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Dr. Timo de Wolff Institute of Mathematics www.math.tamu.edu/~dewolff/Spring16/math302.html MATH 302 – Discrete Mathematics – Section 501 Homework 2 Spring 2016 Due: Wednesday, February 10th, 2016, 4:10 pm. When you hand in your homework, do not forget to add your name and your UIN. Exercise 1. Translate the following sentences into a proposition by first defining suitable atomic formulas and afterwards putting these together with the right connectives. 1. “You are eligible to be President of the U.S.A. only if you are at least 35 years old, were born in the U.S.A., or at the time of your birth both of your parents were citizens, and you have lived at least 14 years in the country.” 2. “You can upgrade your operating system only if you have a 32-bit processor running at 1 GHz or faster, at least 1 GB RAM, and 16 GB free hard disk space, or a 64-bit processor running at 2 GHz or faster, at least 2 GB RAM, and at least 32 GB free hard disk space.” Exercise 2. Prove the following logical equivalences via truth tables. 1. ¬(A ∧ B) ⇔ ¬A ∨ ¬B. 2. (A → B) ∧ (¬A → B) ⇔ B. 3. A ∧ (B ∨ C) ⇔ (A ∧ B) ∨ (A ∧ C). 4. (A → B) ∧ (B → A) ⇔ ¬A ↔ ¬B. Exercise 3. Are the following propositions satisfiable? Give either a solution or prove that they are a contradiction. 1. ¬(¬(A → B) → (¬C → ¬(D ∧ E))) 2. (((A ∨ ¬A) ∧ B) ∨ (A ∧ ¬B) ∨ (¬A ∧ ¬B)) → (C ∧ ¬C) ∨ (D ∧ ¬D) Hint: Applying some of the logical equivalences from the lecture might help you. 1 Exercise 4. Assume the following three propositions are true: 1. “If LeChuck does not have a parrot on his shoulder, then Jack Sparrow speaks French or Billy Bones marries the Governors daughter” 2. “If Billy Bones marries the Governors daughter, then Long John Silver does not like rum.” 3. “Long John Silver likes rum and Jack Sparrow does not speak French or LeChuck has a parrot on his shoulder.” Formalize the statements and conclude that they imply that LeChuck has a parrot on his shoulder. 2