用中文解答下题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 verify the following axioms:

- Closure under ∪ and ∩ operations: For any two sets X, Y in B, X ∪ Y and X ∩ Y are also in B.
- Commutativity of ∪ and ∩ operations: For any two sets X, Y in B, X ∪ Y = Y ∪ X and X ∩ Y = Y ∩ X.
- Associativity of ∪ and ∩ operations: For any three sets X, Y, Z in B, (X ∪ Y) ∪ Z = X ∪ (Y ∪ Z) and (X ∩ Y) ∩ Z = X ∩ (Y ∩ Z).
- Distributivity of ∪ and ∩ operations: For any three sets X, Y, Z in B, X ∪ (Y ∩ Z) = (X ∪ Y) ∩ (X ∪ Z) and X ∩ (Y ∪ Z) = (X ∩ Y) ∪ (X ∩ Z).
- Complementation: For any set X in B, there exists a complement set X' in B such that X ∪ X' = A (the universal set) and X ∩ X' = ∅ (the empty set).
- Identity elements: A and ∅ are the identity elements for ∪ and ∩ operations, respectively.

All of these axioms hold for [B; ∪ , ∩ ,~], so we can conclude that [B; ∪ , ∩ ,~] is a Boolean algebra.

2) The operation tables for the Boolean algebra [B; ∪ , ∩ ,~] can be written as follows:

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

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

~ | a b c
------|-------
a| ∅ b c
b| a ∅ c
c| a b ∅

In the tables, ∪ represents the union operation, ∩ represents the intersection operation, and ~ represents the complement operation. The rows and columns represent the elements in set B (i.e., the power set of A). For example, in the ∪ table, the cell in the first row and second column (b) represents the result of taking the union of sets {a} and {b}, which is {a, b}.