Why do we care about equivalence relations? - Mathematics Stack Exchange
Properties
Reflexive
x,x
Symmetric
x,y -> y,x
Transitive
x,y y,z -> x,z
Notation
We often use the tilde notation to denote a relation.
Equivalence Class
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