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

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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2

3

4

(C)

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

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

$∴$ $R_{1} is not reflexive$

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

$For R_{2}$

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

$∴ R_{2} is reflexive .$

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

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

$∴ R_{2} i s s y m m e t r i c .$

$(i i i) Let (a, b) R (c, d) \Rightarrow a + d = b + c$ ...(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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2

3

4

(D)

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

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2

3

4

(D)

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

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

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

$∴$ $R is reflexive .$

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

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

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

$∴ R$ $is symmetric .$

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

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

$∴ a d f - b c f = 5 k_{1} f$ ...(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

(C)

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]

0%

(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]

0%

(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]

0%

(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]

0%

(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]

0%

(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

Topic Question Set

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]

0%

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]

0%

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

0%

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]

0%

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]

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Topic Question Set

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]

0%

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]

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

0%

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]

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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]

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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]