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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,
6:307:45 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 4nm. 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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 Fall '08
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