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

Q 31 :
Let A = {1, 2, 3, ..., 100} and R be a relation on A such that R = {(a, b) : a = 2b + 1}. Let $(a_{1}, a_{2}), (a_{2}, a_{3}), (a_{3}, a_{4}), . . ., (a_{k}, a_{k + 1})$ be a sequence of k elements of R such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer k, for which such a sequence exists, is equal to : [2025]

8
5
6
7

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4

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

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

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

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

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

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

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

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

$∴$ k = 5.

Q 32 :
Let A = {–3, –2, –1, 0, 1, 2, 3). Let R be a relation on A defined by xRy if and only if $0 < x^{2} + 2 y \leq 4$ . Let $l$ be the number of elements in R. Let m be the minimum number of elements required to be added in R to make it reflexive relation. Then $l$ + m is equal to [2025]

18
20
17
19

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(1)

A = {–3, –2, –1, 0, 1, 2, 3)

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

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

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

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

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

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

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

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

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

$∴$ $l$ = 15

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

$∴$ $l$ + m = 15 + 3 = 18.

Q 33 :
Let A = {–2, –1, 0, 1, 2, 3}. Let R be a relation on A defined by xRy if and only if y = max{x, 1}. Let $l$ be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then $l$ + m + n is equal to [2025]

13
12
14
11

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(2)

We have, A = {–2, –1, 0, 1, 2, 3} and

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

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

For R to be reflexive,

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

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

$∴$ m = 3

For R to be symmetric,

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

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

$∴$ n = 3

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

Q 34 :
Let A = {–3, –2, –1, 0, 1, 2, 3} and R be a relation on A defined by xRy if and only if 2x – y $\in$ {0, 1}. Let $l$ be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then $l$ + m + n is equal to : [2025]

17
16
18
15

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2

3

4

(1)

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

i.e., 2x – y = 0 or 2x – y = 1

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

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

$∴$ $l$ = m + n = 7 + 5 + 5 = 17.

Q 35 :
Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) $\in$ R if and only if max {x, y} $\in$ {3, 4}. Then among the statements

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

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

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

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(2)

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

$∴$ Number of elements in R = 16

$∴$ $S_{1}$ is false.

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

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

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

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

$∴$ $S_{2}$ is true.

Q 36 :
The number of non-empty equivalence relations on the sett {1, 2, 3} is: [2025]

5
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7
4

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(1)

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

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

Q 37 :
Let R = {(1, 2), (2, 3), (3, 3)} be a relation defined on the set {1, 2, 3, 4}. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is: [2025]

7
10
9
8

1

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3

4

(1)

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

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

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

$∴$ Minimum elements that should be added are

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

$∴$ Minimum number of elements required = 7.

Q 38 :
Let $A = {(x, y) \in R \times R : | x + y | \geq 3}$ and $B = {(x, y) \in R \times R : | x | + | y | \leq 3}$

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

18
12
24
15

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(2)

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

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

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

Q 39 :
Let X = R $\times$ R. Define a relation R on X as:

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

Statement I: R is an equivalence relation.

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

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

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

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

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(1)

State I:

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

$∴$ R is reflexive.

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

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

$∴$ R is symmetric.

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

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

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

Hence, relation R is equivalence relation.

$∴$ Statement I is true.

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

So, Statement II is false.

Q 40 :
The relation R = {(x, y) : x, y $\in$ Z and x + y is even} is: [2025]

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

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(4)

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

For reflexive : x + x = 2x is even

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

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

So, the relation is an equivalence.

Topic Question Set

Q 32 :
Let A = {–3, –2, –1, 0, 1, 2, 3). Let R be a relation on A defined by xRy if and only if $0 < x^{2} + 2 y \leq 4$ . Let $l$ be the number of elements in R. Let m be the minimum number of elements required to be added in R to make it reflexive relation. Then $l$ + m is equal to [2025]

Q 35 :
Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) $\in$ R if and only if max {x, y} $\in$ {3, 4}. Then among the statements

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

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

Q 36 :
The number of non-empty equivalence relations on the sett {1, 2, 3} is: [2025]

Q 37 :
Let R = {(1, 2), (2, 3), (3, 3)} be a relation defined on the set {1, 2, 3, 4}. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is: [2025]

Q 38 :
Let $A = {(x, y) \in R \times R : | x + y | \geq 3}$ and $B = {(x, y) \in R \times R : | x | + | y | \leq 3}$

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

Q 40 :
The relation R = {(x, y) : x, y $\in$ Z and x + y is even} is: [2025]

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Topic Question Set

Q 32 : Let A = {–3, –2, –1, 0, 1, 2, 3). Let R be a relation on A defined by xRy if and only if 0<x2+2y≤4. Let l be the number of elements in R. Let m be the minimum number of elements required to be added in R to make it reflexive relation. Then l + m is equal to [2025]

Q 35 : Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) ∈ R if and only if max {x, y} ∈ {3, 4}. Then among the statements (S1) : The number of elements in R is 18, and (S2) : The relation R is symmetric but neither reflexive nor transitive. [2025]

Q 36 : The number of non-empty equivalence relations on the sett {1, 2, 3} is: [2025]

Q 37 : Let R = {(1, 2), (2, 3), (3, 3)} be a relation defined on the set {1, 2, 3, 4}. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is: [2025]

Q 38 : Let A={(x,y)∈R×R:|x+y|≥3} and B={(x,y)∈R×R:|x|+|y|≤3} If C={(x,y)∈A∩B:x=0 or y=0, then ∑(x,y)∈C|x+y| is: [2025]

Q 40 : The relation R = {(x, y) : x, y ∈ Z and x + y is even} is: [2025]

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Q 32 :
Let A = {–3, –2, –1, 0, 1, 2, 3). Let R be a relation on A defined by xRy if and only if $0 < x^{2} + 2 y \leq 4$ . Let $l$ be the number of elements in R. Let m be the minimum number of elements required to be added in R to make it reflexive relation. Then $l$ + m is equal to [2025]

Q 35 :
Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) $\in$ R if and only if max {x, y} $\in$ {3, 4}. Then among the statements

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

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

Q 36 :
The number of non-empty equivalence relations on the sett {1, 2, 3} is: [2025]

Q 37 :
Let R = {(1, 2), (2, 3), (3, 3)} be a relation defined on the set {1, 2, 3, 4}. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is: [2025]

Q 38 :
Let $A = {(x, y) \in R \times R : | x + y | \geq 3}$ and $B = {(x, y) \in R \times R : | x | + | y | \leq 3}$

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

Q 40 :
The relation R = {(x, y) : x, y $\in$ Z and x + y is even} is: [2025]