Week2-2017-solutions.pdf - DMTH137 Session 2 2017 TH We 13...

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DMTH137 – Session 2 – 2017 Week 2 Tutorial Exercises : Propositional Logic DMTH137 Week 2 Tutorials Solutions 1 Tutorial questions for you This section contains the problems you should attempt at home in preparation for your tutorial. 1. True or false? If monkeys can fly, then 1 + 1 = 3. Answer : True. Monkeys cannot fly, so truth or falsity of the final clause is irrelevant. 2. Write the following sentences in symbolic form. Define appropriate terms. (a) This peach is ripe, but I will not eat it now. Answer : p ∧¬ q . where p = peach is ripe and q = I’ll eat it now. (b) If it is not raining and I wake up early, I will go cycling. Answer : ( ¬ p q ) r . where p = it is raining, q = I wake up early and r = I will go cycling. 3. If p is a proposition, what is the truth value of p p ? Why? Answer : Always false. The proposition p cannot be true and false at the same time. 4. Is ( ¬ p ( p q )) q always true? Answer : Yes. Only with p false and q true, is the left side of the implication true; but then so is the right side true. 5. Construct the truth tables for ¬ ( p q ) and ¬ p ∧¬ q . Are they the same, or different? Answer : The same. p q p q ¬ ( p q ) T T T F T F T F F T T F F F F T p q ¬ p ¬ q ¬ p ∧¬ q T T F F F T F F T F F T T F F F F T T T Thus ¬ ( p q ) ≡ ¬ p ∧¬ q , which is one of De Morgan’s Laws. 6. Translate the following logical constructions into natural plain English, where F ( p ) is ‘Printer p is out of service’, B ( p ) is ‘Printer p is busy’, Q ( j ) means ‘Print job j is queued’, and L ( j ) means ‘Print job j has been lost’. (a) p ( F ( p ) B ( p ) ) → ∃ j L ( j ) Solution : When a printer indicates being both busy and ‘out of service’ then a print job has been lost. (b) p B ( p ) ( j Q ( j ) ) Solution : If all printers are busy then there must be a job in the queue. 1
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2 Tutorial questions that your tutor will work through This section contains problems that your tutor will work through with you during your tutorial class. 1. State the converse, contrapositive, and inverse of the following implications: (a) If it rains tonight, then I will stay home.
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