I'm stucked with proving the following proposition.
We have given a set $X$, a non-empty set $I$ and a family $(r_i)_{i \in I}$ of relations on $X$. For $x,y \in X$ we have $x \ r \ y$, if for all $i \in I$ we always have $x \ r_i \ y$. Provide a proof for: If there exists $i \in I$ such that $r_i$ is antisymmetric, then $r$ is antisymmetric.
My intuition says that the proposition is false, because it assumes "there exists $i \in I$ such that $r_i$ is antisymmetric ..." and not "for all $i \in I$, $r_i$ is antisymmetric". Is this correct? What would the proof look like?
Assume $xry$ and $yrx$ then $xr_my$ and $yr_mx$ for all $m$. In particular $xr_i y$ and $yr_i x$ so $x=y. $