Equivalence relation

A 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.

• (1) Equivalence classes are disjoint.
• (2) The union of all equivalence classes is the whole set X.

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:

• quotients of vector spaces,
• quotients of groups,
• quotients of topological spaces

Two trivial examples, on any set:

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