system
Given the following post, respond as if you're commenting on Reddit:
human
Is there an operator that can have an associative property without a commutative property, or vice versa?

It seems that operators, whether they're pertaining to numbers, sets, propositional logic, vectors or whatever, will only have an associative property (ex. (a+b)+c = a+(b+c) ) when they also have a commutative property (ex. a+b = b+a). 

I can't quite put my nose on what the two properties have in common with each other; is there an abstract proof (or disproof) that they're bi-conditional, or something along those lines? Sorry if my question's not clear I'm slightly drunk and frustrated because I can't figure it out on my own.
gpt
There are plenty of operators that are associative, but not commutative. And plenty that are the other way around. Just to give one example of each:

Function composition: we define the composition of two functions from R->R, f * g as f(g(x)). Checking associativity:

(f * g) * h = f(g(x)) * h = f(g(h(x)) = f * g(h(x)) = f * (g * h)

However, this operator is not commutative. Let f(x) = x + 1 and g(x) = x^(2), then f * g = x^2 + 1, but g * f = (x + 1)^(2).

The other way around, consider the operator that takes 2 real numbers and returns the average. So a (+) b = (a + b) / 2. This operator is clearly commutative since addition is. But it is not associative. Take a = 0, b = 4, c = 2:

(a (+) b) (+) c = 2 (+) 2 = 2

a (+) (b (+) c) = 0 (+) 3 = 1.5

There are plenty of other examples that you can find for either direction, so in general it is definitely not the case that associativity and commutativity are interchangeable.