It often happens in mathematics that some notation or term is given different precise definitions. For example, is it possible that 2 + 2 = 1? Yes, if you redefine + and/or = (see arithmetic modulo n, in this case n = 3).

Equality = is usually interpreted as identity, i.e., x = y means that x and y denote the same object. In this case, equality is an equivalence relation, in particular, symmetric (x = y implies y = x). However, there are numerous other interesting binary relations besides identity, some of them are equivalence relations and some are not.

In computer science, a very important relation is rewriting, or reduction, between two programs, or expressions. For example, we can say that (2 + 3) * 4 reduces to 5 * 4, which reduces to 20, but not vice versa. Yes, (2 + 3) * 4 and 5 * 4 still denote the same number 20, but we choose to focus on the simplifying relation to study how expressions are evaluated. Let us denote the simplifying relation by -> and the composition of an arbitrary number of simplifying steps by ->*. Then we can ask questions whether for every expression E the chain E -> E1 -> E2 -> ... is finite, whether E ->* E1 and E ->* E2 imply E1 ->* E' and E2 ->* E' for some E' (this property is called confluence), etc. If we replace -> and ->* by =, these questions become trivial.

So, sometimes it makes sense to study asymmetric refinements of equality. Usually, though, such refinements are not denoted by = or ≡ and are not called equality; these two symbols are reserved for equivalence relations.