Need some help with proofs of sets using logical equivalences

[lookingforhelp]

I have two proofs of sets using two logical equivalences I need some help on. One I finished and I would just like someone to verify that I did it correctly and the other one I'm stuck on.

1. For all sets A,B,C, (A - B) - C=A - (B U C)
Proof: (A - B) - C = (A ∩ B)' ∩ C' [set difference] = A ∩ (B' ∩ C') [associative] = A ∩ (B U C)' [DeMorgan's] = A - (B U C) [set difference]
Is this correct?

2. For all sets A and B, (B' U (B' - A))' = B
Proof: (B' U (B' - A))' = B'' (B' - A)' = B U (B'' - A') [double complement] = B U (B - A')...?
Not sure what else to do with this one.

Thank you for the help!

 

[pka]

I have two proofs of sets using two logical equivalences I need some help on. One I finished and I would just like someone to verify that I did it correctly and the other one I'm stuck on.

1. For all sets A,B,C, (A - B) - C=A - (B U C)
Proof: (A - B) - C = (A ∩ B)' ∩ C' [set difference] = A ∩ (B' ∩ C') [associative] = A ∩ (B U C)' [DeMorgan's] = A - (B U C) [set difference]
Is this correct? YES!

..

 

[pka]

2. For all sets A and B, (B' U (B' - A))' = B
Proof: (B' U (B' - A))' = B'' (B' - A)' = B U (B'' - A') [double complement] = B U (B - A')...?

\(\displaystyle \\(B'\cup (B'\setminus A))'\\(B\cap (B'\cap A')'\\B\cap(B\cup A)\\B\)

 

[lookingforhelp]

\(\displaystyle \\(B'\cup (B'\setminus A))'\\(B\cap (B'\cap A')'\\B\cap(B\cup A)\\B\)

How did you get from step 1 to step 2? Thanks for your help.