Set Theory Proof Homework: Can You Help?

[number0]

Homework Statement

Homework Statement are located in the pdf below. I also upload the file onto an online viewer for pdf

for those people who are afraid to download attachments.

Here is the link: http://view.samurajdata.se/psview.php?id=c1f5a372&page=1

Homework Equations

None.

The Attempt at a Solution

The attempt at a solution is located in the pdf below.

Here is the link again for those who do not wish to download the file: http://view.samurajdata.se/psview.php?id=c1f5a372&page=1

Can anyone see if I did anything wrong in the proof? I am not talking about the format of the proof, just the underlying logic. Thanks!

 

[Mark44]

If you can use deMorgan's Laws, this can be done quite a bit shorter.

B - A = B \cap A^C

So
A \cup (B - A) = A \cup (B \cap A^C) = (A \cup B) \cap (A \cup A^C)

The latter expression is just U, the universal set.

 

[number0]

If you can use deMorgan's Laws, this can be done quite a bit shorter.

B - A = B \cap A^C

So
A \cup (B - A) = A \cup (B \cap A^C) = (A \cup B) \cap (A \cup A^C)

The latter expression is just U, the universal set.

Yes, I know that. I should have mentioned it before hand that I wanted to use the "choose-an-element method" to prove it rather than algebraic manipulation. Other than that, do you think that there was anything wrong with my proof in the pdf?

 

[Mark44]

IMO, you are overcomplicating things by using contradiction proofs. Why don't you just prove the two statements directly?

 

[number0]

IMO, you are overcomplicating things by using contradiction proofs. Why don't you just prove the two statements directly?

To be honest, I do not really know how to prove one of the statements.

That is, A + B is a subset of A + (B - A).

The plus signs means "union."

Attempt:

1) Let x be in A + B.
2) We can say x is in A or x is in B.
3) If x is in A, then x is in A + (B - A).
4) If x is in B, then x is B and x can either be in A or not in A.
5) If x is in B and x is in A, then x is in A + (B - A).
6) If x is in B and x is not in A, then x is in (B - A).

I do not even know if I got it right...

 

[Mark44]

To be honest, I do not really know how to prove one of the statements.

That is, A + B is a subset of A + (B - A).

How about writing the expression on the right as
B \cap A^C

The plus signs means "union."

Attempt:

1) Let x be in A + B.
2) We can say x is in A or x is in B.

x could be in both sets, if it is in their intersection.

Look at three different cases:
1) x is in A but not in B.
2) x is in B but not in A.
3) x is in A \cap B.

For each case, show that x is also in A U (B - A).

3) If x is in A, then x is in A + (B - A).
4) If x is in B, then x is B and x can either be in A or not in A.
5) If x is in B and x is in A, then x is in A + (B - A).
6) If x is in B and x is not in A, then x is in (B - A).

I do not even know if I got it right...