equivalence relation
An equivalence relation is a way of declaring certain elements of a set “the same for present purposes” that behaves like genuine equality. It is a relation ~ satisfying three axioms: reflexivity (every x ~ x), symmetry (if x ~ y then y ~ x), and transitivity (if x ~ y and y ~ z then x ~ z). Everyday sameness — having the same birthday, living in the same country — obeys exactly these three rules.
The payoff is partitioning. An equivalence relation on a set A carves A into disjoint, non-empty equivalence classes, where the class of x is the set of all elements equivalent to x. Every element lies in exactly one class, and two elements are equivalent if and only if they share a class. Partition and equivalence relation are two views of one structure — given either, you can reconstruct the other.
This is the standard machinery for building new objects by “gluing”. The rationals are equivalence classes of integer pairs (a, b) ~ (c, d) when ad = bc; the real numbers can be built as equivalence classes of Cauchy sequences of rationals (two sequences equivalent when their difference tends to 0); residues modulo n are equivalence classes of integers. Whenever you define something “up to” some identification, an equivalence relation is quietly at work.
On Z, “a ~ b iff a − b is divisible by 3” is an equivalence relation. Its three classes are {…,−3,0,3,…}, {…,−2,1,4,…}, {…,−1,2,5,…} — the remainders 0, 1, 2 modulo 3, which partition Z.
Congruence mod 3 splits the integers into exactly three classes.
Dropping just one axiom breaks the partition picture: “x ≤ y” is reflexive and transitive but not symmetric (it is a partial order), and “|x − y| < 1” is reflexive and symmetric but not transitive.