Let R be a relation defined on $\mathbb{N}$ as a R b if $2a+3b$ is a multiple of 5, $a, b \in \mathbb{N}$. Then R is

(B) transitive but not symmetric

(C) symmetric but not transitive

JEE Main · 2023 · 29 Jan 2023 · Shift 2 · Sets Relations and Functions

Let R be a relation defined on

N

as a R b if

2 a + 3 b

is a multiple of 5,

a, b \in N

. Then R is

Answer: (D)

For reflexivity,

2 a + 3 a = 5 a

is a multiple of

5

, so

a R a

. For symmetry, if

2 a + 3 b

is a multiple of

5

then

(2 a + 3 b) + (2 b + 3 a) = 5 (a + b)

shows

2 b + 3 a

is also a multiple of

5

, so

b R a

. For transitivity, if

2 a + 3 b

and

2 b + 3 c

are multiples of

5

, their sum

2 a + 5 b + 3 c

is as well, hence

2 a + 3 c

is a multiple of

5

and

a R c

. Being reflexive, symmetric and transitive, the relation is an equivalence relation.

JEE Main · 2023 · 29 Jan 2023 · Shift 2 · Single correct option