To prove that if $R$ is a relation on a set $A$ and is symmetric and transitive, then $R$ is an equivalence relation, we need to show that $R$ also satisfies the property of reflexivity. An equivalence relation on a set must satisfy three properties: reflexivity, symmetry, and transitivity.

Given:

• $R$ is a relation on $A$.
• $R$ is symmetric: If $(a, b) \in R$, then $(b, a) \in R$.
• $R$ is transitive: If $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$.

To Prove:

• $R$ is reflexive: For every $a \in A$, $(a, a) \in R$.

If we can prove that $R$ is reflexive, then we can conclude that $R$ is an equivalence relation because it will satisfy all three required properties.

Proof

1. Consider any element $a \in A$. We want to show that $(a, a) \in R$.

2. Since $R$ is symmetric and transitive, let us examine the implications:

   • Suppose there exists some $b \in A$ such that $(a, b) \in R$ (or if not, $R$ is empty, in which case reflexivity does not apply since there are no elements in $A$).

3. If $R$