Question:
Test whether the relation is reflexive ,symmetric ,or transitive on the set specified R={(m,n): 3 divides m - n} on{1,2,3, ....,10}
Answer:
1. Let m ∈ Z. Then 3 divides m - m = 0. Therefore mRm holds for all a in Z and R is reflexive.
2. Let m, n ∈ Z and mRn hold. Then 3 divides m - n and therefore 3 divides n - m.
Thus, mRn => nRm and therefore R is symmetric.
3. Let m, n, p ∈ Z and mRn, nRp both hold. Then 3 divides m – n and n – p both.
Therefore, 3 divides m – p = (m – n) + (n – p)
Thus, mRn and nRp => mRp and therefore R is transitive.
So, R is reflexive, symmetric and transitive.
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