Here,
R is a relation defined as,
R = {[(a,b),(c,d)] : ad(b + c) = bc(a + d)}
Reflexivity : By commutative law under addition and multiplication
∴ ab(b+a) = ba(a+b) ∀ a,b ∈ N
(a,b) R (a,b) Hence, R is reflexive.
Symmetric : Let (a,b) R (c,d)
[By commutative law under addition and multiplication]
⇒ (c,d) R (a,b)
Hence, R is symmetric.
Transitivity : Let (a,b) R (c,d) and (c,d) R (e,f)
Now,
(a,b) R (c,d) and (c,d) R (e,f)
Adding both, we get
[c,d ≠0]
Hence, R is transitive.
In this way, R is reflexive, symmetric and transitive.
Therefore, R is an equivalence relation.