Re: Induction on infinite unions/intersections?

[ View Follow Ups ] [ Post Followup ] [ Back to the Course Discussion List ] [ Help ]

Sets, Maps and Symmetry Groups Course Discussion List

Re: Induction on infinite unions/intersections?

Posted by Daniel Goroff on October 06, 1999 at 11:29:27:

In Reply to: Induction on infinite unions/intersections? posted by Kevin Hall on October 05, 1999 at 20:48:17:

It is true that the intersection of closed sets is always closed, no matter how many we take. The proof of this you mention in 2.4 does not use induction, though. Let P(n) denote your statement that

A1 Int A2 Int A3 ... Int An is closed for n in Z+ if A1, A2,...An are closed.

Then the argument you sketch does show that P(n) is true for all n in Z+. But no statement of the form P(n) has anything to say about intersections of infinitely many sets. Induction handles each finite case, infinitely many of them one after another, but no infinite cases. Be careful not to get mixed up over the fact that n, an index of predicates, and i, an index of sets, are both in Z+. We are not doing induction on i, but on n. Does this help? If what you suggested in your post was a valid argument for more than finite n, then it should work for unions, too, and we saw examples that show how infinite unions of closed sets need not be closed.

Follow Ups:

Post a Followup

Name    : 
E-Mail  : 
Subject : 

Comments:
>>
>>It is true that the intersection of closed sets is always closed, no matter how many we take.  The proof of this you mention in 2.4 does not use induction, though.  Let P(n) denote your statement that

>>A1 Int A2 Int A3 ... Int An is closed for n in Z+ if A1, A2,...An are closed.

>>Then the argument you sketch does show that P(n) is true for all n in Z+.  But no statement of the form P(n) has anything to say about intersections of infinitely many sets.  Induction handles each finite case, infinitely many of them one after another, but no infinite cases.  Be careful not to get mixed up over the fact that n, an index of predicates, and i, an index of sets, are both in Z+.  We are not doing induction on i, but on n.  Does this help?  If what you suggested in your post was a valid argument for more than finite n, then it should work for unions, too, and we saw examples that show how infinite unions of closed sets need not be closed.

[ View Follow Ups ] [ Post Followup ] [ Back to the Course Discussion List ] [ HELP ]

This discussion list is provided by The Instructional Computing Group (ICG) of FAS Computer Services