Answer :

Given that, A = {1, 2, 3, 4 }


(a) Reflexive, transitive but not symmetric


Let R be a relation defined by


R = {(1,1),(1,2),(1,4),(2,2),(2,3),(3,2),(3,3),(4,2),(4,4)} on set A.


R is reflexive (1,1),(2,2),(3,3),(4,4) R


R is transitive (1,4) R and (4,2) R (1,2) R


R is not symmetric (1,4) R but (4,1) R


Hence, R is reflexive, transitive but not symmetric.


(b) Symmetric but neither reflexive nor transitive


Let R be a relation defined by


R = {(1,2),(2,1),(2,3),(3,2)} on set A.


R is not reflexive (1,1),(2,2),(3,3),(4,4) R


R is symmetric (1,2) R (2,1) R and (2,3) R (3,2) R


R is not transitive (1,2) R and (2,1) R (1,1) R


Hence, R is symmetric but neither reflexive nor transitive.


(c) Reflexive, symmetric and transitive.


Let R be a relation defined by


R = {(1,1),(1,2),(1,4),(2,1),(2,2),(2,3),(3,2),(3,3),(4,1),(4,4)} on set A.


R is reflexive (1,1),(2,2),(3,3),(4,4) R


R is symmetric (1,2),(1,4),(2,3) R (2,1),(4,1),(3,2) R


R is transitive (1,2) R and (2,1) R (1,1) R


Hence, R is reflexive, symmetric and transitive.


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