mt1-practice - CMPSC 360 Discrete Mathematics for Computer...

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Unformatted text preview: CMPSC 360 Discrete Mathematics for Computer Science Penn State University Midterm #1 Practice Questions Fall 2009 Reminder: The date of the first midterm has been changed. It is Monday, October 5, 6:30-7:45 in 119 Osmond. 1. Put each of the following formulae into Conjuctive and Disjunctive Normal Forms. (a) ¬(a ∧ b) ∨ ¬(c ∧ d) (b) ¬(a ∨ b) ∧ ¬(c ∨ d) 2. Construct a valid argument for the following. Given (1) (2) (3) (4) if I like bacon and don’t like ham then I like sausage. if I like sausage then I like scrapple. I don’t like scrapple but I like bacon. if I like ham then I like eggs. Prove I like eggs. 3. Rewrite each of the following statements using quantifiers and appropriate predicates. Assume the domain for variables is A (Animals). (a) Some birds don’t fly. (b) Any animal that flies has wings. (c) Not every animal that flies is a bird. (d) If something has feathers then it is a bird. (e) All mammals have either fur or hair. (f) No bird has fur. 4. Construct a valid argument for the following in which the universe of people is {Alice, Bob, Carol, Dave}. Given (1) ∃x.likes(x, Alice) ∧ ¬likes(x, Dave) (2) ∀y.likes(y, Alice) → likes(y, Bob) Prove ∃z.likes(z, Bob) ∧ ¬likes(z, Dave) 5. Prove that for all integers n, m, if n and m are both even then 4|nm. Does the converse hold? Justify your answer. 6. Prove that n4 is odd iff n is odd, for all natural numbers n. 7. State the Principle of Mathematical Induction for Natural Numbers. 8. Prove the following holds for all natural numbers n n (i + 1) = i=0 1 n(n + 3) + 2 2 ...
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