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010_2009_midterm - CS3234 Logic and Formal Systems Midterm...

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CS3234 Logic and Formal Systems Midterm Examination Questions 17/09/2009 This examination question booklet has 12 pages, including this cover page, and contains 14 questions. Answer all questions. You have 60 minutes to complete the examination. The examination has two parts: Questions 1–12 are MCQ questions. Use a B2 pencil to fill up the provided MCQ form. In the MCQ form, leave Section A blank, and fill up Sections B and C. Questions 13 and 14 require a written answer. Use the space provided in this question booklet to answer these two questions. Enter your matriculation number here: After finishing, place the MCQ sheet and this question booklet on top of the question sheet and leave both on the table, when you exit the room. 1
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Multiple Choice Part Question 1 : (5 marks) The shortest formula in propositional logic is p where p is a propositional atom. Which one of the following statements is true? 1 A The formula p is valid and satisfiable. 1 B The formula p is invalid and unsatisfiable. 1 C The formula p is invalid and satisfiable. 1 D The formula p is valid and unsatisfiable. 1 E None of the above. Answer 1 : 1 C The formula p evaluates to F under the evaluation p F , and therefore the formula is invalid. It evaluates to T under the valuation p T and therefore the formula is satisifiable. Question 2 : (10 marks) Which one of the following statements is true? 2 A Some sentences in predicate calculus may be satisfied by one model, but not by another one. 2 B All sentences in predicate calculus are either satisfied by all models or by none. 2 C No sentence in predicate calculus is satisified by all models. 2 D Every sentence is satisfied by at least one model. 2 E None of the above. Answer 2 : 2 A Let P = { P } and let us say that P is a unary predicate. Further, let F = { c } and let 2
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us say that c is a constant. The sentence P ( c ) is satisfied in every model M in which c M P M , and unsatisified in every model M in which c M P M . Question 3 : (10 marks) Which one of the following statements is false ? 3 A There are sentences that are satisfied by only finite models. 3 B There are sentences that are satisfied by models of any size but not infinite ones.
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