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. The description above defines a Boolean algebra because it satisfies the following properties:
1. Closure: The union (∪) and intersection (∩) of two sets in the power set P(A) are also in P(A).
2. Identity Elements: The set A acts as the identity element for the union operation (∪) because A ∪ X = A for any set X in P(A). Similarly, the empty set ∅ acts as the identity element for the intersection operation (∩) because ∅ ∩ X = ∅ for any set X in P(A).
3. Complement: The set complement operation is defined, which means for every set X in P(A), there exists a complement set X' such that X ∪ X' = A and X ∩ X' = ∅.
4. Distributive Laws: The Boolean algebra obeys the distributive laws of set theory, which state that for any sets X, Y, and Z in P(A), the following hold:
- X ∩ (Y ∪ Z) = (X ∩ Y) ∪ (X ∩ Z)
- X ∪ (Y ∩ Z) = (X ∪ Y) ∩ (X ∪ Z)
b. Two elements x and y in P(A) such that xy = 0, x ≠ 0, and y ≠ 0 can be found using the given operations. Let's consider the elements of A:
A = {a, b}
The power set P(A) consists of all possible subsets of A. Therefore, to find such elements x and y, we need to identify subsets that satisfy the given conditions.
Since xy = 0, it means that the intersection of sets x and y is empty (∅). In other words, x and y have no common elements.
One possible example is:
x = {a}
y = {b}
Here, x ≠ 0 and y ≠ 0 because they are non-empty subsets. Also, the intersection of x and y is empty, satisfying xy = 0.
a. The description defines a Boolean algebra because it satisfies the three main properties of a Boolean algebra:
1. Closure under union (∪) and intersection (∩): The operations + (union) and · (intersection) defined in the description are closed on the elements of the power set P(A). This means that taking the union or intersection of any two elements in P(A) will always result in another element in P(A).
2. Identity elements: The identity element for union (∪) is the entire set A (represented as 1), and the identity element for intersection (∩) is the empty set ∅ (represented as 0). These identity elements ensure that there exists an element in P(A) that, when combined with any other element using union or intersection, will not change the other element.
3. Complement: The description also includes the complement operation, which allows for the negation of elements in P(A). This operation satisfies the property of complementation in a Boolean algebra, as taking the complement of any element will result in a valid element that is still within P(A).
b. To find two elements x, y in P(A) such that xy = 0, x ≠ 0, and y ≠ 0, we need to consider the power set P(A) and its elements.
Given that A = {a, b}, the power set P(A) would contain all possible subsets of A. Since A has two elements, there are four subsets:
∅ (the empty set)
{a}
{b}
{a, b} (the entire set A)
To find x and y that satisfy the conditions, we need to choose elements such that their intersection is the empty set.
One possible choice for x is {a}, and for y is {b}. In this case, x ∩ y = {a} ∩ {b} = ∅, which satisfies the condition xy = 0. Additionally, x ≠ 0 (x is not the empty set) and y ≠ 0 (y is not the empty set).