Question:
Test whether the relation is reflexive ,symmetric ,or transitive on the set specified R = {(m,n): (m - n) is divisable by 2} on Z
Answer:
1. Let a ∈ Z. Then a – a is divisible by 5. Therefore aRa holds for all a in Z and R is reflexive.
2. Let a, b ∈ Z and aRb hold. Then a – b is divisible by 2 and therefore b – a is divisible by 2.
Thus, aRb => bRa and therefore R is symmetric.
3. Let a, b, c ∈ Z and aRb, bRc both hold. Then a – b and b – c are both divisible by 2.
Therefore a – c = (a – b) + (b – c) is divisible by 2.
Thus, aRb and bRc => aRc and therefore R is transitive.
Since R is reflexive, symmetric and transitive.
So, R is an equivalence relation on Z.
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