Prove that if R is a symmetric relation, so is R^2.

Question

Prove that if R is a symmetric relation, so is R^2. My attempt : The Relation R has (a,b) provided (b,a) is a member of R. So if I go on to find R^2 it will always have element (a,a) that makes R^2 symmetric. Am i correct and does this make sense?

Answer 1

If $R$ is a symmetric relation on a set $S$, then what you've shown so far is that $(a,a) \in R^2$ for all $a \in S$.

What you have not yet shown is that if $(a,b) \in R^2$ then $(b,a) \in R^2$.

Hint: if $(a,b) \in R^2$, then there is a $c \in S$ such that $(a,c)$ and $(c,b) \in R$.