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Surely the basic set operations could be defined that way, but then some weird non-standard operation like ##\omega:\{a,b\}\times{}\{c,d\}\mapsto{}\{a\}## would also be "defined"..

I'm asking this question because I'm wondering if unions and complements, or intersections and complements, are "functionally complete". Can I express ALL definable n-ary set operation using just unions and complements? Like I can express all n-ary boolean functions using just disjunction and negation?

Because if that is the case, the σ-algebra on a set X would be like the subset of the power set of X that is closed under ANY set operation. Or is it that the σ-algebra on X is just the subset of the power set of X closed under enough set operations that allow for all the useful constructions on a σ-algebra to be defined?