Eecs 203 W1 Exam Solutions
EECS 203, Discrete Mathematics Winter 2012, University of Michigan, Ann Arbor
Your name
Date: January 16, 2012
Homework 1 Solutions
Collaborators: Names of collaborators
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) are
Your name
Date: January 16, 2012
Homework 1 Solutions
Collaborators: Names of collaborators
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) are