Discrete Practice Test 1

Discrete Practice Test 1 - lent” you must give an...

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September 17, 2009 Name: Test 1 Math 3336 You have the full class period to complete the test. You cannot use any books or notes. This test is worth 250 points. 1. 50 pts. Prove or disprove whether the formula is a tautology or not: (a) ( p q ) ( ¬ p r ) ( q r ) (b) ( p r ) ( q r ) ( p q ) r (c) ( p r ) ( q r ) ( p q ) r (d) p ( p q ) (e) p ( q r ) ( p q ) r 2. 30 pts. But each wrong answer carries a penalty of -5 pts. Mark as true or false. The implication If P, then Q is equivalent to: a) P is sufficient for Q. b) Q is sufficient for P. c) P is necessary for Q. d) Q is necessary for P. e) P if Q. f) Q only if P. 3. 30 pts. But each wrong answer carries a penalty of -5 pts. Determine whether the following arguments are valid or invalid. (a) If a = b then c = d . But a 6 = b . Thus c 6 = d . (b) If a = b then c = d . But c 6 = d . Thus a 6 = b . (c) a = b only if c = d . One has that c = d . Thus a = b . 4. 40 pts. Find the conjunctive and disjunctive normalform for p q . 5. 30 pts. Decide whether the following formulas are equivalent. In case where your answer is ”not equiva-
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Unformatted text preview: lent” you must give an explanation. (a) ∀ x ( Q ( x ) ∧ P ( x )) and ∀ xQ ( x ) ∧ ∀ xP ( x ) (b) ∀ x ( Q ( x ) ∨ P ( x )) and ∀ xQ ( x ) ∨ ∀ xP ( x ) (c) ∀ x ( Q ( x ) → P ( x )) and ∀ xQ ( x ) → ∀ xP ( x ) 6. 40 pts. Let L ( x, y ) be the predicate for ” x likes to buy y ” and let S ( y ) stand for ” y is on sale”. Formalize: (a) Everybody likes to buy something, but only if it is on sale. (b) There is something everybody likes to buy if it is on sale. (c) If everything is on sale then everybody likes to buy everything. (d) There is somebody who likes to buy everything if it is on sale. 7. 30 pts. Let N be the set of natural numbers and let n | m stand for that n divides m . Formalize: For every n and m there is some g such that g divides n and m, and if q divides n and m, then q divides g....
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