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Question
Answer the following:
Show that the following is an equivalence relation
R in A = {x ∈ Z | 0 ≤ x ≤ 12} given by R = {(a, b)/|a − b| is a multiple of 4}
Solution
A = {x ∈ Z | 0 ≤ x ≤ 12}
R = {(a, b)/|a − b| is a multiple of 4; a, b ∈ A}
|a − a| = 0 is a multiple of 4
∴ aRa ∀ a∈A
∴ R is reflexive
Let aRb
∴ |a − b| is a multiple of 4
∴ |b − a| = |a − b|
∴ |b − a| is a multiple of 4
∴ aRb ⇒ bRa ∀a, b ∈ A
∴ R is symmetric
Let aRb and bRc
∴ |a − b| and |b − c| are multiples of 4
∴ a − b = 4m, b − c = 4n; m, n ∈ Z
a − c = (a − b) + (b − c) = 4m + 4n
= 4(m + n); (m + n) ∈ Z
∴ |a − c| is a multiple of 4
∴ aRc
∴ aRb, bRc ⇒ aRc ∀a, b, c ∈ A
∴ R is transitive
∵ R is reflexive, symmetric and transitive
∴ R is an equivalence relation.
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