State, with evidence, whether each of the following claims is true or false:
a. The conditional probability of A, given B, must be at least as large as the probability of A.
b. An event must be independent of its complement.
c. The probability of A, given B, must be at least as large as the probability of the intersection of A and B.
d. The probability of the intersection of two events cannot exceed the product of their individual probabilities.
e. The posterior probability of any event must be at least as large as its prior probability.