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

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Best answer

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.

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