How do I prove that if a relation $R$ on set $A$ is symmetric, a new relation $R^1 = (A\times A) - R$ is also symmetric?

Question

I am confused about this problem, because I would have assumed that $R^1$ is not symmetric. If $(x,y)\in R$, then $(y,x)\in R$, would neither of these be in $R^1$?

Answer 1

Well if there was some $(x,y) \in R^1$ such that $(y,x) \not\in R^1$, then this implies $R$ is not symmetric.

To answer your question, neither of those would be in $R^1$ since $R^1$ is the complement of $R$.