Name printed Student ID Section or TA s name and time CMSC 250 Quiz 14 ANSWERS Wed May 5 2004 Write all answers legibly in the space provided The number of points possible for each question is indicated in square brackets the total number of points on the quiz is 30 and you will have exactly 15 minutes to complete this quiz You may not use calculators textbooks or any other aids during this quiz 1 10 pnts Explain how the pigeon hole principle can be used to show that the following is true by giving each of the requested items Assume you select at random a set of five not necessarily consecutive positive integers can you be sure you have at least two with the same remainder when divided by 4 The Domain ANSWER The 5 integers chosen The Size of the Domain ANSWER 5 The CoDomain ANSWER The possible remainders when that integer is divided by 4 The Size of the CoDomain ANSWER 4 The Function that Maps this domain to codomain ANSWER f x remainder of x divided by 4 2 5 pnts Is it true or false that the cardinality of the set of the positive odd integers is the same as the cardinality of the set of all integers If it is true prove it by providing any necessary information If it is false explain why ANSWER Yes the cardinalities are the same In order to prove they are you need to have a bijective function that maps from one set to the other On such function is x 4 3 x 1 odd 4 f Z Z where f x x 1 x 4 1 4 Since this function is both one to one and onto it is a bijection and the cardinalities must be equal 3 15 pnts Assume the following definitions of functions Give each of the following in simplest terms where f and g are both R R If the requested action is not possible simply state IMPOSSIBLE and explain why it is not possible Assume f x x2 3 4 and g y 4y a f g ANSWER f g x 16x2 3 4 b g f ANSWER g f x x2 3 c f 1 ANSWER IM P OSSIBLE This is not possible because in order to have an inverse function it must be a bijection This function since if I select y 12 x 2 3 1 which is not real and therefore is not in R d g 1 ANSWER g 1 y y 4
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