question_answer 10)

Give an example of a relation. Which is (i) Symmetric but neither reflexive nor transitive. (ii) Transitive but neither reflexive nor symmetric. (iii) Reflexive and symmetric ut not transitive. (iv) Reflexive and transitive but not symmetric. (v) Symmetric and transitive but not reflexive. 

Answer:

Let A = {5, 6, 7} (i)     Let R = {(6, 7), (7, 6)} Clearly, R is symmetric. As (5, 5) and (6, 7) and (7, 6)  but (6, 6)  R is neither reflexive nor transitive. (ii)    Let R = {(5, 6), (5, 7), (7, 6)} Clearly, R is transitive. As (5, 5)  R and (5, 7)  but (7, 5) R  R is neither reflexive nor symmetric.       (iii) Let R = {(5, 5), (6, 6), (7, 7), (5, 6), (6, 5), (6, 7),       (7, 6)}       Clearly, R is reflexive and symmetric.       As (5, 6) and (6, 7)  but (5, 7)        is not transitive.       (iv)Let R = {(5, 5), (6, 6), (7, 7), (5, 6)}       Clearly, R is reflexive and transitive.       As (5,6)  but (6, 5)        is not symmetric.       (v) Let R = {(5, 6), (6, 5), (5, 5), (6, 6)}       Clearly, R is symmetric and transitive.       As (7, 7)        R is not reflexive.