Let \(\mathrm{A}=\{-4,-3,-2,0,1,3,4\}\) and \(\mathrm{R}=\{(\mathrm{a}, \mathrm{b}) \in \mathrm{A} \times \mathrm{A}: \mathrm{b}=|\mathrm{a}|\) or \(\mathrm{b}^{2}=\mathrm{a}+1\}\) 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 ______.

JEE Main · 2023 · 13 Apr 2023 · Shift 2 · Sets Relations and Functions

Let

A = {- 4, - 3, - 2, 0, 1, 3, 4}

and

R = {(a, b) \in A \times A : b = ∣ a ∣

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 ______.

Answer: 7

From

b = ∣ a ∣

the pairs in

R

are

(- 4, 4), (- 3, 3), (0, 0), (1, 1), (3, 3), (4, 4)

, and from

b^{2} = a + 1

the only pairs keeping both entries in

A

are

(0, 1)

and

(3, - 2)

. So

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

. For reflexivity the missing diagonal pairs are

(- 4, - 4), (- 3, - 3), (- 2, - 2)

, which is

3

pairs. For symmetry the reversals needed are

(4, - 4), (3, - 3), (1, 0), (- 2, 3)

, which is

4

pairs, none already present. Adding these makes

R

reflexive and symmetric, so the minimum number of elements to be added is

3 + 4 = 7

JEE Main · 2023 · 13 Apr 2023 · Shift 2 · Integer answer