Thread: Assymetric equality

I heard a friend who is in computer science talk about asymmetric equality or inequality the other day. He explained that it is when the result from one side of the equal sign toward the other is not the same as if you were to look at it the other way around.
For example,
A = B

If you read it A toward B, it is true, but B toward A it becomes not =. This makes absolutely no sense to me and I think he is pulling my leg.
Google seems to think so too but I did find a (very) advanced math journal that I had to pay 65 bucks for but had my topic in it. Does anyone know if such things are in the realm of the impossible?
Thanks
Chris

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.

Mathematically speaking, saying means that . However, if A and B are events, then you could say that A implies B but B does not imply A. For instance, consider A being the event that a given number is 2 and B being the event that a given number is even. Then A implies B since 2 is even. However, B does not necessarily imply A since there are countably infinite even numbers.