▾ Set of symbols
▾ Alphabet
▾ Deduction system
▾ Theory
▾ Zermelo-Fraenkel set theory
▾ Set
▾ Binary cartesian set product
▾ Binary relation
▾ Relation class
▾ Set of relation classes
Quotient set
Formulation 0
Let $X$ be a D11: Set such that
(i) ${\sim} \subseteq X \times X$ is an D178: Equivalence relation on $X$
(ii) \begin{equation} \forall \, x \in X : {\sim}(x) : = \{ y : (x, y) \in {\sim} \} \end{equation}
The quotient set of $X$ modulo ${\sim}$ is the D11: Set \begin{equation} X / {\sim} : = \{ {\sim}(x) : x \in X \} \end{equation}
Formulation 1
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 \begin{equation} X / {\sim} : = \left\{ \{ y : (x, y) \in {\sim} \} : x \in X\right \} \end{equation}
Formulation 2
Let $X$ be a D11: Set such that
(i) ${\sim} \subseteq X \times X$ is an D178: Equivalence relation on $X$
(ii) \begin{equation} \forall \, x \in X : [x]_{\sim} : = \{ y : (x, y) \in {\sim} \} \end{equation}
The quotient set of $X$ modulo ${\sim}$ is the D11: Set \begin{equation} X / {\sim} : = \{ [x]_{\sim} : x \in X \} \end{equation}
Also known as
Set of equivalence classes
Child definitions
» D181: Canonical set epimorphism
Results
» R4886: Characterisation of a quotient set by proper set partitions