Final answer:
The question involves recursive set definitions and probability within a finite sample space. However, a base case for the recursive definition of set S is missing, and without it, we cannot accurately determine the elements of the set or address the probability questions.
Step-by-step explanation:
The question appears to involve recursive set definitions and elementary probability, specifically in the context of a finite sample space and events derived from it.
The given set S is defined recursively to include elements that are twice any of its existing elements (2S) and the element x already belonging to S (xEs). However, the recursive definition provided is incomplete as there is no base case specified.
A base case is necessary to actually generate the elements of the set S. Without it, we cannot determine if the numbers listed (2, 4, 8, 10, 12, 14, 16, 32) belong to the set.
In the context of probability, the sample space S might be a set of numbers, and events A and B can be subsets of S. Probability questions often ask for the likelihood of certain outcomes, which in this case might relate to the numbers being even, or greater than a certain number.
The probability of event A, for example, could be calculated as P(A) = number of outcomes in A / number of outcomes in S. When two events are considered together, such as in the intersection A AND B, one looks for elements comm
Please refer back to the problem to ensure a proper base case is given for the recursive set definition or provide additional context so that the elements of the set S can be accurately deter