Equivalence relations are often used to group together objects that are similar in some sense. The following are some examples of equivalence relations:
- The equality ("=") relation between real numbers or sets.
- The relation "is congruent to (modulo 5)" between integers.
- The relation "is similar to" on the set of all triangles.
- The relation "has the same birthday as" on the set of all human beings.
- The relation of logical equivalence on statements in first-order logic.
- The relation "is isomorphic to" on models of a set of sentences.
- The relation "is in thermal equilibrium with".
- The relation "has the same image under a function" on the elements of the domain of the function.
- Green's relations are five equivalence relations on the elements of a semi group.