ThmDex – An index of mathematical definitions, results, and conjectures.
 ▼ Set of symbols ▼ Alphabet ▼ Deduction system ▼ Theory ▼ Zermelo-Fraenkel set theory ▼ Set ▼ Binary cartesian set product ▼ Binary relation ▼ Relation class ▼ Set of relation classes
Definition D180
Quotient set

Let $X$ be a D11: Set such that
 (i) ${\sim} \subseteq X \times X$ is an D178: Equivalence relation on $X$ (ii) $$\forall \, x \in X : {\sim}(x) : = \{ y : (x, y) \in {\sim} \}$$
The quotient set of $X$ modulo ${\sim}$ is the D11: Set $$X / {\sim} : = \{ {\sim}(x) : x \in X \}$$

Let $X$ be a D11: Set such that
 (i) ${\sim} \subseteq X \times X$ is an D178: Equivalence relation on $X$
The quotient set of $X$ modulo ${\sim}$ is the D11: Set $$X / {\sim} : = \left\{ \{ y : (x, y) \in {\sim} \} : x \in X\right \}$$

Let $X$ be a D11: Set such that
 (i) ${\sim} \subseteq X \times X$ is an D178: Equivalence relation on $X$ (ii) $$\forall \, x \in X : [x]_{\sim} : = \{ y : (x, y) \in {\sim} \}$$
The quotient set of $X$ modulo ${\sim}$ is the D11: Set $$X / {\sim} : = \{ [x]_{\sim} : x \in X \}$$
Children
 ▶ Canonical set epimorphism