Iniheritance of antisymmetry
Task number: 3373
Let \(R\) be an antisymmetric relation on a set \(X\). Show that every relation \(S\subset R\) is also antisymmetric.
Proceed by contradiction.
If \(S\) is not antisymmetric, then there exist distinct \(x,y\in X\) such that \((x,y)\in S\) and also \((y,x)\in S\).
But then \((x,y)\in R\) while \((y,x)\in R\), and so the relation \(R\) can not be antisymmetric.