1.

Let N be the set of all natural numbers and R be the relation in N × N defined by(a, b) R (c, d) if ad = bc. Show that R is an equivalence relation.

Answer»

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


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