Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Let A = {1, 2, 3}.

A relation R on A is defined as R = {(1, 2), (2, 1)}.

It is seen that $(1,1),(2,2),(3,3) \notin \mathrm{R}$.

∴ R is not reflexive.

Now, as $(1,2) \in R$ and $(2,1) \in R$, then $R$ is symmetric.

Now, $(1,2)$ and $(2,1) \in R$

However,

$(1,1) \notin R$

∴ R is not transitive.

Hence, R is symmetric but neither reflexive nor transitive.