Equivalence Relation

DEFINITION1: Let  X be a set and  R\subset{X\times{X}}. If the relation  R is reflexive, symmetric and transitive, then the relation  R is called an "equivalence relation" and denoted by  R=\sim in general.

DEFINITION2: Let  \sim be an equivalence relation over a set  X and  a\in{X}. The set defined as  \{x\in{X}\:|\:a\sim{x}\}\subset{X} is called the “equivalence class” of  a under  \sim and denoted by  \overline{a},  [a] or  [a]_{\sim}. Since  a\sim{a} for every  a\in{X}, then  a\in{\overline{a}}. So the equivalence class  \overline{a} is non-empty for every  a\in{X}. The family of all the equivalence classes of the relation  \sim is called the “quotient set” of  X by  \sim and denoted by  {^X}/{_\sim}

I.e.,

 {^X}/{_\sim}=\{\overline{a}\:|\:a\in{X}\}\subset{\mathbf{P}(X)}.

DEFINITION3: Let  \sim be an equivalence relation over a set  X and  a,b\in{X}. If  b\in{\overline{a}}, then  b is called a “representative class” of the equivalence class  \overline{a}.

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