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Relations and Functions
Question
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The relation R ⊆ N x N is defined by (a, b) ∈ R if and only if 5 divides ft a. Show that R is an equivalence relation.

Answer

The relation R ⊆ N x N is defined by by (a. b)∈ R if and only if 5 divides b – a.
This means that R is a relation on N defined by , if a. b ∈ N then (a, b) ∈ R if and only if 5 divides b – a.
Let a, b, c belongs to N. Then (i)    a – a = 0 = 5 . 0.
5 divides a – a.
⇒ (a. a) ∈ R .
⇒ R is reflexive.
(ii) Let (a, b) ∈ R.
∴ divides a – b.
⇒ a – b = 5 n for some n ∈ N.
⇒ b – a = 5 (–n).
⇒ 5 divides b – a ⇒ (b, a) ∈ R.
∴ R is symmetric.
(iii) Let (a, b) and (b, c) ∈ R.
5 divides a – b and b – c both
∴ a – 6 = 5 n1 and b – n = 5 n2 for some n1 and n2 ∈ N ∴ (a – b) + (b – c) = 5 n1 + 5 n2⇒ a – c = 5 (n1 + n2)
⇒ 5 divides a – c ⇒ (a, c) ∈ R
∴ R is transitive relation in N.

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