How to prove a relation on a set?

Question

I am looking to show that a relation of a set if reflexive. How do I go about formally writing that out, and how do I show logically that it is reflexive?

Suppose $R$ is a transitive and symmetric relation on a set $A$ and that $∃a ∈ A$ such that $aRx$ for all $x ∈ A$. Prove that $R$ is reflexive.

Thank you!

Answer 1

Let $a \in A$. Then there exists a $x \in A$ with $aRx$. Because of the symmetry we also have $xRa$. Since $R$ is transitive it follows from $aRx$ and $xRa$ that $aRa$. Hence $R$ is reflexive. I hope I helped you :)

Answer 2

So, let $x \in A$. We wish to show that $xRx$. By our third hypothesis, $aRx$. Since __________ $xRa$, and then since $xRa$ and $aRx$, by _________________, we have that $xRx$.

Filling in the blanks will provide a complete proof.