Let A = {a, b} and list the four elements of the power set P (A). We consider the operations + to be ∪, · to be ∩, and complement to be set complement. Consider 1 to be A and 0 to be ∅.
a. Explain why the description above defines a Boolean algebra.
b. Findtwoelementsx,yinP(A)suchthatxy=0,x̸=0andy̸=0.
a. To show that the description defines a Boolean algebra, we need to verify that the set P(A) with the defined operations forms an algebraic structure satisfying certain properties.
- Closure: For any two sets X and Y in P(A), the union X ∪ Y and the intersection X ∩ Y are also elements of P(A). This is because the power set P(A) contains all possible subsets of A, and the union and intersection of any subsets of A will also be subsets of A.
- Identity Elements: The set A itself acts as the identity element for the union operation (∪), since X ∪ A = A ∪ X = A for any X in P(A). The empty set ∅ acts as the identity element for the intersection operation (∩), as X ∩ ∅ = ∅ ∩ X = ∅ for any X in P(A).
- Associativity: For any three sets X, Y, and Z in P(A), the union operation is associative: (X ∪ Y) ∪ Z = X ∪ (Y ∪ Z). Similarly, the intersection operation is associative: (X ∩ Y) ∩ Z = X ∩ (Y ∩ Z).
- Complement: For any set X in P(A), the complement of X (denoted as X') exists in P(A). The complement of a set X is the set of all elements in A that are not in X. In other words, X ∩ X' = X' ∩ X = ∅.
- Distributivity: The operations ∪ (union) and ∩ (intersection) distribute over each other, meaning (X ∪ Y) ∩ Z = (X ∩ Z) ∪ (Y ∩ Z) and (X ∩ Y) ∪ Z = (X ∪ Z) ∩ (Y ∪ Z).
- Complement Laws: For any set X in P(A), X ∪ X' = A and X ∩ X' = ∅.
- Annihilator Laws: For any set X in P(A), X ∪ ∅ = X and X ∩ A = X.
These properties satisfy the requirements of a Boolean algebra, making P(A) with the defined operations a Boolean algebra.
b. To find two elements x and y in P(A) such that xy = 0 (where 0 represents the empty set ∅), we can use the fact that the intersection of any set with the empty set is always the empty set.
Let's choose x = ∅ and y = A.
- x = ∅: The empty set ∅ has no elements, and any intersection with another set will result in the empty set. Therefore, x ∩ A = ∅ ∩ A = ∅.
- y = A: The set A contains both elements 'a' and 'b'. The intersection of A with any set will result in the set itself. Therefore, y ∩ A = A ∩ A = A.
Thus, x = ∅ and y = A satisfy xy = ∅, x ≠ ∅, and y ≠ ∅.