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Show that the relation R in the set A =...

Show that the relation R in the set `A = {1, 2, 3, 4, 5,6,7}`given by `R = {(a , b) : |a - b| " is even" }`, is an equivalence relation.

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(i) Since `|a – a|` is even,
∴ (a, a) ∈ R
∴ R is reflexive.
(ii) Let (a, b) ∈ R Then |a – b| is even
∴ |b – a| is even
∴ (b, a) ∈ R and R is symmetric.
(iii) Let (a, b), (b, c) ∈ R
Then a – b = ±2m, b – c = ±2n
∴ a – c = ±2(m + n), where m, n are integers.
∴ (a, c) ∈ R and hence R is transitive
Thus, R is an equivalence relation.
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