Prove that if a subset C of R*R is symmetric with respect to both the x-axis and y-axis, then it is symmetric with respect to the origin.

Answer 1

A reflection about both the x and y axes will move a point to the opposite side of the origin (180 degrees away), and the same distance away from it.

Hence there is symmetry of a set about the origin, if there is symmetry about the x and y axis.

This is probably no a "proof" in words that set theorists would prefer.