解答下题Let A = {a, b, c} and B = P(A). 1) Prove that [B; ∪ , ∩ ,~] is a Boolean algebra. 2) Write out the operation tables for the Boolean algebra.

1) To prove that [B; ∪ , ∩ ,~] is a Boolean algebra, we need to show that it satisfies the following axioms:

- Closure under union and intersection: For any two sets X, Y ∈ B, X ∪ Y and X ∩ Y ∈ B.
- Associativity of union and intersection: For any three sets X, Y, Z ∈ B, (X ∪ Y) ∪ Z = X ∪ (Y ∪ Z) and (X ∩ Y) ∩ Z = X ∩ (Y ∩ Z).
- Commutativity of union and intersection: For any two sets X, Y ∈ B, X ∪ Y = Y ∪ X and X ∩ Y = Y ∩ X.
- Distributivity of intersection over union and vice versa: For any three sets X, Y, Z ∈ B, X ∩ (Y ∪ Z) = (X ∩ Y) ∪ (X ∩ Z) and X ∪ (Y ∩ Z) = (X ∪ Y) ∩ (X ∪ Z).
- Identity elements of union and intersection: There exist two sets 0, 1 ∈ B such that for any set X ∈ B, X ∪ 0 = X and X ∩ 1 = X.
- Complementation: For any set X ∈ B, there exists a set ¬X ∈ B such that X ∪ ¬X = 1 and X ∩ ¬X = 0.

All of these axioms can be easily verified for [B; ∪ , ∩ ,~], therefore it is a Boolean algebra.

2) The operation tables for the Boolean algebra [B; ∪ , ∩ ,~] are:

| ∪ | a b c ∅
--|----------------
a | a a a a
b | a b b b
c | a b c c
∅ | a b c ∅

| ∩ | a b c ∅
--|----------------
a | a b c ∅
b | b b ∅ ∅
c | c ∅ c ∅
∅ | ∅ ∅ ∅ ∅

| ~ | a b c ∅
--|------------------------
| ∅ {b,c} {a,c} {a,b}
a | {b,c} ∅ {b} {c}
b | {a,c} {a} ∅ {c}
c | {a,b} {a} {b} ∅
∅ | {a,b,c} {a,b,c} {a,b,c} ∅

Note: ∅ represents the empty set, and {} represents the set with no elements.