Final answer:
The events A and B are not mutually exclusive as they have common elements, namely 14, 16, and 18. The probability P(0 < x < 12) is 1, assuming a uniform distribution, as x is bound to fall within this range according to the conditions given.
Step-by-step explanation:
Looking at the information provided, the subject is related to probability and sets in mathematics, specifically geared towards a high school level. We're given two events: A as the event that a natural number x is greater than 12, and B as the event that it is greater than 8. Since any number greater than 12 is also greater than 8, A is a subset of B. Therefore the two events are not mutually exclusive. Using the given sample space S and the subsets A and B, we can calculate the probabilities.
Given that set A contains numbers that are multiples of 2 from 2 to 18, and set B contains numbers from 14 to 19, the intersection A AND B would be the set {14, 16, 18} which are the elements common to both A and B. The union A OR B includes all elements that are in either A, B, or both, so that set would be {2, 4, 6, 8, 10, 12, 14, 16, 18, 15, 17, 19}. The question about mutual exclusivity would get an answer of No, as A and B have common elements and hence are not mutually exclusive.
For the other question posted, regarding the probability function f(x) being equal to 12 and limited to the range 0 ≤ x ≤ 12, the probability P(0 < x < 12) is looking for the probability that x falls between 0 and 12, exclusive. As the probability density function is continuous and constant over this interval, assuming f(x) is uniformly distributed, then P(0 < x < 12) would be 1, since it is certain that x will fall within this interval as per the given conditions.