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.

  • $\begingroup$ Are you confused about what a class of sets is, or about what "closed" means, or about what these operators are? $\endgroup$ – Alex Becker Sep 6 '12 at 2:37
  • $\begingroup$ Actually, all of those. Would it have been better to ask these separately?.. $\endgroup$ – James Sep 6 '12 at 2:39
  • $\begingroup$ I don't think you have to ask them separately. I just wanted clarification about your question. $\endgroup$ – Alex Becker Sep 6 '12 at 2:42

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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    $\begingroup$ What is the more elegant/formal definition of class? $\endgroup$ – Sigur Sep 6 '12 at 2:45
  • $\begingroup$ @Sigur: The term is being used informally here, so the question doesn’t really apply. $\endgroup$ – Brian M. Scott Sep 6 '12 at 2:50
  • $\begingroup$ I agree. It was only to get a reference here, instead of make another question. It is not necessary. Thanks. $\endgroup$ – Sigur Sep 6 '12 at 2:54
  • $\begingroup$ Does the union operation on a class 𝒞 = {{set1},{set2},{set3},...} also return the union of all the sets in the class, or does it only return the union of pairs of sets like in your example? $\endgroup$ – James Sep 6 '12 at 3:08
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    $\begingroup$ @James: Unless otherwise specified, it’s understood to be the binary union operation: $\mathscr{C}$ is closed under union if $A\cap B\in\mathscr{C}$ whenever $A,B\in\mathscr{C}$. No claim is being made about unions of more than two sets from the class. (One can then prove by induction that for any positive integer $n$, $A_1\cup A_2\cup\dots\cup A_n\in\mathscr{C}$ whenever $A_1,\dots,A_n\in\mathscr{C}$, but one cannot conclude the class is closed under taking unions of infinitely many of its members.) $\endgroup$ – Brian M. Scott Sep 6 '12 at 3:14

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