This site is being phased out.

# Equivalence relation

From Mathematics Is A Science

Jump to navigationJump to searchA relation on set X is called an *equivalence relation* if it satisfies the following conditions:

*Reflexivity*: $A \sim A$ for all $A \in X$.*Symmetry*: $A \sim B => B \sim A$ for all $A,B \in X$.*Transitivity*: $A \sim B, B \sim C => A \sim C$ for all $A,B,C \in X$.

The *equivalence class* $[A]$ of $A\in X$ is the set of all elements equivalent to $A$:
$$[A]=\{B\in X: B \sim A\}.$$

**Theorem.**

In other words this is a partition. See also quotient sets.

Examples:

- integers with the same remainder, see modular arithmetic;
- integers as equivalence classes of finite sets;
- rational numbers as equivalence classes of pairs of integers;
- functions with the same derivative, see antiderivative;
- connected components;
- homotopic maps, see homotopy;
- cycles that form a boundary, see homology as an equivalence relation (and bordism);
- homeomorphic topological spaces, see topological equivalence;
- topological spaces of the same homotopy class, see homotopy equivalence;
- isomorphic vector spaces and groups.

In all of these examples, the equivalence relation respects the extra structure the set possesses (algebra, topology etc). To understand that see:

Two trivial examples, on any set:

- $x \sim y$ for all $x,y$;
- $x \sim y$ iff $x=y$;