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

Step 1: Check for Reflexivity

A relation $R$ on a set $A$ is reflexive if every element in $A$ is related to itself. This means that for every element $a$ in the set $A$, the ordered pair $(a,a)$ must be present in the relation $R$. The given set is $A = \{1{,}2{,}3\}$. We need to check if $(1{,}1)$, $(2{,}2)$, and $(3{,}3)$ are in $R$.

Step 2: Check for Symmetry

A relation $R$ is symmetric if, whenever an element $a$ is related to an element $b$, then $b$ is also related to $a$. In other words, if $(a,b)$ is in $R$, then $(b,a)$ must also be in $R$. We will check the pairs in $R$ to see if their reverse pairs are also present.

Step 3: Check for Transitivity

A relation $R$ is transitive if, whenever $a$ is related to $b$ and $b$ is related to $c$, then $a$ is also related to $c$. This means if $(a,b)$ and $(b,c)$ are in $R$, then $(a,c)$ must also be in $R$. We will examine the pairs in $R$ to verify this condition.

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