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
Definition D179
Equivalence class
Formulation 0
Let $X$ be a D11: Set such that
(i) \begin{equation} X \neq \emptyset \end{equation}
(ii) ${\sim} \subseteq X \times X$ is an D178: Equivalence relation on $X$
The equivalence class of $x \in X$ with respect to ${\sim}$ is the D11: Set \begin{equation} \{ y : (x, y) \in {\sim} \} \end{equation}
Children
D4430: Constancy class
Results
R714: Element belongs to its own equivalence class
R716: Equivalence class is not empty