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Question
Answer the following:
Check if R : Z → Z, R = {(a, b)/2 divides a – b} is equivalence relation.
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Solution
i. Since, 2 divides a – a.
∴ (a, a) ∈ R
∴ R is reflexive.
ii. Let (a, b) ∈ R
Then 2 divides a – b
∴ 2 divides b – a
∴ (b, a) ∈ R
∴ R is symmetric.
iii. Let (a, b) ∈ R, (b, c) ∈ R
Then, a – b = 2m, b – c = 2n,
∴ a – c = 2(m + n), where m, n are integers
∴ 2 divides a – c
∴ (a, c) ∈ R
∴ R is transitive.
Thus, R is an equivalence relation.
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