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CS70_sp01_mt1_sol - CS 70 Midterm 1 Solutions Midterm Exam...

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Midterm Exam 1 Solutions CS70 Blum/Wagner, 9 March 2001 This is a CLOSED BOOK examination. One page of notes is permitted. Calculators are permitted. Do all your work on the pages of this examination. Give reasons for all your answers. Problem 1 (Short answer grab-bag) [10 Points] For parts (a)-(c), justify your answer briefly. (a) [2 Points] True or False: (P => Q) => (Q => P) always holds, for all propositions P, Q. False. Counter example: P=False, Q=True. This is a converse error. (b) [3 Points] True or False: ((P v Q) => Q) => (Q => (P v Q)) always holds, for all propositions P, Q. True. (Q => (P v Q)) = True for all P,Q, and (anything => True) is true. (Don't be fooled into thinking this is another converse error. It's not.) (Common error: Not realizing that (False => False) is True.) (c) [3 Points] Recall that two sest S,T are said to be disjoint if = Ø. Can two events A and B be simultaneously independent and disjoint? Yes. An example: If A = Ø, then A isect B = Ø and Pr[A]=0. So, Pr[A isect B] = 0 = 0 × Pr[B] = Pr[A]Pr[B], and A,B are independent and disjoint.
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