Equivalence relations and classes

Equivalence relations “group together” the elements that share a property. Each element $a$ identifies its equivalence class $[a]=\{x\in A : x\mathbin{R} a\}.$ The set of classes forms a partition of $A$: each element belongs to exactly one class.

Example — Remainder of division by $3$

On $\mathbb{Z}$ we set $a\mathbin{R} b$ if $a-b$ is a multiple of $3$. It is an equivalence relation (the three properties can be checked). There are three classes: $[0]=\{\ldots,-3,0,3,6,\ldots\},\quad [1]=\{\ldots,-2,1,4,7,\ldots\},\quad [2]=\{\ldots,-1,2,5,8,\ldots\}.$ Every integer belongs to exactly one class, determined by the remainder ($0$, $1$ or $2$) upon division by $3$. This is the first step of modular arithmetic.

Links

Topics: Set theory
Concepts: Equivalence class · Partition · Equivalence relation