This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
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 (cant 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!...
View Full Document