Equivalence classes: group objects without making the rule ambiguous

Reflexivity says every object is related to itself. Symmetry says that relatedness can be reversed. Transitivity says that two compatible links can be joined. None of these conditions follows automatically from the everyday word similar. Being within one meter is reflexive and symmetric for positions on a line, but not transitive: a chain of short gaps can connect endpoints farther than a meter apart. Such a relation cannot define consistent classes of nearby points.

For a genuine equivalence relation, the class of an object consists of everything related to it. Reflexivity ensures the object belongs to its own class. If two classes share an element, symmetry and transitivity imply that every member of either class belongs to the other, so the classes are equal. Therefore two equivalence classes are either identical or disjoint; they never partially overlap. This is a consequence to prove, not an extra axiom to memorize.

A partition is a family of nonempty, pairwise disjoint subsets whose union is the whole set. Starting from a partition, define two objects to be related exactly when they lie in the same block. Every object has a block, so reflexivity holds; sharing a block is symmetric; and disjointness makes a shared middle object's block unique, giving transitivity. Thus partitions and equivalence relations are two descriptions of the same organizing information.

The quotient set is the set of classes, not the union of their elements. For integers modulo three, it has three elements: the zero-remainder class, the one-remainder class, and the two-remainder class. Each of those elements is itself an infinite set of integers. Choosing the representatives zero, one, and two is convenient, but choosing three, four, and five would name the same three classes. Representatives are labels, not the underlying class objects.

To define a function on classes, start with a rule on representatives and prove equivalent representatives give the same result. The map taking an integer's class modulo three to its remainder is well-defined. The proposed map taking that class to the integer's square is not: zero and three represent the same class but have different squares. The notation using brackets cannot repair this dependence; a proof of independence is required.

Some operations become well-defined when their outputs are also classes. Sending a class to the class of the representative's square works modulo three: if a minus b is divisible by three, then a squared minus b squared equals their difference times their sum and is also divisible by three. The output integers may differ, but their classes agree. Always identify whether equality is being demanded for original objects or for quotient objects; those are different standards.