Equivalence relation

Apr 3, 2023

Why do we care about equivalence relations? - Mathematics Stack Exchange

x,x

x,y -> y,x

x,y y,z -> x,z

We often use the tilde notation $a∼b$ to denote a relation.

Let ∼ be an equivalence relation on a set X and let x ∈ X. Then [x] = {y ∈ X : y ∼ x} is called the equivalence class of x
We will see that an equivalence relation gives rise to a partition via equivalence classes
Also, whenever a partition of a set exists, there is some natural underlying equivalence relation