EECS 203, Discrete Mathematics Winter 2012, University of Michigan, Ann Arbor

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Date: January 16, 2012

Homework 1 Solutions
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Problem 1 [Ch 1.1 #22] Write each of these statements in the form “if p, then q” in English. a) If you get promoted, then you wash the boss’s car. b) If the winds are from the south, then there will be a spring thaw. c) If you bought the computer less than a year ago, then the warranty is good. d) If Willy cheats, then he gets caught. e) If you access the website, then you must pay a subscription fee. f) If you know the right people, then you will be elected. g) If Carol is on a boat, then she gets seasick.

Problem 2 [Ch 1.1 #32d] Construct a truth table for the compound proposition (p∧q) → (p∨q). p T T F F q T F T F (p ∧ q) T F F F p∨q T T T F (p ∧ q) → (p ∨ q) T T T T

Problem 3 [Ch 1.2 #12] Are these system speciﬁcations consistent? “If the ﬁle system is not locked, then new messages will be queued. If the ﬁle system is not locked, then the system is functioning normally, and conversely. If new messages are not queued, then they will be sent to the message buﬀer. If the ﬁle system is not locked, then new messages will be sent to the message buﬀer. New messages will not be sent to the message buﬀer.”. This system is consistent. We use: L = “The ﬁle system is locked,” Q = “New messages will be queued,” N = “The system is functioning normally,” B = “New messages will be sent to the message buﬀer.” Then the given speciﬁcations are ¬L → Q, ¬L ↔ N, ¬Q → B, ¬L → B, and ¬B. If we want consistency, then we had better have B false in order that ¬B be true. This ﬁrst conditional statement therefore is of the form F → T , which is true. Finally, the biconditional ¬L ↔ N can be satisﬁed by taking N to be false. Thus this set of speciﬁcations is consistent. Note that there is just this one satisfying truth assignment. 1

Problem 4 [Ch 1.3 #22] Show that (p → q) ∧ (p → r) and p → (q ∧ r)...

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