Answer to Question #280874 in Discrete Mathematics for Ashwini

Question

Define a relation R on {a,b,c, int i* e . a Reflexive but not symmetric ↳ Symmetric but not transitive <> Transitive but not reflexive

Answer 1

Let us define a relation $R$ on $A=\{a,b,c,d,e\}.$

a) The relation $R=\{(a,a),(b,b),(c,c),(d,d),(e,e),(a,b)\}$ is reflexive because of $(x,x)\in R$ for each $x\in A,$ but it is not symmetric because of $(b,a)\notin R.$

b) The relation $R=\{(a,b),(b,a)\}$ is symmetric but not transitive because of $(a,b)\in R$ and $(b,a)\in R$ but $(a,a)\notin R.$

c) The relation $R=\{(a,b),(b,a),(a,a),(b,b)\}$ is transitive because of $(x,y),(y,z)\in R$ imply $(x,z)\in R,$ but it is not reflexive becase of $(c,c)\notin R.$