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Equivalence Relation**

A relation \(\sim\) is called an equivalence relation on a set \(X\) if it is reflexive, symmetric and transitive, i.e. \(\forall x\in X:x\sim x\), \(\forall x,y\in X:x\sim y\Leftrightarrow y\sim x\), and \(\forall x,y,z\in X:(x\sim y\land y\sim z)\Rightarrow x\sim z\). Given an element \(x\in X\), its 'equivalence class' is the subset \([x]:=\{y\in X|x\sim y\}\), sometimes denoted \([x]_\sim\). The equivalence classes of elements of \(X\) under \(\sim\) form a partition of \(X\).