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On the set A = {a, b, c}, three relations are defined: R1 = {(a,a), (a,b), (a,c), (b,b), (b,c), (c,a), (c,b), (c,c)} R2 = {(a,b), (b,a), (a,c), (c,a)} R3 = {(a,b), (b,c), (c,a)} Which one of the following correctly describes the properties (reflexive, symmetric, transitive) of R1, R2 and R3?

1. R1: reflexive, not symmetric, not transitive; R2: symmetric, not reflexive, not transitive; R3: neither reflexive, symmetric nor transitive
2. R1: reflexive, symmetric and transitive; R2: symmetric and transitive; R3: transitive only
3. R1: symmetric only; R2: reflexive and symmetric; R3: reflexive and transitive
4. R1: reflexive and transitive only; R2: reflexive and symmetric; R3: symmetric only

Correct answer: R1: reflexive, not symmetric, not transitive; R2: symmetric, not reflexive, not transitive; R3: neither reflexive, symmetric nor transitive

Solution

R1 contains (a,a),(b,b),(c,c) so it is reflexive. It is not symmetric: (a,b) is present but (b,a) is not. It is not transitive: (b,c) and (c,a) are present but (b,a) is missing. R2 has no (x,x) pair so it is not reflexive; it is symmetric since each pair appears with its reverse ((a,b)&(b,a), (a,c)&(c,a)); it is not transitive since (a,b) and (b,a) are present but (a,a) is not. R3 has no diagonal pair (not reflexive); (a,b) present but (b,a) absent (not symmetric); (a,b),(b,c) present but (a,c) absent (not transitive).

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