Question

Let X be a finite set. Define * on any A, B ∈ ℙ(X) by A * B = (A-B) ∪ (B-A). The set ℙ(X) is closed under * and hence * defines an association

Answer 1

The operation * defined on the set of subsets of X, denoted as ℙ(X), is called symmetric difference. To find A * B, we take the set difference A - B, which contains the elements in A but not in B, and the set difference B - A, which contains the elements in B but not in A. Then we combine these sets by taking their union.

Step-by-step explanation:

The operation * defined on the set of subsets of X, denoted as ℙ(X), is called symmetric difference. The symmetric difference of two sets A and B is the set of elements that are in A or B, but not in both.

To find A * B, we take the set difference A - B, which contains the elements in A but not in B, and the set difference B - A, which contains the elements in B but not in A. Then we combine these sets by taking their union.

For example, let X = {1, 2, 3} and A = {1, 2} and B = {2, 3}. A * B = (A - B) ∪ (B - A) = ({1} ∪ {3}) = {1, 3}.