COP 3100 – Intro to Discrete StructuresDr. Sahar [email protected] of Computer ScienceLab 012Question 1Translate the following statement into propositional logic using the propositions provided:“You are eligible to be President of the U.S.A only if you are at least 35 years old, were bornin the U.S.A., or at the time of your birth both of your parents were citizens. And also youhave lived at least 14 years in the country.”Express your answer using:• e (“You are eligible to be President of the U.S.A ”)• a (“You are at least 35 years old”)• b (“You were born in the U.S.A.”)• p (“At the time of your birth both of your parents were citizens”)• r (“You have lived at least 14 years in the country”)3SolutionTranslate the following statement into propositional logic using the propositions provided:“You are eligible to be President of the U.S.A only if you are at least 35 years old, were bornin the U.S.A., or at the time of your birth both of your parents were citizens. And also youhave lived at least 14 years in the country.”Express your answer using:• e (“You are eligible to be President of the U.S.A ”)• a (“You are at least 35 years old”)• b (“You were born in the U.S.A.”)• p (“At the time of your birth both of your parents were citizens”)• r (“You have lived at least 14 years in the country”)Solution:e --> ( a ∧ ( b ∨ p ) ∧ r )n Using a truth table, show that 𝑝 ∧ 𝑞 ∧ 𝑟 ↔ 𝑝 ∧ 𝑞 ∧ 𝑟always have the same value.¨ You just proved something. What is it?4Question 25Solution𝒑 𝒒 𝒓 𝒑 ∧ 𝒒 𝒒 ∧ 𝒓 𝒑 ∧ 𝒒 ∧ 𝒓 𝒑 ∧ 𝒒 ∧ 𝒓 𝒑 ∧ 𝒒 ∧ 𝒓 ↔ 𝒑 ∧ 𝒒 ∧ 𝒓T T T T T T T TT T F T F F F TT F T F F F F TT F F F F F F TF T T F T F F TF T F F F F F TF F T F F F F TF F F F F F F TThis proves that conjunction is associative.n Use a truth table to show that:p ∨ ( ¬ q ∧ r ) and ( p ∨ ¬ q ) ∧ ( p ∨ r ) are logically equivalent6Question 37Solutionn Let p, q, and r be the propositions p : You have the flu. q : You miss the final exam.r : You pass the course. Express each of the following propositions as an English sentence.(a) p→ q(b) ¬ q ↔ r(c) q→ ¬ r(d) ( p ∧ q ) ∨ ( ¬ q ∧ r ) 8Question 4n Let p, q, and r be the propositions p : You have the flu. q : You miss the final exam.r : You pass the course. Express each of the following propositions as an English sentence.(a) p→ q "If you have the flu, then you will miss the final exam." or, "You will miss the final exam if you have the flu."(b) ¬ q ↔ r "You pass the course if and only if you do not miss the final exam."(c) q→ ¬ r "If you miss the final exam, then you will not pass the course."(d) ( p ∧ q ) ∨ ( ¬ q ∧ r ) "Either you have the flu and miss the final exam or you don't miss the final exam and pass the course."9Solutionn Using a truth table, show that 𝑝 ∧ 𝑞 → 𝑟 ↔ ¬𝑟 → (¬𝑝 ∨ ¬𝑞) .10Question 511Solution𝒑 𝒒 𝒓 ¬𝒑 ¬𝒒 ¬𝒓 𝒑 ∧ 𝒒 𝒑 ∧ 𝒒 → 𝒓 (¬𝒑 ∨ ¬𝒒) ¬𝒓 → (¬𝒑 ∨ ¬𝒒)T T T F F F T T F TT T F F F T T F F FT F T F T F F T T TT F F F T T F T T TF T T T F F F T T TF T F T F T F T T TF F T T T F F T T TF F F T T T F T T
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