Set Theory Proof Help | Prove (A-C) U (B-C) = (A U B) - C

[INdeWATERS]

I need to prove the following:

(A-C) U (B-C) = (A U B) - C

I know that the union means that I have to do a proof by cases to show that these two sets are equal.

But where do I start?!

thanks

 

[gb7nash]

Element-wise proof:

http://www.btinternet.com/~g8yoa/odl/F2_maths_Q1.htm

I'll get you started, but you should get the idea:

We want to show that (A-C) U (B-C) ⊆ (A U B) - C and (A U B) - C ⊆ (A-C) U (B-C).

Show (A-C) U (B-C) ⊆ (A U B) - C:

Fix x ∈(A-C) U (B-C). We want to arrive at x ∈(A U B) - C.

Case 1: x ∈(A-C)

Since x ∈(A-C), x ∈A and x ∉ C. Since x ∈ A and A ⊆ A U B, x ∈ A U B. Since x ∈ A U B and x ∉ C, we finally have that x ∈(A U B) - C.

Case 2:x ∈(B-C)

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Then you have to show that (A U B) - C ⊆ (A-C) U (B-C). It's very similar though.

 

[INdeWATERS]

Thank you for the help! I am in the process of writing up my proof...

Is is possible to use the terminology "without loss of generality" in this proof? I know that involving that term can help me save a lot of writing...

Thanks again

 

[gb7nash]

I wouldn't do it. At your level, you probably want to leave every detail in it (and I don't know how rigorous your teacher is). You can probably see though that showing case 2 is analogous to case 1.

 

[INdeWATERS]

thank you for your help. Now, I am having issues with a different proof, as follows.
U = universal set , P(U) = power set of universal set

For all sets A, B, C ∈ P(U), if A ⊆ C and B ⊆ C, then A ⊆ B or B ⊆ A.

I am pretty sure the statement is false and so I have to disprove it, i.e. prove the negation. I am stuck on how to negate. My attempts are as follows...

(1) There exist sets A, B, C ∈ P(U) such that A ⊆ C or B ⊆ C and A ⊄ B and B ⊄ A.
(2) There exist sets A, B, C ∈ P(U) such that if A ⊆ C or B ⊆ C then A ⊄ B and B ⊄ A.

Would the contrapositive of the statement be easier to work with??
For all sets A, B, C ∈ P(U), if A ⊆ B or B ⊆ A then, A ⊆ C and B ⊆ C.

Thank you for your time and help!

 

[INdeWATERS]

Also, how does my proof look for the original post?
Prove set equality: (A-C) U (B-C) = (A U B) - C

Proof: Show (A-C) U (B-C) ⊆ (A U B) - C and (A U B) - C ⊆ (A-C) U (B-C).

Let A, B, and C be arbitrary sets. First, assume (A-C) U (B-C). Let x ∈ (A-C) U (B-C). Without loss of generality, suppose x ∈ (A-C). Since x ∈ (A-C), we have x ∈ A and x ∉ C. So, x ∈ A and by a theorem proved in class A ⊆ (A U B). Thus, x ∈ (A U B) and x ∉ C means x ∈ (A U B) - C. Therefore, (A-C) U (B-C) ⊆ (A U B) - C. For the reverse inclusion, let x ∈ (A U B) - C. Then x ∈ (A U B) and x ∉ C. Without loss of generality, suppose x ∈ A. Since x ∈ A and x ∉ C then, x ∈ (A - C). It follows that, x ∈ (A U B) - C. Hence, (A U B) - C ⊆ (A-C) U (B-C). Therefore, by definition of set equality, (A-C) U (B-C) = (A U B) - C.

 

[gb7nash]

Also, how does my proof look for the original post?
Prove set equality: (A-C) U (B-C) = (A U B) - C

Proof: Show (A-C) U (B-C) ⊆ (A U B) - C and (A U B) - C ⊆ (A-C) U (B-C).

Let A, B, and C be arbitrary sets. First, assume (A-C) U (B-C). Let x ∈ (A-C) U (B-C). Without loss of generality, suppose x ∈ (A-C). Since x ∈ (A-C), we have x ∈ A and x ∉ C. So, x ∈ A and by a theorem proved in class A ⊆ (A U B). Thus, x ∈ (A U B) and x ∉ C means x ∈ (A U B) - C. Therefore, (A-C) U (B-C) ⊆ (A U B) - C. For the reverse inclusion, let x ∈ (A U B) - C. Then x ∈ (A U B) and x ∉ C. Without loss of generality, suppose x ∈ A. Since x ∈ A and x ∉ C then, x ∈ (A - C). It follows that, x ∈(A-C) U (B-C). Hence, (A U B) - C ⊆ (A-C) U (B-C). Therefore, by definition of set equality, (A-C) U (B-C) = (A U B) - C.

Highlighted in red is what I think you should take out. Bolded is a change you should make. It looks like you understand what to do and put the other thing by mistake.

Also, you might want to do the other case for both sides, even if it is more writing. It depends how rigorous your teacher is, so just use good judgement here.