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TAB TABLE 1 Rules of Inference. Rule of Inference Tautology Name Ide (PA(-9) Modus ponens AL (Apq)) Modus tollens AC AL An ((p - )^( g r) (p) Hypothetical syllogism ((p Vq) 4-p) 9 Disjunctive syllogism AL P (PV) Addition PV png (png) P Simplification ((p) ^ () (PAG) Conjunction PV (pvq)^(PVT) (Vr) Resolution - V V AC Discrete Mathematics HW2 Look at all problems. Submit only the questions in BLUE, Due date: Wed Feb 19th at start of the lecture: 1. What rule of inference is used in each of these arguments? a) Alice is a mathematics major. Therefore, Alice is either a mathematics major or a computer science major b) Jerry is a mathematics major and a computer science major. Therefore, Jerry is a mathematics major. c) If it is rainy, then the pool will be closed. It is rainy. Therefore, the pool is closed. d) If it snows today, the university will close. The university is not closed today. Therefore, it did not snow today e) If I go swimming, then I will stay in the sun too long. If I stay in the sun too long, then I will sunburn Therefore, if I go swimming, then I will sunburn. HIV r, ( rs) (pvt), ( us), t, q, and conclusion 6. Show that the argument form with premises u p is valid by using rules of inference. 7. Prove that if m + n and n + p are even integers, where m, n, and p are integers, then m + p is even. What kind of proof did you use? 8. Show that if n is an integer and n +5 is odd, then n is even using a) a proof by contraposition. b) a proof by contradiction. 9. Prove that n +1 2 2n when n is a positive integer with 1 sn s 4. Kuwait University/College of Engineering and Petroleum / Department of Computer Engineering CpE-203: Discrete Mathematics TABLE 6 Legical Equivalences TULE 1 Set de TABLE 2 Rules of Inference for Quantified Statements h PATE pv AU VP asoslation Dews AU PVTT PATE ..Pd Parc YP Universal generalization PE Pdforse dement PVVP Commutations Acas ( pari) AAAA Plc formelement .: Pir) Existential generalization AU BU=AU BUC ABC 4 Prem LAU LAC ABBAN Barbie law PVV A) Derties PAVAVAT) TABLE 1 Useful Properties of the Floor and Ceiling Functions. is an integer, is a real number) -PA De Mass UTAT PVP PAPV AULA ABA AUTEU Ania (la) L a nd only if