Microsoft Word - Document1 Question 1. Consider the statement below. The program crashes when the index is out of bounds. Let p = the program crashes and q = the index is out of bounds. 1. (a)...

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Microsoft Word - Document1 Question 1. Consider the statement below. The program crashes when the index is out of bounds. Let p = the program crashes and q = the index is out of bounds. 1. (a) Translate the statement into propositional logic and rewrite it in English using if and then. 2. (b) Write the converse of the statement in propositional logic and two different ways in English. 3. (c) Write the inverse of the statement in propositional logic and two different ways in English. 4. (d) Write the contrapositive of the statement in propositional logic and two different ways in English. 5. (e) Give an example of when the original statement would be proven false. 6. (f) Given an example of when the converse would be proven false. Question 2. Fill in the truth table below with the missing values. ? ? ? ? ¬(?∧?)→(?∨¬?) T T T T T T T F T T F T T T F F T F T T T F T F T F F T T F F F F T T T F T T F F T F T F T F F F F T T F F T F F F F T F F F F Question 3. Write three different (but logically equivalent) propositional expressions that would fit the truth table below. At least one of your answers should be in DNF and at least one of them should use at least one implication. ? ? ? ??? T T T F T T F T T F T F T F F F F T T T F T F T F F T F F F F F Question 4. Determine if ¬?↔(?∧¬?)≡(?∨?)∧(?∨?)or not by using a truth table. Question 5. Given that ?=?,?=?,and ?=?, evaluate each expression until you get it into its most simplified form. Show your work. (a)¬(?→?)∨?∧¬? (b)?↔¬?→?∨?∧¬(?→?) Question 6. Given that ?=0, ?=0, and ?=1, evaluate each expression until you get it into its most simplified form. Show your work. (a) !(?==!?&&(?||?)) (b)!?||?!=?&&!? Question 7. Determine if each compound proposition is a tautology, a contradiction, or a contingency, and prove your answer. (a)¬(?∧(?∨¬?))→? (b)(?∧?)∨?↔(¬?∨¬?)∧¬? Question 8. For each of the following proofs, provide a justification for each numbered step in the proof. (a) Proof that ¬(?↔?)≡?⊕? ¬(?↔?) ≡¬((?→?)∧(?→?))(truth table/definition of biconditional) ≡¬(?→?)∨¬(?→?)(1) ≡¬(¬?∨?)∨¬(¬?∨?)(2) ≡(¬¬?∧¬?)∨(¬¬?∧¬?)(3) ≡(?∧¬?)∨(?∧¬?)(4) ≡?⊕?(truth table/definition of xor) (b) Proof that ¬?→(?→?)↔?≡(¬?∨?)∧(¬?∨?)¬?→(?→?)↔? ≡¬¬?∨(?→?)↔?(1) ≡?∨(?→?)↔?(2) ≡?∨(¬?∨?)↔?(3) ≡(?∨?)∨¬?↔?(4) ≡?∨¬?↔?(5) ≡(?∨¬?→?)∧(?→?∨¬?)(6) ≡[¬(?∨¬?)∨?]∧[¬?∨?∨¬?](7) ≡[(¬?∧¬?)∨?]∧[¬?∨?∨¬?](8) ≡[(¬?∨?)∧(¬?∨?)]∧[¬?∨?∨¬?](9) ≡[(¬?∨?)∧?]∧[¬?∨?∨¬?](10) ≡(¬?∨?)∧[¬?∨?∨¬?](11) ≡(¬?∨?)∧[¬?∨¬?∨?](12) ≡(¬?∨?)∧(¬?∨?)(13) Question 9. Prove each of the following by using logical equivalences. Make sure you justify each line in your proof. (a)(¬?→?)→¬?≡¬?∧¬?∨¬? (b) (?∧?)→(?∨?)is a tautology Question 10. Put the expression ¬(?→?)→¬(?→?∨?)into (a) DNF and (b) CNF. In both cases, try to make them as simple as possible (i.e. as short as possible). Show all of your work. Please note that even if your answers for (a) and (b) are the same, you should still show work for both.