Equivalence relations are often used to group together objects that are similar in some sense. The following are some examples of equivalence relations:

  1. The equality ("=") relation between real numbers or sets.
  2. The relation "is congruent to (modulo 5)" between integers.
  3. The relation "is similar to" on the set of all triangles.
  4. The relation "has the same birthday as" on the set of all human beings.
  5. The relation of logical equivalence on statements in first-order logic.
  6. The relation "is isomorphic to" on models of a set of sentences.
  7. The relation "is in thermal equilibrium with".
  8. The relation "has the same image under a function" on the elements of the domain of the function.
  9. Green's relations are five equivalence relations on the elements of a semi group.