# What is the domain of a set operation (e.g., union and intersection)?

My understanding: The set operations (such as union and intersection) are operations and so functions with a domain and codomain. A function's domain and codomain are always sets. The input to a set operation is either a set or a tuple of sets.

My thoughts: My thoughts about what the domain of a set operation would be (based on the above understanding) has brought me to "the set of all sets". This is clearly wrong.

My Question: What is in fact the domain of a set operation? Am I to instead understand the domain of such an operation to be a class? This would cause me to revise my current understanding of a function to allow the domain and codomain to be classes.

Thanks.

• Operations can be defined on proper classes. – Dan Rust Sep 6 '16 at 23:31
• Let me make sure I understand you: the definition of "function" can be extended (beyond the usual definition I see in which a function's domain and codomain are always sets) by allowing a function's domain and codomain to be sets or proper classes? – Josh Flick Sep 6 '16 at 23:54
• " "the set of all sets". This is clearly wrong." We have a loophole. Instead of calling it a "set" of all sets, call it a "class" of all sets. Frustratingly it really is that simple. we're only forbidden to define sets in terms of sets to avoid paradoxes. We use classes to talk of collections of sets The domain is the class of all ordered pairs of sets. (Note: for an binary operator to be viewed as a function it takes pairs of values as inputs and the domain is the cross product of sets or classes.) – fleablood Sep 7 '16 at 0:17
• You understand me, @fleablood. I find the notion of classes very frustrating indeed! Admittedly, however, this finding is based on a rather cursory glance. I will be studying them in more detail after this. – Josh Flick Sep 7 '16 at 0:25
• "the set of all sets" is verbotten for purposes of definition only. We don't want to be able to define a set so that it will have itself as a member. But we definitely want to be able to talk about sets of sets (just not sets of sets with themselves as members or subsets). As a binary operation set union and intersection the domain is, naturally the set of all sets and ... oops. Except it's a perfectly legitimate concept It's only a problem when we define things by their inclusion. So "collection of sets" seems okay. It IS the answer, after all. – fleablood Sep 7 '16 at 1:32

In most contexts short of serious mathematical logic the set operations are on subsets of a particular given set $S$ - the universe in that particular context. Then the domain of the set operations is the (well defined) power set of $S$ - the set of all its subsets.