ps03 - MAT 300 Mathematical Structures Dr L Mantini Problem...

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Unformatted text preview: MAT 300, Mathematical Structures Dr. L. Mantini Problem Set 3 Solutions Spring 2010 Problems 2.1 : 2, 5, 7, 8; 2.2 : 2, 3, 7, 9, 10; 2.3 : 3, 5, 6, 9, 11, 12 2.1.2 Analyze the logical forms of the following statements (other answers are possible). (a) Anyone who has bought a Rolls Royce with cash must have a rich uncle. Solution: Let R ( x ) mean that x bought a Rolls Royce with cash, U ( y,x ) mean that y is the uncle of x , and W ( y ) mean that y is wealthy (can’t use R for rich). The statement is then R ( x ) ⇒ ∃ y ( U ( y,x ) ∧ W ( y )). (b) If anyone in the dorm has the measles, then everyone who has a friend in the dorm will have to be quarantined. Solution: Let M ( x ) mean x has the measles, D ( x ) means x lives in the dorm, F ( x,y ) means x is friends with y , and Q ( x ) means that x will have to be quarantined. Then the statement might be written ∃ x ( M ( x ) ∧ D ( x )) ⇒ ∀ z ( ∃ y ( D ( y ) ∧ F ( z,y )) ⇒ Q ( z )) . (c) If noone failed the test, then everyone who got an A will tutor someone who got a D. Solution: Let F ( x ) mean x failed the test; A ( x ) mean x got an A on the test; D ( x ) mean x got a D on the test; and T ( x,y ) mean x tutors y . Then the statement is ( ∀ x, ¬ F ( x )) ⇒ ∀ x ( A ( x ) ⇒ [ ∃ yD ( y ) ∧ T ( x,y )]) . (d) If anyone can do it, Jones can. Solution: Let C ( x ) mean x can do it. The sentence is ∃ xC ( x ) ⇒ C (Jones). (e) If Jones can do it, anyone can. Solution: Notation as in (h), then the sentence is C (Jones) ⇒ ∀ x,C ( x ). 2.1.5 (2 points, 1 per part) Translate the given symbolic statements into idiomatic mathe- matical English. Solution: (a) ∀ x [( P ( x ) ∧ ¬ ( x = 2)) ⇒ O ( x )], where P ( x ) means “ x is prime” and O ( x ) means “ x is odd.” This means “Every prime which is not equal to 2 is odd.” (b) ∃ x [ P ( x ) ∧∀ y ( P ( y ) ⇒ y ≤ x )], where P ( x ) means “ x is a perfect number” means “There exists a perfect number x which is greater than every other perfect number y ” or “There exists a largest perfect number.” A perfect number has the sum of its divisors, not including itself, equal to itself. For example, 6 = 1 + 2 + 3 is the smallest perfect number; 28 = 1 + 2 + 4 + 7 + 14 is the next one. It is not known if there is a largest perfect number, but it is doubtful as the known list includes some very, very large numbers. This is an active area of modern number theory research!theory research!...
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This note was uploaded on 04/04/2010 for the course MAT 300 taught by Professor Thieme during the Spring '07 term at ASU.

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ps03 - MAT 300 Mathematical Structures Dr L Mantini Problem...

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