Give an example of a relation on a set $A = \{5, 6, 7\}$ which is symmetric but neither reflexive nor transitive.

(A) Let $A = \{5, 6, 7\}$.
Define a relation $R$ on $A$ as $R = \{(5, 6), (6, 5)\}$.
$1$. Reflexivity: $A$ relation $R$ is reflexive if $(a, a) \in R$ for all $a \in A$. Here,$(5, 5) \notin R$,$(6, 6) \notin R$,and $(7, 7) \notin R$. Thus,$R$ is not reflexive.
$2$. Symmetry: $A$ relation $R$ is symmetric if $(a, b) \in R \implies (b, a) \in R$. Here,$(5, 6) \in R$ and its reverse $(6, 5) \in R$. Thus,$R$ is symmetric.
$3$. Transitivity: $A$ relation $R$ is transitive if $(a, b) \in R$ and $(b, c) \in R \implies (a, c) \in R$. Here,$(5, 6) \in R$ and $(6, 5) \in R$,but $(5, 5) \notin R$. Thus,$R$ is not transitive.
Conclusion: The relation $R = \{(5, 6), (6, 5)\}$ on set $A$ is symmetric but neither reflexive nor transitive.