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.

  • Hint

    Proceed by contradiction.

  • Solution

    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.

Difficulty level: Easy task (using definitions and simple reasoning)
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