Is the statement on the UNION and INTERSECTION of Indexed Sets always true?

[SmashtheVan]

Homework Statement

Can't quite figure out the LaTeX for Indexed Sets, so bear with me:

From "Book of Proof" Section 1.8 #11 http://www.people.vcu.edu/~rhammack/BookOfProof/index.html

Is the UNION of A_a, where a is in I, a subset of the INTERSECTION of A_a always true for any collection of sets A_a with index set I?

Homework Equations

The Attempt at a Solution

The answer listed in the solutions is "Yes, this is always true."

However, I contest that it is false.
My reasoning:

Given I= {1,2,3} and A_a=[a, 2a]

This gives me A₁= [1,2], A₂= [2,4], and A₃= [3,6]

The UNION is therefore [1,6] and the INTERSECTION is the nullset: {}

I find that the INTERSECTION is a subset of the UNION, but not the other way around, as the book asks.

Am I correct in my work, or is the statement given by the solutions manual correct?

 

[CompuChip]

Yes, the union contains all the elements that are in some of the sets; the intersection contains all the elements that are in all of the sets - in general there are fewer of those.

In fact the opposite is easy to show: suppose that x lies in the intersection*, then x lies in all A_a, so in particular it lies in one of them and therefore sits in the union as well. Hence, the intersection is a subset of the union. *) For the nitpickers/mathematicians: if no such x exists, i.e. the intersection is empty, then the proof is trivial because the empty set is a subset of any set by definition.

 

[gb7nash]

I agree with you. It's almost never the case that the union of any number of sets is a subset of the intersection of the same sets. The only way this is possible is if all sets have the same elements (in which case the union is equal to the intersection).

 

[SmashtheVan]

Thanks everyone.

When I saw the solution it didn't make sense to me based on the previous exercises I got right using Unions/Intersections, so I gave myself that example to work it out mathematically. Must have been just a mixup by the author over the Union and Intersection symbols. Just looking for a 2nd opinion.