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What is this question asking?

"Which of the following classes of sets are closed under each of the following operations: union, intersection, power set operation?" An example class given is the class of all finite sets.

I guess I am asking for the meaning of classes of sets, closed classes, and what the operations do to these classes.

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    I don't think you have to ask them separately. I just wanted clarification about your question.2012-09-06

1 Answers 1

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Added after clarification of the question: In this context the term class of sets is being used very informally. You should think of it simply as some well-defined collection of sets, in the sense that it’s possible to decide unambiguously whether a set belongs to the collection or not. For instance, once you have a definition of finite set, it’s possible in principle to determine whether any given set is finite and hence whether it belongs to the class of finite sets or not.

The operations don’t do anything to the classes: they are operations on the members of the classes, and we’re interested in whether the results of those operations are themselves always members of the same class.

When we say that a collection $\mathscr{C}$ of sets is closed under some operation, we just mean that when you apply that operation to members of $\mathscr{C}$, you always get a member of $\mathscr{C}$. Suppose, for instance, that $\mathscr{C}$ is the class of all finite sets. If $A,B\in\mathscr{C}$, is it true that $A\cup B\in\mathscr{C}$? Yes: the union of two finite sets is finite. Therefore $\mathscr{C}$ is closed under $\cup$. Is it true that $A\cap B\in\mathscr{C}$? Again, yes: the intersection of two finite sets is finite, so $\mathscr{C}$ is closed under $\cap$. Finally, is it true that $\wp(A)\in\mathscr{C}$? Once again the answer is yes: a finite set has only finitely many subsets, so its power set is again finite, and $\mathscr{C}$ is closed under $\wp$. (In fact if $A$ has $n$ members, $\wp(A)$ has $2^n$ members.)

Suppose now that $\mathscr{C}$ is the class of all infinite sets. Then $\mathscr{C}$ is closed under union and power set (why?), but not under intersection: the set of positive integers and the set of negative integers are infinite sets whose intersection is not infinite. (It’s as far from infinite as you can get!)

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    Okay, that is what I gathered from your answer.2012-09-06