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Question
Show that the relation S in the set A = [x ∈ Z : 0 ≤ x ≤ 12] given by S = [(a, b) : a, b ∈ Z, ∣a − b∣ is divisible by 3] is an equivalence relation.
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Solution
A = {c ∈ Z : 0≤ x ≤ 12} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
R = {(a, b) : |a - b| is divisible by 3}
For any element a ∈ A, we have (a, a) ∈ R as |a - a| = 0 is divisible by 3.
∴ R is reflexive.
Now, let (a, b) ∈ R ⇒ |a - b|is divisible 3.
⇒ |- (a - b)| = |b - a| is divisible by 3
⇒ (b, a) ∈ R
∴ R is symmetric.
Now, let (a, b), (b, c) ∈ R.
⇒ |a - b| is divisible by 3 and |b - c| is divisible by 3.
⇒ (a - b) is divisible by 3 and (b - c) is divisible by 3.
⇒ (a - c) = (a - b) + (b - c) is divisible by 3.
⇒ |a - c| is divisible by 3.
⇒ (a, c) ∈ R
∴ R is transitive.
Hence, R is an equivalence relation.
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