Definition One: A relation over a set $X$ is symmetric if for all $a,b$ $\in X$, $(a,b)\in R$ if and only if $(b,a)\in R$.
Definition Two: A relation over a set $X$ is symmetric if for all $a,b$ $\in X$, $(a,b)\in R$ if $(b,a)\in R$.
Am I correct to believe that, if I want to prove a relation over a set $X$ is symmetric, then it does not matter if I use definition one or definition two? I think that those two definitions are the exact same despite the fact that one has a lone "if" and then other has "if and only if." Although this may seem really obvious, I want to double check that my thinking is correct.
Def. $1$ clearly implies Def. $2$.
You can easily show that Def. $2$ implies Def. $1$: we are already given that $(b,a)\in R\rightarrow(a,b)\in R$. Thus $(a,b)\in R\rightarrow (b,a)\in R$, which follows from interchanging $a,b$ in Def. $2$.