Consider a finite set 5 of atomic variables, and a set of premises P , each of the form x-+ y, where x and y are each literals (members of S or their negations). If two of t hese premises are x -+ y and y -+ z, for example, we can derive the consequence x -+ z by Hypothetical Syllogism. Suppose that we exhaustively check every pair of premises and consequences, and add any new consequences we can derive in this way, until we can go no furt her. (a) Argue informally that this process cannot go on forever. (b) Suppose we derive a consequence x -+ ,x, where x is a literal. What conclusion can we draw about settings of the variables making all the premises true? (c) Suppose we derive both x -+ , x and ,x-+ x? What conclusion can we t hen draw about settings making all the premises true? (d) Suppose that the process stops without the situation in part (c) occurring. Argue that there is at least one setting of the variables making all the premises true.

Amachine that produces a special type of transistor (a component of computers) has a 2% defective rate. the production is considered a random process where each transistor is independent of the others. (a) what is the probability that the 10th transistor produced is the first with a defect? (b) what is the probability that the machine produces no defective transistors in a batch of 100? (c) on average, how many transistors would you expect to be produced before the first with a defect? what is the standard deviation? (d) another machine that also produces transistors has a 5% defective rate where each transistor is produced independent of the others. on average how many transistors would you expect to be produced with this machine before the first with a defect? what is the standard deviation? (e) based on your answers to parts (c) and (d), how does increasing the probability of an event a↵ect the mean and standard deviation of the wait time until success?