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# tut2 - Modus Tollens to make a valid argument • “If...

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Problems to Week 3 Tutorial — MACM101 1. Verify that p ( q ( p q )) is a tautology. (If a similar problem wasn’t solved in Tutorial 1.) 2. Is ( p q ) ( q ( p q ) a contradiction? 3. Verify that ( p q ) ( q r ) ( r p ) ( p q ) ( q r ) ( r p ) . 4. Negate the following statement and simplify the result p q ( ¬ p ¬ q r ). 5. Let “Nand” be the logic connective defined by p q ⇐⇒ ¬ ( p q ). Express ¬ , , using only Nand. (difficult problem, not for everyone) 6. Verify that the Rule of Proof by Cases is a valid argument. (Use the corresponding tautology.) 7. Verify that the following is a tautology by showing that it is impossible for the conclusion to have truth value 0 while the premises have truth value 1: (( p q ) ( r s ) ( p r )) ( q s ) . 8. For each of the following pairs of statements use Modus Ponens or

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Unformatted text preview: Modus Tollens to make a valid argument. • “If Janice has trouble starting her car, then her daughter Angela will check Janice’s spark plugs. Janice had trouble starting her car.” • “If Brady solved the ﬁrst problem correctly, then the answer he obtained is 137. Brady’s answer to the ﬁrst problem is not 137.” 9. Give the reasons for each step needed to show that the following ar-gument is valid Premises: p,p → q,s ∨ r,r → ¬ q Conclusion: s . 1 Steps Reasons 1. p 2. p → q 3. q 4. r → ¬ q 5. q → ¬ r 6. ¬ r 7. s ∨ r 8. s 10. Solve problems 5,6 for the previous Tutorial using rules of inference. (diﬃcult problem, not for everyone) 2...
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