Answer the following questions about set theory and probability theory. a) Suppose the set X is a subset of the set Y. Is it the case that X is an element of Y? Explain your answer. b) Is it possible for there to be two sets X and Y such that X is a subset of Y, and (X ∩ Y) is empty? Explain your answer. c) Suppose X is a non-empty subset of Y. What is Pr(Y | X)? Explain your answer.

Label each of the following statements as either true or false. 2. for all nonempty sets A and B.

Consumer Preference In a population of 100,000 consumers, there are 20,000 users of Brand A, 30,000 users of Brand B, and 50,000 who use neither brand. During any month, a Brand A user has a 20 probability of switching to Brand B and a 5 of not using either brand. A Brand B user has a 15 probability of switching to Brand A and a 10 probability of not using either brand. A nonuser has a 10 probability of purchasing Brand A and a 15 probability of purchasing Brand B. How many people will be in each group a in 1 month, b in 2 months, and c in 18 months?

Answer the remaining questions about set theory and probability theory.   d) Give an informal argument showing that if X is a subset of Y and Y is a subset of X, then X and Y are identical.