Homework 2 M325K PDF

M325K – Homework 2 1. Translate the following English sentences into logical formulas. Use letters for propositions and the appropriate symbols for Boolean operators. Clearly define the proposition variables. a) If you exercise then you will be healthy. b) It is raining outside but there is no li...

M325K – Homework 2 1. Translate the following English sentences into logical formulas. Use letters for propositions and the appropriate symbols for Boolean operators. Clearly define the proposition variables. a) If you exercise then you will be healthy. b) It is raining outside but there is no lightning. c) Steve is healthy and wealthy but not wise. d) Amy is a lawyer or a teacher. 2. Use truth tables to indicate whether each of the following formulas is valid, satisfiable but not valid, or unsatisfiable. For each formula that is satisfiable but not valid, write down satisfying truth assignments and unsatisfying truth assignments. Include all columns needed for the sentence in the truth table. a) 𝑃 → 𝑃 b) 𝑄 → (𝑄 ⋀(𝑃 → 𝑄)) c) (𝑃 ⋀ 𝑄) ⋀(𝑃 ⋀ 𝑄) 3. Use truth tables to determine whether the following statements are true. Include all columns needed for both sides. Be sure to conclude. a) 𝑃 ⋁ 𝑃 ⋀ 𝑄 ≡ 𝑃 ⋁ 𝑄 b) (𝑃 ⋁ 𝑄) ⋁ 𝑅 ≡ 𝑃 ⋁(𝑄 ⋁ 𝑅) c) 𝑃 → (𝑄 ⋁ 𝑅) ≡ 𝑃 ⋀ 𝑄 → 𝑅 4. Answer each of the following questions with an explanation. Suppose that n is an integer greater than 1. a) If you have n proposition variables how many rows are needed in the truth table? b) Is n! greater than the answer in part (a)? c) Is the answer in part (a) greater than n2 + 4n + 2? d) Discuss and compare parts (a), (b), and (c) when n is 10.