Prove that every antisymmetric relation is weakly antisymmetric

Question

antisymmetric: if, for all x,y∈X, if xRy holds, then yRx does not
weakly antisymmetric: if, for all x,y∈X, if xRy and yRx hold, then x=y

Answer 1

Warning: what you refer to as 'weakly antisymmetric' is what is more commonly known as 'antisymmetric'.

Let $R$ be an antisymmetric relation on a set $X$. Assume that $R$ is not weakly antisymmetric. Then there exist $x,y\in X$ such that $xRy$ and $yRx$, but $x\ne y$. However, $xRy$ and $yRx$ contradict antisymmetry. Thus, $R$ is weakly antisymmetric.