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.
(N/A) Let $A = \{1, 2, 3\}$.
$A$ relation $R$ on $A$ is defined as $R = \{(1, 2), (2, 1)\}$.
$1$. Reflexivity: For $R$ to be reflexive,$(a, a) \in R$ for all $a \in A$. Here,$(1, 1), (2, 2), (3, 3) \notin R$. Therefore,$R$ is not reflexive.
$2$. Symmetry: For $R$ to be symmetric,if $(a, b) \in R$,then $(b, a) \in R$. Since $(1, 2) \in R$ and $(2, 1) \in R$,the condition holds. Therefore,$R$ is symmetric.
$3$. Transitivity: For $R$ to be transitive,if $(a, b) \in R$ and $(b, c) \in R$,then $(a, c) \in R$. Here,$(1, 2) \in R$ and $(2, 1) \in R$,but $(1, 1) \notin R$. Therefore,$R$ is not transitive.
Hence,$R$ is symmetric but neither reflexive nor transitive.
$A$ relation $R$ on $A$ is defined as $R = \{(1, 2), (2, 1)\}$.
$1$. Reflexivity: For $R$ to be reflexive,$(a, a) \in R$ for all $a \in A$. Here,$(1, 1), (2, 2), (3, 3) \notin R$. Therefore,$R$ is not reflexive.
$2$. Symmetry: For $R$ to be symmetric,if $(a, b) \in R$,then $(b, a) \in R$. Since $(1, 2) \in R$ and $(2, 1) \in R$,the condition holds. Therefore,$R$ is symmetric.
$3$. Transitivity: For $R$ to be transitive,if $(a, b) \in R$ and $(b, c) \in R$,then $(a, c) \in R$. Here,$(1, 2) \in R$ and $(2, 1) \in R$,but $(1, 1) \notin R$. Therefore,$R$ is not transitive.
Hence,$R$ is symmetric but neither reflexive nor transitive.
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