equivalence relation

A binary relation R on a set A is an equivalence relation if it is also a

• reflexive relation
• symmetric relation
• transitive relation

There are multiple ways that equivalence relations are notated, but the most common are “a ~ b” and “a ≡ b” when it is obvious that a relation R is being referenced. Otherwise the equivalence may be denoted “a ~_R b“, “a ≡_R b“, or “aRb“.

So for every a, b, c in set A and

• a ~ a (Reflexivity)
• if a ~ b if and only if b ~ a (Symmetry) 
• if a ~ b and b ~ c, then a ~ c (Transitivity)