Question

Let a be defined to be the set 1,2,3,4,5,6,7,8. For x⊆a, f(x)=a-x. Recall that for a finite set a, p(a) denotes the power set of a which is the set of all subsets of a. What is the function f(x) when x is a subset of a?

Answer 1

The function f(x) defined as f(x) = a - x returns the set difference between the set a and a subset x. This function, when applied to any subset of the set a, provides the complementary elements of a. It can be applied to all elements of the power set p(a) of a.

Step-by-step explanation:

Understanding the Function f(x) for a Subset of a

For a set a, which consists of the numbers 1 through 8, the function f(x) is defined where x is any subset of a. The function is given by f(x) = a - x, meaning it takes the set difference between set a and the subset x. If we apply the function f to a particular subset of a, say {2, 4}, we would get f({2, 4}) = {1, 3, 5, 6, 7, 8}, because we subtract the elements of x from a.

Considering power sets, p(a) represents the power set of a, which consists of all possible subsets of a, including the empty set and the set a itself. The function f can be applied to any element of the power set. Applying f to the entire power set would give us a collection of sets, each being the complement within a of the corresponding subset.

For example, applying f to the empty set, which is a subset of a, we get the full set a since there are no elements to subtract. Conversely, applying f to the full set a returns the empty set because a - a is empty.