> [!NOTE] **Definition** (Equivalence class of an element) >Suppose that $E$ is an [[Equivalence Relation|equivalence relation]] on a set $X.$ Given $x \in X$, we define the set $[x]_{E} = \{ y \in X \mid xEy \}$ to be the *equivalence class* of $x$. > [!Example] > [[Congruence Modulo n|Congruence Classes]] # Applications The set of equivalence classes, denoted $X/E,$ is called the *[[Quotient Set|quotient]]* (*quotient set* or *quotient space*) of $X$ by $E.$ The [[Quotient Map|quotient map]] given by $E$ is the function that maps each $x$ to the equivalence of $x.$