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

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]

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(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]

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(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]

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(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]

0%

(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]

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(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]

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(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.

1

2

3

4

(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.

Topic Question Set

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]

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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]

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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]

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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]

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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]

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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]

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Topic Question Set

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]

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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]

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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]

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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]

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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]

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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]

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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]

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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]