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(C) An equivalence relation $R$ on a set $A$ satisfies three properties: $1$. Reflexivity: $(a, a) \in R$ for all $a \in A$. $2$. Symmetry: If $(a, b)

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Equivalence

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(C) An equivalence relation $R$ on a set $A$ satisfies three properties: $1$. Reflexivity: $(a, a) \in R$ for all $a \in A$. $2$. Symmetry: If $(a, b) \in R$,then $(b, a) \in R$. $3$. Transitivity: If $(a, b) \in R$ and $(b, c) \in R$,then $(a, c) \in R$. Now,consider $R^{-1} = \{(b, a) : (a, b) \in R\}$. $1$. Reflexivity: Since $(a, a) \in R$,then $(a, a) \in R^{-1}$. Thus,$R^{-1}$ is reflexive. $2$. Symmetry: If $(a, b) \in R^{-1}$,then $(b, a) \in R$. Since $R$ is symmetric,$(a, b) \in R$,which implies $(b, a) \in R^{-1}$. Thus,$R^{-1}$ is symmetric. $3$. Transitivity: If $(a, b) \in R^{-1}$ and $(b, c) \in R^{-1}$,then $(b, a) \in R$ and $(c, b) \in R$. Since $R$ is transitive,$(c, a) \in R$,which implies $(a, c) \in R^{-1}$. Thus,$R^{-1}$ is transitive. Since $R^{-1}$ satisfies all three properties,it is an equivalence relation.

If $R$ is an equivalence relation on a set $A$,then $R^{-1}$ is

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