JEE MAIN - Mathematics (2023 - 29th January Evening Shift - No. 2)

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

an equivalence relation

non reflexive

symmetric but not transitive

transitive but not symmetric

a R b if 2a + 3b = 5m, m $$\in$$ $$l$$

(1) $$(a,a) \in R$$ as $$2a + 3a = 5a,a \in N$$

Hence, R is reflexive

(2) If $$(a,b) \in R$$ then $$2a + 3 = 5m$$

Now, $$5(a + b) = 5n$$

$$3a + 2b + 2a + 3b = 5n$$

$$\therefore$$ $$3a + 2b = 5(n - m)$$

$$\therefore$$ $$(b,a) \in R$$

$$\therefore$$ R is symmetric

(3) If $$(a,b) \in R$$ and $$(b,c) \in R$$ then

$$2a + 3b = 5m,2b + 3c = 5n$$

$$ \Rightarrow 2a + 5b + 3c = 5(m + n)$$

$$ \Rightarrow 2a + 3c = 5(m = n - b)$$

$$\therefore$$ $$(a,c) \in R$$

$$\therefore$$ R is transitive

Hence, R is equivalence relation.

Option (1) is correct.