An equivalence relation on a set $A$ is a relation that is reflexive, symmetric and transitive.

These three properties make the relation behave like a generalized notion of “being the same for the purpose we care about.”

Equivalence classes

For $a\in A$, its equivalence class is

$$[a]=\lbrace x\in A:x\sim a\rbrace.$$

Any two equivalence classes are either identical or disjoint.

Partitions

A partition of a set divides it into nonempty disjoint subsets whose union is the whole set.

Every equivalence relation determines a partition into its equivalence classes, and every partition determines an equivalence relation by declaring two elements equivalent when they belong to the same block.

Example: congruence modulo $n$

For integers, define

$$a\sim b$$

when $n$ divides $a-b$. The equivalence classes collect integers with the same remainder modulo $n$.

Equivalence relations let mathematics replace individual objects by classes of objects that are indistinguishable under a chosen criterion.