Identity

Identity is a primary, axiomatic concept. It refers to the "this"-ness of an object, the fact that it is this and not that.

Mathematics conceptualizes identity using the equals ("=") symbol.

Equivalence relations

Things can be identical in some respects, but not others.

Standard mathematics formalizes this idea as follows.

Let $S$ be a set. An equivalence relation is a relation $\sim$ on $S$ satisfying three axioms. For any $a,b,c\in S$,

1. (Reflexivity) $a\sim a$,
2. (Symmetry) If $a\sim b$ then $b\sim a$,
3. (Transitivity) If $a\sim b$ and $b\sim c$, then $a\sim c$.

If $a\in S$, we denote the equivalence class of $a$ to be the set $[a]:=\{b\in S:b\sim a\}$. Define the quotient of $S$ by the equivalence relation $\sim$, or $S/\sim$, as the set of all equivalence classes of elements in $S$.

Note that if $a\sim b$, then $[a]=[b]$.