Let $\mathrm{A}=\{1,2,3,4,5,6,7\}$. Then the relation $\mathrm{R}=\{(\mathrm{x}, \mathrm{y}) \in \mathrm{A} \times \mathrm{A}: \mathrm{x}+\mathrm{y}=7\}$ is

(A) transitive but neither symmetric nor reflexive

(B) reflexive but neither symmetric nor transitive

JEE Main · 2023 · 08 Apr 2023 · Shift 2 · Sets Relations and Functions

Let

A = {1, 2, 3, 4, 5, 6, 7}

. Then the relation

R = {(x, y) \in A \times A : x + y = 7}

Answer: (D)

The relation

R = {(x, y) \in A \times A : x + y = 7} = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}

. Check reflexivity:

(1, 1) \in / R

since

1 + 1 \neq = 7

, so not reflexive. Check symmetry: if

(a, b) \in R

then

a + b = 7

, so

b + a = 7

and

(b, a) \in R

. Thus symmetric. Check transitivity:

(1, 6) \in R

and

(6, 1) \in R

, but

(1, 1) \in / R

, so not transitive. Therefore, the relation is symmetric but neither reflexive nor transitive.

JEE Main · 2023 · 08 Apr 2023 · Shift 2 · Single correct option