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# examples - Midterm on Part I Logic Exercise 1 We study A...

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Midterm - on 11/08/2007 Part I: Logic Exercise 1 We study : A = [ ¬ ( p q ) ¬ ( q p ) ( p q )] p q p q ¬ ( p q ) p q A T T T F T F T F T F F T F T T F F T F F F T F F From column 6, we get that A is not a tautology nor a contradiction. Exercise 2 A and B are inhabitants of Smullyan islands, and A says ”If I am a knight, then B is a knave”, while B says ”If I am a knave, then A is a knight”. The answer is: The two propositions are contradictory: there is no answer. We proceed as follows: We show first that A is a knight. We use a proof by contradiction. We suppose that A is a knave. In this case, the proposition ”If I am a knight, then B is a knave” is false. Since this proposition is a conditional of the form p q , it is false when p is true and q is false. p is true means A is a knight. But this contradicts A is a knave. So our hypothesis is false, and therefore A is a knight. We show that B is a knight. Again we use a proof by contradiction. We suppose that B is a knave. In this case, the proposition ”If I am a knave, then A is a knight” is false, i.e. B is a knave and A is a knave. But we have shown that A is a knight, and therefore we have reached a contradiction, and B is a knight. We show that B is a knave. A is a knight, and therefore the proposition ”A is a knight and B is a knave is true”. Since this is a proposition of the form p q that is true, with p true, then q is true, i.e. B is a knave We have therefore reached a contradiction, and the problem has no answer. Exercise 3 We want to prove that : ((( p q ) r ) ( r s )) ( ¬ s ¬ p ) is a tautology: 1

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p q r s p q a = ( p q ) r r s b = a ( r s ) c = ¬ s ¬ p b c T T T T T T T T T T T T T F T T F F F T T T F T T F T F T
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