# final - Discrete Structures CSCI-150 Spring 2015 Final Exam...

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Discrete Structures. CSCI-150. Spring 2015. Final Exam. You are allowed to use a formula sheet: It should be a single standard sheet of paper (of Letter/A4 size). The formula sheet has to be handed in after the exam. For every problem, provide a reasonably complete argument supporting your answer. Answers with no explanation may be insufficient to get full credit. Problem 1 (3 pts) Prove the equivalence: ( a b ) ( b c ) ≡ ¬ b c Hint: you may need to use the identity law: ( ¬ x ) ( ¬ x ) T . Problem 2 (3 pts) Given the following recurrently defined sequence of integers: a 0 = 3 , a n = 5 a n - 1 + 8 Prove by induction that all elements in this sequence are congruent to 3 modulo 4, or in other words: n 0 : a n 3 ( mod 4) Problem 3 (3 pts) Show that if n is an integer than the remainder ( n 2 rem 4) = 1 or 0 . Hint: every integer is either even or odd. Problem 4 (3 pts) (a) Draw the Hasse diagram for the following partially ordered set: ( 10 , 14 , 15 , 21 , 30 , 42 , 70 , 105 , 210 , ) , where the partial order relation is the divisibility relation: x y

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