Show that the relation $R$ in the set $\{1, 2, 3\}$ given by $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)\}$ is reflexive but neither symmetric nor transitive.
(N/A) relation $R$ on a set $A$ is reflexive if $(a, a) \in R$ for all $a \in A$. Here,$A = \{1, 2, 3\}$. Since $(1, 1), (2, 2), (3, 3) \in R$,the relation $R$ is reflexive.
$A$ relation $R$ is symmetric if $(a, b) \in R$ implies $(b, a) \in R$. Here,$(1, 2) \in R$,but $(2, 1) \notin R$. Therefore,$R$ is not symmetric.
$A$ relation $R$ is transitive if $(a, b) \in R$ and $(b, c) \in R$ implies $(a, c) \in R$. Here,$(1, 2) \in R$ and $(2, 3) \in R$,but $(1, 3) \notin R$. Therefore,$R$ is not transitive.
$A$ relation $R$ is symmetric if $(a, b) \in R$ implies $(b, a) \in R$. Here,$(1, 2) \in R$,but $(2, 1) \notin R$. Therefore,$R$ is not symmetric.
$A$ relation $R$ is transitive if $(a, b) \in R$ and $(b, c) \in R$ implies $(a, c) \in R$. Here,$(1, 2) \in R$ and $(2, 3) \in R$,but $(1, 3) \notin R$. Therefore,$R$ is not transitive.
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