Let $R$ be a relation on a group $A$, how to prove that if $R$ is symmetric and transitive relation then $R$ is also reflexive relation?
$R$ is symmetric relation, so for all $x,y ∈ A$ if $xRy$ then $yRx$.
$R$ is transitive relation, so for all $x,y,z ∈ A$ if $xRy$ and also $yRz$ then $xRz$.
By the definitions we get that:
for all $x,y,z ∈ A$ if $xRy$ and also $yRz$ then $yRx$ , $xRz$ , $zRy$ , $zRx$.
I have no idea how to continue from here to prove that $R$ is also reflexive relation, i.e all $x ∈ A$ are in the relation $R$: $xRx$.
I tried this:
Because $xRy$ and $yRx$, (as we saw above by definitions), I thougt that $xRx$, because $y$ exists in both terms. but its not like that because $xR∘Rx$, i.e $(x,x) ∈ R∘R$.
Can someone help with an idea?