Equivalence Relation

An equivalence relation on a set $X$ is a Subset of $X \times X$, i.e., a collection $R$ of ordered pairs of elements of $X$, satisfying certain properties. Write ``$xRy$'' to mean $(x,y)$ is an element of $R$, and we say ``$x$ is related to $y$,'' then the properties are

1. Reflexive: $aRa$ for all $a \in X$,

2. Symmetric: $aRb$ Implies $bRa$ for all $a,b \in X$

3. Transitive: $aRb$ and $bRc$ imply $aRc$ for all $a,b,c \in X$,

where these three properties are completely independent. Other notations are often used to indicate a relation, e.g., $a \equiv b$ or $a \sim b$.

See also Equivalence Class, Teichmüller Space

References

Stewart, I. and Tall, D. The Foundations of Mathematics. Oxford, England: Oxford University Press, 1977.