Question

Consider the set $A=\{a,b,c\}$.
Give an example of a relation R and A which is
(i) reflexive and symmetric but not reflexive.
(ii) symmetric and transitive but not reflexive.
(iii) reflexive and transitive but not symmetric.

Answer 1

(i) Consider the relation $R=\left\{\right(a,a),(a,b),(b,a),(c,b),(b,b),(c.c\left)\right\}$ on
A.Here, R is relation which is reflexive and symmetric but not transitive.
$(a,b)\in R$ and $(b,c)\in R$ but $(a,c)\in R$
(ii) Consider the relation $R=\left\{\right(a,a),(a,b),(b,a),(b.b\left)\right\}$ on A.
Here R is symmetric and transitive but not reflexive.
$(c,c)\notin R$
(iii) Consider the relation $R=\left\{\right(a,a),(b,b),(c,c),(a,b\left)\right\}$ on A.
Here R is reflexive and transitive but not symmetric.
$(a,b)\in Rbut(b,a)\notin R$