Give an algebraic proof to show that: For all sets A and B, [tex](B^c U (B^c - A))^c = B[/tex].


This is a set theory problem, of course. Right now, I have the following proof worked out:

We wish to show that for all sets A and B, [tex](B^c U (B^c - A))^c = B[/tex]. We will proceed algebraically.

Distributing the compliment, [tex]B U (B \ A^c).[/tex]

Under the set difference law, this can be written as B ∪ (B ∩ A).

Under the distributive law, this can be written as (B ∪ B) ∩ A.

Under the identity law, (B ∪ B) = B. Then, B ∩ A is left.


I don't know what I did wrong, but I also don't know B ∩ A could equal B. Any guidance?