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.

Given the set $\{1, 2, 3\}$ where R $= \{(1, 2), (2, 1)\}$

(i) Reflexive

$1,\;2,\;3 \in \{ 1,\;2,\;3\} ,\;(1,\;1)\not \in R,\;(2,\;2)\not \in R,\;(3,\;3)\not \in R$

Therefore, R is not reflexive.

(ii) Symmetric

$1,\;2 \in \{ 1,\;2,\;3\} ,\;(1,\;2) \in R \Rightarrow (2,\;1) \in R$

Therefore, R is symmetric.

(iii) Transitive

$1,\;2,\;3 \in \{ 1,\;2,\;3\} ,$
Consider, $(1,\;2) \in R,\;(2,\;3)\not \in R,\;(1,\;3)\not \in R$

R is not transitive.
Hence, R is symmetric but neither reflexive nor transitive.