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Sure. I was considering operators to be axioms, e.g. 'union is the set of all distinct elements' is certainly a definition/axiom we created. It's been a while since I had Discrete, I'm not sure what the exact definition of 'axiom' is.
I think mathematics in terms of physics, because that's where I did my heavy learning. So when I think 'operator' I think in terms of QM operators, which are defined actions, instead of assumed properties. I see the two as separate, although I suppose they probably seem less so when viewed through a purely mathematical lens.