1. Set Theory

Hello!!! Please help meeeeeee!! I am just doing some homework problems for my test and I come across this one I don't understand.

1) Is it possible to find set A and B such that both A in B and A subset of B are true? Give an example or prove this is impossible.

2. Originally Posted by jenjen
Hello!!! Please help meeeeeee!! I am just doing some homework problems for my test and I come across this one I don't understand.

1) Is it possible to find set A and B such that both A in B and A subset of B are true? Give an example or prove this is impossible.

A={null_set}, B={null_set, {null_set}}.

That is A is the set whose only element is the null set,
and B is the set with two elements, the null set, and the set
whose only element is the null set (which is A).

RonL

3. elmentary constructions on sets

Thank you Captainblack. I hope you are still there because I just came across one more question.

1) Let A, B, C be subsets of some fixed set S.
Prove that (A-B)-C = (A-C) - (B-C)

Thank you so much.

4. Originally Posted by jenjen
Thank you Captainblack. I hope you are still there because I just came across one more question.

1) Let A, B, C be subsets of some fixed set S.
Prove that (A-B)-C = (A-C) - (B-C)

Thank you so much.
[x in (A-B)-C] iff [x in (A-B) and x not in C] iff [x in A and x not in B and x not in C)

[y in (A-C)-(B-C)] iff [y in (A-C) and y not in (B-C)] iff [(y in A and y not in C) and y not in (B-C)]

But [(y not in C) and (y not in (B-C))] iff [y not in B], so:

[y in (A-C)-(B-C)] iff [y in A and y not in C and y not in B]

The final steps are trivial, what we have shown is that [x in (A-B)-C] iff
[x in (A-C)-(B-C)]

Informally the process involved in a proof is just to expand what the two
expressions mean, and you evantualy find that they mean the same thing.

RonL