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s09_a5

# s09_a5 - (d History students are Arts students Prove that(d...

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UNIVERSITY OF WATERLOO School of Computer Science CS245 Logic and Computation Spring 2009 ASSIGNMENT 5 (Due: Thursday, July 9, 2009 at 5:00pm) Use makeCover to produce a cover page for your assignment and hand in your assignment in the CS245 assignment box. Assignments are to be done individually. 1. (6) Prove that: 2 2 n - 1 is divisible by 3 for all n 0. 2. (10 marks—1 for each formalization; 6 for proof) Using appropriate sets, formalize the following sentences. Your answers must not contain logical quan- tifiers. (a) Computer Science students are not History students. (b) No student is a Psychology student and not an Arts student. (c) Any student who is not a Computer Science student is a Psychology student.
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Unformatted text preview: (d) History students are Arts students. Prove that (d) logically follows from (a)-(c) using natural deduction. In your proof you may use any of the inference rules from propositional and predicate logic and you may use any of the deﬁnitions given on the summary sheet for set theory. 3. (9 marks) Assume that S and T are binary relations. Prove the following predicates or give counter examples showing why they are not valid. • dom ( S ∩ T ) ⊆ ( dom S ) ∩ ( dom T ) • ( dom S ) ∩ ( dom T ) ⊆ dom ( S ∩ T ) 4. (9 marks) Prove or disprove the following property of binary relations. • id ( dom R ) = R ; R ∼...
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