Homework Help: Help with dimensions

1. Feb 21, 2008

pte419

1. For subsets U1, U2, U3 of a finite set, prove that

dim(U1+U2+U3) = dimU1 + dimU2 + dimU3 - dim(U1∩U2) - dim(U1∩U3) - dim(U2∩U3) + dim(U1∩U2∩U3)

2. dim(U1+U2) = dimU1 + dimU2 - dim(U1∩U2)

3. I found that U1+U2 theorem in my book, and I think I should use that, but I'm not sure where to start...

2. Feb 21, 2008

EnumaElish

Let V = U1 + U2. Now apply the theorem to V + U3.

Unless you are asked to prove 2 before proving 1. If this is the case please make it clear.

Last edited: Feb 21, 2008
3. Feb 21, 2008

StatusX

The two equations are also true, and easier to see, with the vectors spaces replaced by finite sets and the dimensions replaced by the sizes of the sets. It's possible, by picking certain bases, to make the two problems equivalent. But, as enumaelish hints, induction is probably easier.

4. Feb 21, 2008

pte419

proof

I am only asked to prove equation one, but in doing that, I am allowed to use equation 2.

5. Feb 21, 2008

pte419

Thanks guys, I've got one more question.

I did: dim(V+U3)
and I've ended up with:
dim(V+U3) = dimU1 + dimU2 - dim(U1∩U2) + dimU3 - dim(V∩U3)

Is there a property I can use to show that dim(V∩U3) is equivalent to the terms I still need to include for the proof? I can't find anything helpful in my book...

6. Feb 21, 2008

Vid

$$dim(V\cap U_{3}) = dim((U_{1}+U_{2})\cap U_{3}) = dim((U_{1}\cap U_{3}) + (U_{2}\cap U_{3}))$$

Break that term up again using the second part and you're done.