Let A = {–3, –2, –1, 0, 1, 2, 3). Let R be a relation on A defined by xRy if and only if . Let 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 + m is equal to

Question

Let A = {–3, –2, –1, 0, 1, 2, 3}. Let R be a relation on A defined by $x R y$ if and only if $0 < x^{2} + 2 y \leq 4$ . Let $l$ be the number of elements in R and 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]

Smart Achievers · Accepted Answer

(1)

A = {–3, –2, –1, 0, 1, 2, 3}

xRy if and only if $0 < x^{2} + 2 y \leq 4$

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.

Solution

Q.
Let A = {–3, –2, –1, 0, 1, 2, 3}. Let R be a relation on A defined by $x R y$ if and only if $0 < x^{2} + 2 y \leq 4$ . Let $l$ be the number of elements in R and 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]

1 18

2 20

3 17

4 19

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