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(chatquestion)...ADDIS ABABA UNIVERSITY DEPARTMENT OF MATHEMATICS Mathematics for Natural Sciences (Math 1011) Worksheet 1 Part I: Propositional Logic 1. Which of the following sentences are propositions? What are the truth values of those that are propositions? a) 5<9 and there are infinitely many prime numbers. b) x-y provided that x y. c) How many courses are you taking this semester? d) The first human kind lived in Ethiopia. 2. What is the negation of each of the following propositions? a) Summer in A.A is warm and not hot. b) I will not go to class if I have a critical wok to do. 3. Consider the following propositions r: Rabbits have been seen in the area. w: Walking on the path is safe. b: Berries are ripe along the path. Write the following propositions using r, b and w and logical connectives. a) Berries are ripe along the path, but rabbits have not been seen in the area. b) For walking on the path to be safe, it is necessary but not sufficient that berries not b ripe along the path and for rabbits not to have been seen in the area. c) If berries are ripe along the path, then walking is safe if and only if rabbits have not been seen in the area. d) Walking is not safe on the path whenever rabbits have been seen in the area and berries are ripe along the path. 4. Given the propositions p: You have a job to do. q: You miss the final examination. r: You pass the course. Express each of the following popositions as an ordinary English sentence. c) (p -r) V (q= r) b) -(pAq) a) -qr Determine the truth value of p if a) (q = -p) Vr is False. C) (p V-q)) = (-r V p) is False. b) -q and p =q are True. d) -q and qAr are True. p 6. If -r= - (p Aq) is false, find the truth value of [(p r) V q] (-pA r). b) (qvr) p= p)( p) 7. Show that a) (p Aq) = r (q r) p Using truth table decide whether each of the following is a tautology, a contradiction or neither. c) (-q=-p) = (-p)= q) A-(-4-p) b) (p a) [pA (p q)] =

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