JEE Main 2023 — Sets And Relations Question with Solution

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.