Consider the following two binary relations on the set A = \ a, b,...

Since, $( b, c ) \in {R}_{1}$ but $( c, b ) \notin {R}_{1}$

Hence, ${R}_{1}$ is not symmetric.

In ${R}_{1}: \left(b, c\right) \in {R}_{1}$ and \left(c, a\right) \in {R}_{1}$but \left(b, a\right) \notin {R}_{1}$

So, ${R}_{1}$ is not transitive.

${R}_{2}$ is symmetric.

In ${R}_{2}: \left(b, a\right) \in {R}_{2}$ and \left(a, c\right) \in {R}_{1}$but \left(b, c\right) \notin {R}_{2}$

So, ${R}_{2}$ is not transitive.