Mathematics Relation

Equivalence relations permeate mathematics with several salient examples readily available:

1. Residue classes [a]N consist of all numbers congruent (equivalent) modulo N
2. a negative number is a set of all equivalent pairs (a,b) of integers with a3. a rational number is a set of all equivalent pairs (a,b) of integers, where two pairs (a,b) and (c,d) are equivalent iff ad = bc
4. an irrational number is an equivalence class of sequences r1, r2, r3, ... of rational numbers, where two sequences {ri} and {si} are equivalent iff they have limits as i and their limits coincide.