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Question
Answer the following:
Show that the relation R in the set A = {1, 2, 3, 4, 5} Given by R = {(a, b)/|a − b| is even} is an equivalence relation.
Solution
R = {(a, b)/|a − b| is even, a, b ∈ A}, where
A = {1, 2, 3, 4, 5}
|a − a| = 0 is even
∴ aRa ∀ a ∈ A
∴ R is reflexive
Let aRb
∴ |a − b| is even
∴ |a − b| = |b − a|
∴ |b − a| is even
∵ bRa
∴ aRb ⇒ bRa ∀a, b ∈ A
∴ R is symmetric
Let aRb and bRc
∴ |a − b| and |b − c| are even
If b is even, then a and c both are even
∴ |a − c| is even
If b is odd, then a and c both are odd
∴ |a − c| is even
∴ 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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