Let N be the set of all natural numbers and R be the relation in N × N defined by

Normally relationship are between element of ordered pairs but how can we establish relationship between ordered pairs. Is it as per definition equivalence relation? Please clarify.

Here, relation R defined on NxN and defined by if ad = bc, then (a, b) R (c, d) (means (a, b) is related to (c, d) if ad = bc).

For any (a, b) ∈ N × N ; ab = ba

⇒ (a, b) R (a, b) Thus R is reflexive

Let (a, b) R (c, d) for any a, b, c, d ∈ N

∴ ad = bc

⇒ cb = da ⇒ (c, d) R (a, b)

∴ R is symmetric

Let (a, b) R (c,d),d and (c, d) R (e, f) for a, b, c, d, e, f ∈ N

Then ad = bc and cf = de

⇒ a d c f = b c d e or af = be ⇒ (a, b) R (e, f)

∴ R is transitive

So R is an equivalence Relation