If $R$ is symmetric and transitive then prove that $R$ is reflexive

Question

Let $R$ be a relation on $A$ s.t $$\forall a \in A, \exists b\in A, \quad aRb$$ Prove that if $R$ is symmetric and transitive then $R$ is reflexive.

I don't know how to solve it, I just did that $$\text{Symmetric : } \forall a,b \in A,\quad aRb \implies bRa$$ $$\text{Transitive : } \forall a,b,c \in A \quad aRb \text{ and } bRc \implies aRc$$

and we want to prove $$\forall a \in A,\quad aRa.$$

Well, put the three conditions together to deduce the statement.. Use transitivity for $c=a$.. — Berci, Nov 13 '21 at 15:34

score 3 · Accepted Answer · answered Nov 13 '21 at 15:40

3

Let $a\in A$. Then by hypothesis, $aRb$ for some $b\in A$.

Thus $bRa$ by symmetry. Hence, $aRb$ and $bRa$ imply $aRa$ by transitivity. Done.

answered Nov 13 '21 at 15:40

Wuestenfux

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If $R$ is symmetric and transitive then prove that $R$ is reflexive

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