Eduprobe
Question:
Consider the following two binary relations on the set A = {a, b, c}: R1 = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)} and R2 = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)}. Then which of the following statements is true?
Both R1 and R2 are transitive
Both R1 and R2 are not symmetric
R2 is symmetric but it is not transitive
R1 is not symmetric but it is transitive
Solution:
The correct option is R2 is symmetric but it is not transitive.
R2 is symmetric as for any (a1, a2) ∈ R2, we have (a2, a1) ∈ R2.
But R1 is not symmetric as (b, c) ∈ R1 but (c, b) ∉ R1.
For checking transitivity, we observe for R2 that (b, a) ∈ R2, (a, c) ∈ R2 but (b, c) ∉ R2. Similarly, for R1, (b, c) ∈ R1, (c, a) ∈ R1 but (b, a) ∉ R1. So neither R1 nor R2 is transitive. So, the correct answer is option C.