2 choices.

$\Rightarrow 3 \times 2 = 6$

$a_{2}$ divides $b_{1}$

Each element has 2 choices.

$\Rightarrow 3 \times 2 = 6$

$∴ Total number of relations is 6 \times 6 = 36$

Q 3 :

Let A = {1, 3, 4, 6, 9} and B = {2, 4, 5, 8, 10}. Let R be a relation defined on A x B such that $R = {((a_{1}, b_{1}), (a_{2}, b_{2})) : a_{1} \leq b_{2}$ and $b_{1} \leq a_{2}} .$ Then the number of elements in the set R is [2023]

52
180
26
160

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2

3

4

(4)

We have, $A = {1, 3, 4, 6, 9}$ ; $B = {2, 4, 5, 8, 10}$

$R = {((a_{1}, b_{1}), (a_{2}, b_{2})) : a_{1} \leq b_{2} and b_{1} \leq a_{2}}$

At $a_{1} = 1$ , there are 5 choices for $b_{2}$ ;

$a_{1} = 3$ , there are 4 choices for $b_{2}$ ;

$a_{1} = 4$ , there are 4 choices for $b_{2}$ ;

$a_{1} = 6$ , there are 2 choices for $b_{2}$ ;

$a_{1} = 9$ , there is 1 choice for $b_{2}$ ;

$∴$ $Total ways for (a_{1} \leq b_{2}) = 16$

Now, at $b_{1} = 2$ , there are 4 choices for $a_{2}$ ;

$b_{1} = 4$ , there are 3 choices for $a_{2}$ ;

$b_{1} = 5$ , there are 2 choices for $a_{2}$ ;

$b_{1} = 8$ , there is 1 choice for $a_{2}$ .

$Total ways for (b_{1} \leq a_{2}) = 10$

$∴ Required number of ways = 16 \times 10 = 160$

Q 4 :

Let R be a relation on R, given by

R = ${(a, b) : 3 a - 3 b + \sqrt{7}$ is an irrational number $} .$

Then R is [2023]

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

1

2

3

4

(3)

$R = {(a, b) : 3 a - 3 b + \sqrt{7}$ is an irrational number}

Reflexive: For $(a, a)$ , we have $3 a - 3 a + \sqrt{7} = \sqrt{7},$ which is an irrational number. $∴ R$ is reflexive.

Symmetric: Let $(a, b) \in R$ , i.e., $3 a - 3 b + \sqrt{7}$ is an irrational number.

Now, we need to check $(b, a) \in R$ or not.

Let $3 a = \sqrt{7}$ and $3 b = 8$

Then $3 a - 3 b + \sqrt{7} = \sqrt{7} - 8 + \sqrt{7} = 2 \sqrt{7} - 8,$ which is an irrational number.

But $3 b - 3 a + \sqrt{7} = 8 - \sqrt{7} + \sqrt{7} = 8,$ which is not an irrational number.

$∴$ For $(a, b) \in R$ , $(b, a) \notin R .$

$∴ R$ is not symmetric.

Transitive: Let $(a, b)$ and $(b, c) \in R .$

Let $3 a = 8, 3 b = 2 \sqrt{7}, 3 c = \sqrt{7}$

Then $3 a - 3 b + \sqrt{7} = 8 - 2 \sqrt{7} + \sqrt{7} = 8 - \sqrt{7},$ which is an irrational number.

Also, $3 b - 3 c + \sqrt{7} = 2 \sqrt{7} - \sqrt{7} + \sqrt{7} = 2 \sqrt{7},$ which is an irrational number.

But $3 a - 3 c + \sqrt{7} = 8 - \sqrt{7} + \sqrt{7} = 8,$ which is not an irrational $\Rightarrow (a, c) \notin R .$

$∴$ $R$ is not transitive.

Q 5 :

Let P(S) denote the power set of S = {1, 2, 3, ...., 10}. Define the relations $R_{1}$ and $R_{2}$ on P(S) as $A R_{1} B$ if $(A \cap B^{C}) \cup (B \cap A^{C}) = ϕ$ and $A R_{2} B$ if $A \cup B^{C} = B \cup A^{C}, \forall A, B \in P (S)$ . Then [2023]

both $R_{1}$ and $R_{2}$ are not equivalence relations
only $R_{2}$ is an equivalence relation
only $R_{1}$ is an equivalence relation
both $R_{1}$ and $R_{2}$ are equivalence relations

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3

4

(4)

Q 6 :

The relation $R = {(a, b) : g c d (a, b) = 1, 2 a \neq b, a, b, \in Z}$ is [2023]

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

1

2

3

4

(4)

Reflexive : $\gcd (a, a) = a \neq 1 for a \neq 1 .$

Symmetric: Let $a = 2, b = 1$

$\gcd (a, b) = 1; b \neq 2 a$

Thus, $(2, 1) \in R; \gcd (1, 2) = 1 .$ But $2 = 2 \times 1$

$Therefore (1, 2) \notin R . Hence, R is not symmetric.$

Transitive: $(2, 3) \in R$

$(3, 8) \in R .$ But $(2, 8) \notin R .$

$Hence R is not transitive.$

$R is neither symmetric nor transitive.$

Q 7 :

Let R be a relation defined on $ℕ$ as $a R b$ if $2 a + 3 b$ is a multiple of 5, $a, b \in ℕ$ . [2023]

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

1

2

3

4

(3)

If 'R' be a relation defined on $N$ as $a R b$ is $2 a + 3 b$ is a multiple of 5, $a, b \in N$ .

(i) $a R a \Rightarrow 5 a$ is a multiple of 5

R is a reflexive relation.

(ii) $a R b, 2 a + 3 b = 5 α$ (let)

Now, $b R a \Rightarrow 2 b + 3 a = 2 b + (\frac{5 α - 3 b}{2}) \cdot 3$

$= \frac{15}{2} α - \frac{5}{2} b = \frac{5}{2} (3 α - b) = \frac{5}{2} (2 a + 2 b - 2 α) = 5 (a + b - α)$

So, R is a symmetric relation.

(iii) $a R b \Rightarrow 2 a + 3 b = 5 α$ ; $b R c \Rightarrow 2 b + 3 c = 5 β$

Now, $2 a + 5 b + 3 c = 5 (α + β)$

$\Rightarrow 2 a + 5 b + 3 c = 5 (α + β) or 2 a + 3 c = 5 (α + β - b)$

$\Rightarrow a R c$

So, R is a transitive relation.

Hence Relation 'R' is an equivalence relation.

Q 8 :

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 [2023]

3
7
4
5

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2

3

4

(2)

Given relation, $R = {(a, b), (b, c)}$ and set = ${a, b, c}$

For symmetric, since $(a, b), (b, c) \in R$

So, $(b, a), (c, b)$ be in R. For transitive, since $(a, b), (b, c) \in R$

So, $(a, c)$ should be in R. Then $(c, a)$ should also be in R since $(a, c)$ lies in R.

Since $(a, b), (b, a) \in R$ , so $(a, a) \in R$ should also lie in R.

Since $(c, b), (b, c) \in R$ , so $(c, c)$ should also lie in R.

Since $(b, c), (c, b) \in R$ , so $(b, b)$ should also lie in R.

$∴$ Elements to be added are:

$(b, a), (c, b), (a, c), (c, a), (a, a), (c, c), (b, b)$ .

Total number of elements to be added = 7.

Q 9 :

Let R be a relation on N x N defined by (a, b) R(c, d) if and only if $a d (b - c) = b c (a - d)$ . Then R is [2023]

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

1

2

3

4

(2)

We have, $(a, b) R (c, d) \Leftrightarrow a d (b - c) = b c (a - d)$

$\Rightarrow \frac{b - c}{b c} = \frac{a - d}{a d} and \frac{1}{c} - \frac{1}{b} = \frac{1}{d} - \frac{1}{a} \Rightarrow \frac{1}{a} - \frac{1}{b} = \frac{1}{d} - \frac{1}{c}$

For reflexive: $(a, b) R (a, b) \Rightarrow \frac{1}{a} - \frac{1}{b} = \frac{1}{b} - \frac{1}{a}$ which is false.

Hence, it is not reflexive.

For symmetric: $(a, b) R (c, d) \Rightarrow \frac{1}{a} - \frac{1}{b} = \frac{1}{d} - \frac{1}{c}$

$\Rightarrow \frac{1}{c} - \frac{1}{d} = \frac{1}{b} - \frac{1}{a} \Rightarrow (c, d) R (a, b)$

Hence, it is symmetric.

For transitive: $(a, b) R (c, d) \Rightarrow \frac{1}{a} - \frac{1}{b} = \frac{1}{d} - \frac{1}{c}$

$and (c, d) R (e, f) \Rightarrow \frac{1}{c} - \frac{1}{d} = \frac{1}{f} - \frac{1}{e}$

$\Rightarrow \frac{1}{a} - \frac{1}{b} = \frac{1}{e} - \frac{1}{f} \neq (a, b) R (e, f)$

Hence, it is not transitive.

Q 10 :

Among the relations

$S = {(a, b) : a, b \in R - {0}, 2 + \frac{a}{b} > 0}$ and

$T = {(a, b) : a, b \in R, a^{2} - b^{2} \in Z}$ [2023]

S is transitive but T is not
both S and T are symmetric
neither S nor T is transitive
T is symmetric but S is not

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2

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(4)

For relation $T = a^{2} - b^{2} \in I$

Then, $(b, a)$ on relation $R \Rightarrow b^{2} - a^{2} \in I$

$∴ T is symmetric,$ $S = {(a, b) : a, b \in R - {0}, 2 + \frac{a}{b} > 0}$

$2 + \frac{a}{b} > 0 \Rightarrow \frac{a}{b} > - 2 \Rightarrow \frac{b}{a} > - \frac{1}{2}$

If $(b, a) \in S$ , then $2 + \frac{b}{a}$ is not necessarily positive.

$∴$ $S is not symmetric.$

Q 11 :

Let A = {1, 2, 3, 4, .... ,10} and B = {0, 1, 2, 3, 4}. The number of elements in the relation $R = {(a, b) \in A \times A : 2 {(a - b)}^{2} + 3 (a - b) \in B}$ is __________ . [2023]

(18)

$A = {1, 2, 3, 4, \dots, 10}$ ; $B = {0, 1, 2, 3, 4}$

$R =$ ${(a, b) \in A \times A : 2 {(a - b)}^{2} + 3 (a - b) \in B}$

$Now, 2 {(a - b)}^{2} + 3 (a - b) = (a - b) [2 (a - b) + 3] \in B$

$\Rightarrow a = b or a - b = - 2$

$When a = b \Rightarrow 10 ordered pairs$

$When a - b = - 2 \Rightarrow 8 ordered pairs. Total = 18$

Q 12 :

Let A = {0, 3, 4, 6, 7, 8, 9, 10} and R be the relation defined on A such that $R = {(x, y) \in A \times A : x - y$ is odd positive integer or $x - y = 2} .$ The minimum number of elements that must be added to the relation R, so that it is a symmetric relation, is equal to __________ . [2023]

(19)

$A = {0, 3, 4, 6, 7, 8, 9, 10}$

$R = {(x, y) \in A \times A : x - y$ is an odd positive integer or $x - y = 2$ }

$∴$ Possible pairs to be added are

${(0, 3), (0, 7), (0, 9), (3, 4), (3, 6), (3, 8), (3, 10), (4, 7), (4, 9), (6, 7), (6, 9), (7, 8), (7, 10), (8, 9), (9, 10), (4, 6), (6, 8), (7, 9), (8, 10)}$

$∴$ $We need to add a minimum of 19 elements to form it symmetric.$

Q 13 :

The number of relations, on the set {1, 2, 3} containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _________ . [2023]

(3)

Let we have set $A = {1, 2, 3}$

$By definition, we can say that$

$For reflexivity: (1, 1), (2, 2), (3, 3) \in R ...(i)$

$For symmetry: (2, 1), (1, 2) \in R ...(ii)$

$For transitivity: (1, 2) \in R and (2, 3) \in R \Rightarrow (1, 3) \in R ...(iii)$

$But according to the question,$

$For not symmetric: (2, 1) and (3, 2) \notin R ...(iv)$

$So, R_{1} = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}$

$R_{2} = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3), (2, 1)}$

$and R_{3} = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3), (3, 2)}$

Q 14 :

Let A = {-4, -3, -2, 0, 1, 3, 4} and $R = {(a, b) \in A \times A : b = | a | o r b^{2} = a + 1}$ be a relation on A.

Then the minimum number of elements, that must be added to the relation R so that it becomes reflexive and symmetric, is ________ . [2023]

(7)

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

$For reflexive, number of elements added to the relation R is (- 2, 2), (- 4, - 4), (- 3, - 3)$

$For symmetric, number of elements added to the relation is (4, - 4), (3, - 3), (- 2, 3), (1, 0)$

$So, total number of elements = 3 + 4 = 7$

Q 15 :

Let A = {1, 2, 3, 4} and R be a relation on the set A x A defined by $R = {((a, b), (c, d)) : 2 a + 3 b = 4 c + 5 d} .$

Then the number of elements in R is ___________ . [2023]

(6)

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

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

$R = {((a, b), (c, d)) : 2 a + 3 b = 4 c + 5 d}$

$On checking the elements of A \times A, we get:$

${((1, 4), (1, 2)), ((2, 3), (2, 1)), ((3, 1), (1, 1)), ((3, 4), (2, 2)), ((4, 2), (1, 2)), ((4, 3), (3, 1))} will satisfy R .$

$∴ Number of elements in R = 6$

Q 16 :

The minimum number of elements that must be added to the relation R = {(a, b), (b, c), (b, d)} on the set {a, b, c, d} so that it is an equivalence relation is _____________ . [2023]

(13)

Given $S = {(a, b), (b, c), (b, d)}$

$\begin{matrix} Reflexive : & (a, a), & (b, b), & (c, c), & (d, d) \\ Symmetric : & (b, a), & (c, b), & (d, b) \\ Transitive : & (a, c), & (a, d), & (d, c) \\ ↓ & ↓ & ↓ \\ (c, a) & (d, a) & (c, d) \end{matrix}$

So, 13 elements must be added.

Q 17 :

Let a relation R on $N \times N$ be defined as:

$(x_{1}, y_{1}) R (x_{2}, y_{2})$ if and only if $x_{1} \leq x_{2} or y_{1} \leq y_{2} .$

Consider the two statements:

(I) R is reflexive but not symmetric.

(II) R is transitive.

Then which one of the following is true? [2024]

Both (I) and (II) are correct.
Neither (I) nor (II) is correct.
Only (I) is correct.
Only (II) is correct.

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(3)

$(x_{1}, y_{1}) R (x_{2}, y_{2})$

$\Rightarrow x_{1} \leq x_{2} or y_{1} \leq y_{2}$

For reflexive :

$(x_{1}, y_{1}) R (x_{1}, y_{1})$

$\Rightarrow x_{1} \leq x_{1} or y_{1} \leq y_{1} which is true.$

So, R is reflexive.

For symmetric:

$When (x_{1}, y_{1}) R (x_{2}, y_{2})$

$\Rightarrow x_{1} \leq x_{2} or y_{1} \leq y_{2}$

$⇏ x_{2} \leq x_{1} or y_{2} \leq y_{1}$

They may or may not be true.

For example (1, 2) and (3, 4)

$1 \leq 3 and 2 \leq 4 but 3 ≰ 1 and 4 ≰ 2 .$

$∴$ R is not symmetric.

For transitive :

Take pairs as (3, 9), (4, 6), (2, 7)

$(3, 9) R (4, 6) as 4 \geq 3$

$(4, 6) R (2, 7) as 7 \geq 6$

$But (3, 9) R (n o t) (2, 7), as neither 2 \geq 3 nor 7 \geq 9$

So, R is not transitive.

Q 18 :

Let the relations $R_{1}$ and $R_{2}$ on the set $X = {1, 2, 3, . . ., 20}$ be given by $R_{1} = {(x, y) : 2 x - 3 y = 2}$ and $R_{2} = {(x, y) : - 5 x + 4 y = 0}$ . If M and N be the minimum number of elements required to be added in $R_{1}$ and $R_{2}$ , respectively, in order to make the relations symmetric, then M+N equals [2024]

10
8
16
12

1

2

3

4

(1)

$R_{1} = {(x, y) : 2 x - 3 y = 2}$

$R_{1} = {(4, 2), (7, 4), (10, 6), (13, 8), (16, 10), (19, 12)}$

So, 6 elements are needed to make $R_{1}$ symmetric

$\Rightarrow M = 6$

$R_{2} = {(x, y) : - 5 x + 4 y = 0}$

$R_{2} = {(4, 5), (8, 10), (12, 15), (16, 20)}$

So, 4 elements are needed to make $R_{2}$ symmetric

$\Rightarrow N = 4$

$∴ M + N = 6 + 4 = 10$

Q 19 :

Let $A = {1, 2, 3, 4, 5}$ . Let R be a relation on A defined by $x R y$ if and only if $4 x \leq 5 y$ . Let m be the number of elements in R and n be the minimum number of elements from $A \times A$ that are required to be added to R to make it a symmetric relation. Then m+n is equal to: [2024]

24
26
25
23

1

2

3

4

(3)

$Given, A = {1, 2, 3, 4, 5}$

$R = {(x, y) : 4 x \leq 5 y, x, y \in A}$

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

$∴ n (R) = 16 = m$

$Elements needed to make R symmetric = {(2, 1), (3, 1), (4, 1), (5, 1), (3, 2), (4, 2), (5, 2), (4, 3), (5, 3)} i.e., 9 elements$

$∴ n = 9$

$So, m + n = 16 + 9 = 25$

Q 20 :

Let A = {2, 3, 6, 8, 9, 11} and B = {1, 4, 5, 10, 15}. Let R be a relation on $A \times B$ defined by (a, b) R(c, d) if and only if 3ad−7bc is an even integer. Then the relation R is [2024]

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

1

2

3

4

(2)

We have, $(a, b) R (c, d) \Rightarrow 3 a d - 7 b c$ is an even integer.

For reflexive : $(a, b) R (a, b) \Rightarrow 3 a b - 7 a b = - 4 a b$ , which is an even integer.

For symmetric, $(a, b) R (c, d) \Rightarrow 3 a d - 7 b c$ is an even integer.

$\Rightarrow 3 a d - 7 b c + 4 b c + 4 a d$ is also an even integer

( $∵$ even + even = even number)

$\Rightarrow 7 a d - 3 b c$ is an even integer

$\Rightarrow 3 c b - 7 d a$ is also an even integer

$\Rightarrow (c, d) R (a, b)$

For transitive, $(a, b) R (c, d) a n d (c, d) R (e, f)$

$\Rightarrow 3 a d - 7 b c$ and $3 c f - 7 d e$ is an even integer.

For $a = 2, b = 5, c = 6, d = 8, e = 9, f = 1$

$3 a f - 7 b c = 3 \times 2 \times 1 - 7 \times 5 \times 9 = 6 - 315 = - 309,$ which is not an even integer.

$∴$ Given relation is not transitive.

Q 21 :

Consider the relations $R_{1}$ and $R_{2}$ defined as $a R_{1} b \Leftrightarrow a^{2} + b^{2} = 1$ for all $a, b \in R$ and $(a, b) R_{2} (c, d) \Leftrightarrow a + d = b + c$ for all $(a, b), (c, d) \in N \times N .$ Then [2024]

$R_{1}$ and $R_{2}$ both are equivalence relations
Only $R_{1}$ is an equivalence relation
Only $R_{2}$ is an equivalence relation
Neither $R_{1}$ nor $R_{2}$ is an equivalence relation

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(3)

$Consider, (x, x) \in R_{1}$

$Then, x^{2} + x^{2} \neq 1 for any x \in R$

$∴$ $R_{1} is not reflexive$

$Hence, R_{1} is not an equivalence relation .$

$For R_{2}$

$(i) L e t (x, y) R (x, y) \Rightarrow x + y = y + x, which is true for all x, y \in N .$

$∴ R_{2} is reflexive .$

$(i i) Let (x, y) R (z, p) \Rightarrow x + p = y + z \Rightarrow p + x = z + y$

$\Rightarrow z + y = p + x \Rightarrow (z, p) R (x, y)$

$∴ R_{2} i s s y m m e t r i c .$

$(i i i) Let (a, b) R (c, d) \Rightarrow a + d = b + c$ ...(i)

$(c, d) R (e, f) \Rightarrow c + f = d + e$ ...(ii)

$From (i) and (ii), we have a + d + c + f = b + c + d + e$

$a + f = b + e \Rightarrow (a, b) R (e, f)$

$∴ R_{2} is transitive .$

$S o, R_{2} is an equivalence relation .$

Q 22 :

Let S = {1, 2, 3,…,10}. Suppose M is the set of all the subsets of S, then the relation $R = {(A, B) : A \cap B \neq ϕ; A, B \in M}$ is: [2024]

symmetric and transitive only
reflexive only
symmetric and reflexive only
symmetric only

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(4)

Given, $A \in M \Rightarrow A \cap A = ϕ, if A = ϕ$

So, R is not reflexive.

If $(A, B) \in R \Rightarrow A \cap B \neq ϕ \Rightarrow B \cap A \neq ϕ \Rightarrow (B, A) \in R$

So, R is symmetric.

If $(A, B)$ and $(B, C) \in R \Rightarrow A \cap B \neq ϕ$ and $B \cap C \neq ϕ$

$\Rightarrow A \cap C$ is not necessarily non-empty set.

Hence, $A \cap C$ not necessarily belongs to R.

So, R is not a transitive relation.

$∴$ Given relation is symmetric and reflexive only.

Q 23 :

Let R be a relation on $Z \times Z$ defined by (a, b) R (c, d) if and only if $a d - b c$ is divisible by 5. Then R is [2024]

Reflexive but neither symmetric nor transitive
Reflexive, symmetric and transitive
Reflexive and transitive but not symmetric
Reflexive and symmetric but not transitive

1

2

3

4

(4)

$Reflexive : For (a, b) R (a, b)$

$\Rightarrow a b - a b = 0 is divisible by 5 .$

$So, (a, b) R (a, b)$ $\forall a, b \in Z$

$∴$ $R is reflexive .$

$Symmetric : For (a, b) R (c, d), if a d - b c is divisible by 5 .$

$Then, b c - a d is also divisible by 5 .$

$So, (c, d) R (a, b) \forall a, b, c, d \in Z$

$∴ R$ $is symmetric .$

$Transitive : For (a, b) R (c, d) \Rightarrow a d - b c is divisible by 5 and (c, d) R (e, f) \Rightarrow c f - d e is divisible by 5$

$L e t a d - b c = 5 k_{1} and c f - d e = 5 k_{2}, where k_{1} and k_{2} are integers .$

$∴ a d f - b c f = 5 k_{1} f$ ...(i)

$and c f b - d e b = 5 k_{2} b$ ...(ii)

$Solving (i) and (ii), we get$

$a d f - b c f + c f b - d e b = 5 k_{1} f + 5 k_{2} b$

$\Rightarrow a d f - d e b = 5 (k_{1} f + k_{2} b) \Rightarrow d (a f - b e) = 5 (k_{1} f + k_{2} b)$

$a f - b e is not divisible by 5 for \forall a, b, c, d, e, f \in Z$

$So, R is not transitive .$

$∴$ $R is Reflexive and symmetric but not transitive .$

Q 24 :

If 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 is _______ . [2024]

12
15
10
8

1

2

3

4

(3)

To make R an equivalence relation, R should be reflexive, symmetric, and transitive.

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

So, minimum number of elements in R should be 10.

Q 25 :

Let A = {2, 3, 6, 7} and B = {4, 5, 6, 8}. Let R be a relation defined on $A \times B$ by $(a_{1}, b_{1}) R (a_{2}, b_{2})$ if and only if $a_{1} + a_{2} = b_{1} + b_{2}$ . Then the number of elements in R is ______________. [2024]

(25)

$Given, A = {2, 3, 6, 7} and B = {4, 5, 6, 8}$

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

$\begin{matrix} (2, 4) R (6, 4) & (3, 6) R (7, 4) \\ (2, 4) R (7, 5) & (3, 5) R (7, 5) \\ (2, 5) R (7, 4) & (6, 5) R (7, 8) \\ (3, 4) R (6, 5) & (6, 8) R (7, 5) \\ (3, 5) R (6, 4) & (7, 6) R (7, 8) \\ (3, 4) R (7, 6) & (6, 4) R (6, 8) \\ (6, 6) R (6, 6) \end{matrix}] \times 2$

Hence, total number of elements = $13 \times 2 - 1 = 25$

Q 26 :

Let A = {1, 2, 3,…,20}. Let $R_{1}$ and $R_{2}$ be two relation on A such that $R_{1}$ = {(a, b) : b is divisible by a} $R_{2}$ = {(a, b) : a is an integral multiple of b} Then, number of elements in $R_{1} - R_{2}$ is equal to _______ . [2024]

(46)

$R_{1}$ = {(1, 1), (1, 2)…(1, 20), (2, 2), (2, 4),…(2, 20), (3, 3), (3, 6)…(3, 18), (4, 4), (4, 8),…(4, 20), (5, 5), (5, 10), (5, 15), (5, 20), (6, 6), (6, 12), (6, 18), (7, 7), (7, 14), (8, 8), (8, 16), (9, 9), (9, 18), (10, 10), (10, 20), (11, 11), (12, 12), (13, 13), (14, 14), (15, 15), (16, 16), (17, 17), (18, 18), (19, 19), (20, 20)}

$R_{2}$ = {(20, 1), (20, 2), (20, 4), (20, 5), (20, 10), (20, 20), (19, 19), (19, 1), (18, 1), (18, 2), (18, 3), (18, 6), (18, 9), (18, 18), (17, 1), (17, 17), (16, 1), (16, 2), (16, 4), (16, 8), (16, 16), (15, 1), (15, 3), (15, 5), (15, 15), (14, 1), (14, 2), (14, 7), (14, 14), (13, 1), (13, 13), (12, 1), (12, 2), (12, 3), (12, 4), (12, 6), (12, 12), (11, 1), (11, 11), (10, 1), (10, 2), (10, 5), (10, 10), (9, 1), (9, 3), (9, 9), (8, 1), (8, 2), (8, 4), (8, 8), (7, 1), (7, 7), (6, 1), (6, 2), (6, 3), (6, 6), (5, 1), (5, 5), (4, 1), (4, 2), (4, 4), (3, 1), (3, 3), (2, 1), (2, 2), (1, 1)}

$R_{1} \cap R_{2}$ = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (7, 7), (8, 8), (9, 9), (10, 10), (11, 11), (12, 12), (13, 13), (14, 14), (15, 15), (16, 16), (17, 17), (18, 18), (19, 19), (20, 20)}

Number of elements in $R_{1} - R_{2} = 66 - 20 = 46$ elements.

Q 27 :

The number of symmetric relations defined on the set {1, 2, 3, 4} which are not reflexive is _______. [2024]

(960)

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

$∴ n (A) = 4$

$∴$ Number of reflexive and symmetric relation $= 2^{\frac{n^{2} - n}{2}} = 2^{6}$

Number of symmetric relations $= 2^{\frac{n (n + 1)}{2}} = 2^{10}$

$∴$ Number of relations which is symmetric but not reflexive $= 2^{10} - 2^{6} = 1024 - 64 = 960$

Q 28 :

Let A = {1, 2, 3, 4} and R = {(1, 2), (2, 3), (1, 4)} be a relation on A. Let S be the equivalence relation on A such that $R \subset S$ and the number of elements in S is $n$ . Then, the minimum value of $n$ is ________. [2024]

(16)

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

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

S to be equivalence, it should be reflexive, symmetric and transitive.

For reflexive add (1, 1), (2, 2), (3, 3), (4, 4).

For symmetric add (2, 1), (3, 2), (4, 1).

For transitive (1, 2), (2, 3) $\Rightarrow$ (1, 3), so add (1, 3) also add (3, 1) for symmetric

and (4, 1), (1, 2) $\Rightarrow$ (4, 2), so add (4, 2) also add (2, 4) for symmetric.

∴ S = {(1, 2), (2, 3), (1, 4), (1, 1), (2, 2), (3, 3), (4, 4), (2, 1), (3, 2), (4, 1), (3, 1), (1, 3), (2, 4), (4, 2), (3, 4), (4, 3)}

∴ n(S) = 16

Q 29 :

Let A = {1, 2, 3, ..........., 100}. Let R be a relation on A defined by (x, y) ∈ R if and only if 2x = 3y. Let $R_{1}$ be a symmetric relation on A such that $R \subset R_{1}$ and the number of elements in $R_{1}$ is $n$ . Then, the minimum value of $n$ is ______. [2024]

(66)

We have, A = {1, 2, 3, ..., 100}

R = {(3, 2), (6, 4), (9, 6), (12, 8), ....(99, 66)}

n(R) = 33

Since, $R_{1}$ is symmetric and $R \subset R_{1}$

∴ n( $R_{1}$ ) = 66

Q 30 :

Let A be the set of all functions f : Z $\to$ Z and R be a relation on A such that R = {(f, g) : f(0) = g(1) and f(1) = g(0)}. Then R is : [2025]

Transitive but neither reflexive nor symmetric.
Symmetric and transitive but not reflexive.
Symmetric but neither reflexive nor transitive.
Reflexive but neither symmetric nor transitive.

1

2

3

4

(3)

We have, R = {(f, g) : f(0) = g(1) and f(1) = g(0)}

For reflexive: (f, f) $\in$ R

$\Rightarrow$ f(0) = f(1) must hold $\forall$ f

But f(0) $\neq$ f(1)

Therefore, R is not reflexive.

For symmetric: If (f, g) $\in$ R $\Rightarrow$ (g, f) $\in$ R

Now, g(0) = f(1) and g(1) = f(0), which is true $\forall$ f

$∴$ R is symmetric

For transitive : IF (f, g) $\in$ R and (g, h) $\in$ R $\Rightarrow$ (f, h) $\in$ R

Now, (f, g) $\in$ R $\Rightarrow$ f(0) = g(1) and f(1) = g(0)

(g, h) $\in$ R $\Rightarrow$ g(0) = h(1) and g(1) = h(0)

For (f, h) $\in$ R, we need f(0) = h(1) and f(1) = h(0)

But, f(0) = g(1) = h(0) $\neq$ h(1) and f(1) = g(0) = h(1) $\neq$ h(0)

Hence, R is not transitive.

Topic Question Set

Q 1 :

Let A = {1, 2, 3, 4, 5, 6, 7}. Then the relation $R = {(x, y) \in A \times A : x + y = 7}$ is [2023]

Q 2 :

Let A = {2, 3, 4} and B = {8, 9, 12}. Then the number of elements in the relation

$R = {((a_{1}, b_{1}), (a_{2}, b_{2})) \in (A \times B, A \times B) : a_{1}$ divides $b_{2}$ and $a_{2}$ divides $b_{1}}$ is [2023]

Q 3 :

Let A = {1, 3, 4, 6, 9} and B = {2, 4, 5, 8, 10}. Let R be a relation defined on A x B such that $R = {((a_{1}, b_{1}), (a_{2}, b_{2})) : a_{1} \leq b_{2}$ and $b_{1} \leq a_{2}} .$ Then the number of elements in the set R is [2023]

Q 4 :

Let R be a relation on R, given by

R = ${(a, b) : 3 a - 3 b + \sqrt{7}$ is an irrational number $} .$

Then R is [2023]

Q 5 :

Let P(S) denote the power set of S = {1, 2, 3, ...., 10}. Define the relations $R_{1}$ and $R_{2}$ on P(S) as $A R_{1} B$ if $(A \cap B^{C}) \cup (B \cap A^{C}) = ϕ$ and $A R_{2} B$ if $A \cup B^{C} = B \cup A^{C}, \forall A, B \in P (S)$ . Then [2023]

(4)

Q 6 :

The relation $R = {(a, b) : g c d (a, b) = 1, 2 a \neq b, a, b, \in Z}$ is [2023]

Q 7 :

Let R be a relation defined on $ℕ$ as $a R b$ if $2 a + 3 b$ is a multiple of 5, $a, b \in ℕ$ . [2023]

Q 8 :

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 [2023]

Q 9 :

Let R be a relation on N x N defined by (a, b) R(c, d) if and only if $a d (b - c) = b c (a - d)$ . Then R is [2023]

Q 10 :

Among the relations

$S = {(a, b) : a, b \in R - {0}, 2 + \frac{a}{b} > 0}$ and

$T = {(a, b) : a, b \in R, a^{2} - b^{2} \in Z}$ [2023]

Q 11 :

Let A = {1, 2, 3, 4, .... ,10} and B = {0, 1, 2, 3, 4}. The number of elements in the relation $R = {(a, b) \in A \times A : 2 {(a - b)}^{2} + 3 (a - b) \in B}$ is __________ . [2023]

Q 13 :

The number of relations, on the set {1, 2, 3} containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _________ . [2023]

Q 14 :

Let A = {-4, -3, -2, 0, 1, 3, 4} and $R = {(a, b) \in A \times A : b = | a | o r b^{2} = a + 1}$ be a relation on A.

Then the minimum number of elements, that must be added to the relation R so that it becomes reflexive and symmetric, is ________ . [2023]

Q 15 :

Let A = {1, 2, 3, 4} and R be a relation on the set A x A defined by $R = {((a, b), (c, d)) : 2 a + 3 b = 4 c + 5 d} .$

Then the number of elements in R is ___________ . [2023]

Q 16 :

The minimum number of elements that must be added to the relation R = {(a, b), (b, c), (b, d)} on the set {a, b, c, d} so that it is an equivalence relation is _____________ . [2023]

Q 17 :

Let a relation R on $N \times N$ be defined as:

$(x_{1}, y_{1}) R (x_{2}, y_{2})$ if and only if $x_{1} \leq x_{2} or y_{1} \leq y_{2} .$

Consider the two statements:

(I) R is reflexive but not symmetric.

(II) R is transitive.

Then which one of the following is true? [2024]

Q 20 :

Let A = {2, 3, 6, 8, 9, 11} and B = {1, 4, 5, 10, 15}. Let R be a relation on $A \times B$ defined by (a, b) R(c, d) if and only if 3ad−7bc is an even integer. Then the relation R is [2024]

Q 21 :

Consider the relations $R_{1}$ and $R_{2}$ defined as $a R_{1} b \Leftrightarrow a^{2} + b^{2} = 1$ for all $a, b \in R$ and $(a, b) R_{2} (c, d) \Leftrightarrow a + d = b + c$ for all $(a, b), (c, d) \in N \times N .$ Then [2024]

Q 22 :

Let S = {1, 2, 3,…,10}. Suppose M is the set of all the subsets of S, then the relation $R = {(A, B) : A \cap B \neq ϕ; A, B \in M}$ is: [2024]

Q 23 :

Let R be a relation on $Z \times Z$ defined by (a, b) R (c, d) if and only if $a d - b c$ is divisible by 5. Then R is [2024]

Q 24 :

If 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 is _______ . [2024]

Q 25 :

Let A = {2, 3, 6, 7} and B = {4, 5, 6, 8}. Let R be a relation defined on $A \times B$ by $(a_{1}, b_{1}) R (a_{2}, b_{2})$ if and only if $a_{1} + a_{2} = b_{1} + b_{2}$ . Then the number of elements in R is ______________. [2024]

Q 26 :

Let A = {1, 2, 3,…,20}. Let $R_{1}$ and $R_{2}$ be two relation on A such that $R_{1}$ = {(a, b) : b is divisible by a} $R_{2}$ = {(a, b) : a is an integral multiple of b} Then, number of elements in $R_{1} - R_{2}$ is equal to _______ . [2024]

Q 27 :

The number of symmetric relations defined on the set {1, 2, 3, 4} which are not reflexive is _______. [2024]

Q 28 :

Let A = {1, 2, 3, 4} and R = {(1, 2), (2, 3), (1, 4)} be a relation on A. Let S be the equivalence relation on A such that $R \subset S$ and the number of elements in S is $n$ . Then, the minimum value of $n$ is ________. [2024]

Q 29 :

Let A = {1, 2, 3, ..........., 100}. Let R be a relation on A defined by (x, y) ∈ R if and only if 2x = 3y. Let $R_{1}$ be a symmetric relation on A such that $R \subset R_{1}$ and the number of elements in $R_{1}$ is $n$ . Then, the minimum value of $n$ is ______. [2024]

Q 30 :

Let A be the set of all functions f : Z $\to$ Z and R be a relation on A such that R = {(f, g) : f(0) = g(1) and f(1) = g(0)}. Then R is : [2025]

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Topic Question Set

Q 1 : Let A = {1, 2, 3, 4, 5, 6, 7}. Then the relation R={(x,y)∈A×A:x+y=7} is [2023]

Q 2 : Let A = {2, 3, 4} and B = {8, 9, 12}. Then the number of elements in the relation R={((a1,b1),(a2,b2))∈(A×B,A×B):a1 divides b2 and a2 divides b1} is [2023]

Q 3 : Let A = {1, 3, 4, 6, 9} and B = {2, 4, 5, 8, 10}. Let R be a relation defined on A x B such that R={((a1,b1),(a2,b2)):a1≤b2 and b1≤a2}. Then the number of elements in the set R is [2023]

Q 4 : Let R be a relation on R, given by R = {(a,b):3a-3b+7 is an irrational number}. Then R is [2023]

Q 5 : Let P(S) denote the power set of S = {1, 2, 3, ...., 10}. Define the relations R1 and R2 on P(S) as AR1B if (A∩BC)∪(B∩AC)=ϕ and AR2B if A∪BC=B∪AC,∀A,B∈P(S). Then [2023]

(4)

Q 6 : The relation R={(a,b):gcd(a,b)=1,2a≠b,a,b,∈Z} is [2023]

Q 7 : Let R be a relation defined on ℕ as a R b if 2a+3b is a multiple of 5, a,b∈ℕ. [2023]

Q 8 : 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 [2023]

Q 9 : Let R be a relation on N x N defined by (a, b) R(c, d) if and only if ad(b-c)=bc(a-d). Then R is [2023]

Q 10 : Among the relations S={(a,b):a,b∈R-{0},2+ab>0} and T={(a,b):a,b∈R,a2-b2∈Z} [2023]

Q 11 : Let A = {1, 2, 3, 4, .... ,10} and B = {0, 1, 2, 3, 4}. The number of elements in the relation R={(a,b)∈A×A:2(a-b)2+3(a-b)∈B} is __________ . [2023]

Q 12 : Let A = {0, 3, 4, 6, 7, 8, 9, 10} and R be the relation defined on A such that R={(x,y)∈A×A:x-y is odd positive integer or x-y=2}. The minimum number of elements that must be added to the relation R, so that it is a symmetric relation, is equal to __________ . [2023]

Q 13 : The number of relations, on the set {1, 2, 3} containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _________ . [2023]

Q 14 : Let A = {-4, -3, -2, 0, 1, 3, 4} and R={(a,b)∈A×A:b=|a|or b2=a+1} be a relation on A. Then the minimum number of elements, that must be added to the relation R so that it becomes reflexive and symmetric, is ________ . [2023]

Q 15 : Let A = {1, 2, 3, 4} and R be a relation on the set A x A defined by R={((a,b),(c,d)):2a+3b=4c+5d}. Then the number of elements in R is ___________ . [2023]

Q 16 : The minimum number of elements that must be added to the relation R = {(a, b), (b, c), (b, d)} on the set {a, b, c, d} so that it is an equivalence relation is _____________ . [2023]

Q 17 : Let a relation R on N×N be defined as: (x1,y1)R(x2,y2) if and only if x1≤x2 or y1≤y2. Consider the two statements: (I) R is reflexive but not symmetric. (II) R is transitive. Then which one of the following is true? [2024]

Q 18 : Let the relations R1 and R2 on the set X={1,2,3,...,20} be given by R1={(x,y):2x-3y=2} and R2={(x,y):-5x+4y=0}. If M and N be the minimum number of elements required to be added in R1 and R2, respectively, in order to make the relations symmetric, then M+N equals [2024]

Q 19 : Let A={1, 2, 3, 4, 5}. Let R be a relation on A defined by xRy if and only if 4x≤5y. Let m be the number of elements in R and n be the minimum number of elements from A×A that are required to be added to R to make it a symmetric relation. Then m+n is equal to: [2024]

Q 20 : Let A = {2, 3, 6, 8, 9, 11} and B = {1, 4, 5, 10, 15}. Let R be a relation on A×B defined by (a, b) R(c, d) if and only if 3ad−7bc is an even integer. Then the relation R is [2024]

Q 21 : Consider the relations R1 and R2 defined as aR1b⇔a2+b2=1 for all a,b∈R and (a,b)R2(c,d)⇔a+d=b+c for all (a,b),(c,d)∈N×N. Then [2024]

Q 22 : Let S = {1, 2, 3,…,10}. Suppose M is the set of all the subsets of S, then the relation R={(A,B):A∩B≠ϕ;A,B∈M} is: [2024]

Q 23 : Let R be a relation on Z×Z defined by (a, b) R (c, d) if and only if ad-bc is divisible by 5. Then R is [2024]

Q 24 : If R is the smallest equivalence relation on the set {1, 2, 3, 4} such that {(1,2),(1,3)}⊂R, then the number of elements in R is _______ . [2024]

Q 25 : Let A = {2, 3, 6, 7} and B = {4, 5, 6, 8}. Let R be a relation defined on A×B by (a1,b1)R(a2,b2) if and only if a1+a2=b1+b2. Then the number of elements in R is ______________. [2024]

Q 26 : Let A = {1, 2, 3,…,20}. Let R1 and R2 be two relation on A such that R1 = {(a, b) : b is divisible by a} R2 = {(a, b) : a is an integral multiple of b} Then, number of elements in R1-R2 is equal to _______ . [2024]

Q 27 : The number of symmetric relations defined on the set {1, 2, 3, 4} which are not reflexive is _______. [2024]

Q 28 : Let A = {1, 2, 3, 4} and R = {(1, 2), (2, 3), (1, 4)} be a relation on A. Let S be the equivalence relation on A such that R⊂S and the number of elements in S is n. Then, the minimum value of n is ________. [2024]

Q 29 : Let A = {1, 2, 3, ..........., 100}. Let R be a relation on A defined by (x, y) ∈ R if and only if 2x = 3y. Let R1 be a symmetric relation on A such that R⊂R1 and the number of elements in R1 is n. Then, the minimum value of n is ______. [2024]

Q 30 : Let A be the set of all functions f : Z → Z and R be a relation on A such that R = {(f, g) : f(0) = g(1) and f(1) = g(0)}. Then R is : [2025]

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Q 1 :

Let A = {1, 2, 3, 4, 5, 6, 7}. Then the relation $R = {(x, y) \in A \times A : x + y = 7}$ is [2023]

Q 2 :

Let A = {2, 3, 4} and B = {8, 9, 12}. Then the number of elements in the relation

$R = {((a_{1}, b_{1}), (a_{2}, b_{2})) \in (A \times B, A \times B) : a_{1}$ divides $b_{2}$ and $a_{2}$ divides $b_{1}}$ is [2023]

Q 3 :

Let A = {1, 3, 4, 6, 9} and B = {2, 4, 5, 8, 10}. Let R be a relation defined on A x B such that $R = {((a_{1}, b_{1}), (a_{2}, b_{2})) : a_{1} \leq b_{2}$ and $b_{1} \leq a_{2}} .$ Then the number of elements in the set R is [2023]

Q 4 :

Let R be a relation on R, given by

R = ${(a, b) : 3 a - 3 b + \sqrt{7}$ is an irrational number $} .$

Then R is [2023]

Q 5 :

Let P(S) denote the power set of S = {1, 2, 3, ...., 10}. Define the relations $R_{1}$ and $R_{2}$ on P(S) as $A R_{1} B$ if $(A \cap B^{C}) \cup (B \cap A^{C}) = ϕ$ and $A R_{2} B$ if $A \cup B^{C} = B \cup A^{C}, \forall A, B \in P (S)$ . Then [2023]

Q 6 :

The relation $R = {(a, b) : g c d (a, b) = 1, 2 a \neq b, a, b, \in Z}$ is [2023]

Q 7 :

Let R be a relation defined on $ℕ$ as $a R b$ if $2 a + 3 b$ is a multiple of 5, $a, b \in ℕ$ . [2023]

Q 8 :

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 [2023]

Q 9 :

Let R be a relation on N x N defined by (a, b) R(c, d) if and only if $a d (b - c) = b c (a - d)$ . Then R is [2023]

Q 10 :

Among the relations

$S = {(a, b) : a, b \in R - {0}, 2 + \frac{a}{b} > 0}$ and

$T = {(a, b) : a, b \in R, a^{2} - b^{2} \in Z}$ [2023]

Q 11 :

Let A = {1, 2, 3, 4, .... ,10} and B = {0, 1, 2, 3, 4}. The number of elements in the relation $R = {(a, b) \in A \times A : 2 {(a - b)}^{2} + 3 (a - b) \in B}$ is __________ . [2023]

Q 13 :

The number of relations, on the set {1, 2, 3} containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _________ . [2023]

Q 14 :

Let A = {-4, -3, -2, 0, 1, 3, 4} and $R = {(a, b) \in A \times A : b = | a | o r b^{2} = a + 1}$ be a relation on A.

Then the minimum number of elements, that must be added to the relation R so that it becomes reflexive and symmetric, is ________ . [2023]

Q 15 :

Let A = {1, 2, 3, 4} and R be a relation on the set A x A defined by $R = {((a, b), (c, d)) : 2 a + 3 b = 4 c + 5 d} .$

Then the number of elements in R is ___________ . [2023]

Q 16 :

The minimum number of elements that must be added to the relation R = {(a, b), (b, c), (b, d)} on the set {a, b, c, d} so that it is an equivalence relation is _____________ . [2023]

Q 17 :

Let a relation R on $N \times N$ be defined as:

$(x_{1}, y_{1}) R (x_{2}, y_{2})$ if and only if $x_{1} \leq x_{2} or y_{1} \leq y_{2} .$

Consider the two statements:

(I) R is reflexive but not symmetric.

(II) R is transitive.

Then which one of the following is true? [2024]

Q 20 :

Let A = {2, 3, 6, 8, 9, 11} and B = {1, 4, 5, 10, 15}. Let R be a relation on $A \times B$ defined by (a, b) R(c, d) if and only if 3ad−7bc is an even integer. Then the relation R is [2024]

Q 21 :

Consider the relations $R_{1}$ and $R_{2}$ defined as $a R_{1} b \Leftrightarrow a^{2} + b^{2} = 1$ for all $a, b \in R$ and $(a, b) R_{2} (c, d) \Leftrightarrow a + d = b + c$ for all $(a, b), (c, d) \in N \times N .$ Then [2024]

Q 22 :

Let S = {1, 2, 3,…,10}. Suppose M is the set of all the subsets of S, then the relation $R = {(A, B) : A \cap B \neq ϕ; A, B \in M}$ is: [2024]

Q 23 :

Let R be a relation on $Z \times Z$ defined by (a, b) R (c, d) if and only if $a d - b c$ is divisible by 5. Then R is [2024]

Q 24 :

If 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 is _______ . [2024]

Q 25 :

Let A = {2, 3, 6, 7} and B = {4, 5, 6, 8}. Let R be a relation defined on $A \times B$ by $(a_{1}, b_{1}) R (a_{2}, b_{2})$ if and only if $a_{1} + a_{2} = b_{1} + b_{2}$ . Then the number of elements in R is ______________. [2024]

Q 26 :

Let A = {1, 2, 3,…,20}. Let $R_{1}$ and $R_{2}$ be two relation on A such that $R_{1}$ = {(a, b) : b is divisible by a} $R_{2}$ = {(a, b) : a is an integral multiple of b} Then, number of elements in $R_{1} - R_{2}$ is equal to _______ . [2024]

Q 27 :

The number of symmetric relations defined on the set {1, 2, 3, 4} which are not reflexive is _______. [2024]

Q 28 :

Let A = {1, 2, 3, 4} and R = {(1, 2), (2, 3), (1, 4)} be a relation on A. Let S be the equivalence relation on A such that $R \subset S$ and the number of elements in S is $n$ . Then, the minimum value of $n$ is ________. [2024]

Q 29 :

Let A = {1, 2, 3, ..........., 100}. Let R be a relation on A defined by (x, y) ∈ R if and only if 2x = 3y. Let $R_{1}$ be a symmetric relation on A such that $R \subset R_{1}$ and the number of elements in $R_{1}$ is $n$ . Then, the minimum value of $n$ is ______. [2024]

Q 30 :

Let A be the set of all functions f : Z $\to$ Z and R be a relation on A such that R = {(f, g) : f(0) = g(1) and f(1) = g(0)}. Then R is : [2025]