The minimum number of elements that must be added to the relation $\mathrm{R}=\{(\mathrm{a}, \mathrm{b}),(\mathrm{b}, \mathrm{c})\}$ on the set $\{\mathrm{a}, \mathrm{b}, \mathrm{c}\}$ so that it becomes symmetric and transitive is:

JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Sets Relations and Functions

The minimum number of elements that must be added to the relation

R = {(a, b), (b, c)}

on the set

{a, b, c}

so that it becomes symmetric and transitive is:

Answer: (B)

Begin with

R = {(a, b), (b, c)}

. Symmetry forces

(b, a)

and

(c, b)

; transitivity on

(a, b), (b, c)

forces

(a, c)

, and symmetry then forces

(c, a)

. With all three elements now mutually linked, transitivity on pairs such as

(a, b), (b, a)

(b, c), (c, b)

and

(c, a), (a, c)

forces the loops

(a, a), (b, b), (c, c)

. The relation must therefore contain every ordered pair of

{a, b, c}

, that is all nine pairs; since two are already present, the number that must be added is

9 - 2 = 7.

JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Single correct option