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STOR 215 Sample Questions for Midterm 1 Here are some sample questions for the rst midterm exam Questions on the actual midterm will be di erent Remember that all exams in the course are closed book and closed notes and calculators are not permitted Most questions will have multiple parts and in many cases di erent parts of the same question are unrelated so if you cannot solve one part of a question you should still try the other parts 1 a Construct a truth table for the proposition p q b De ne what it means for a compound proposition p to be a tautology c Find the Cartesian product of the sets Don Sue and P F and its cardinality d Find the following cid 98 7 6 cid 99 e Let f A B be a function For S A carefully de ne the image f S cid 100 3 25 cid 101 cid 98 3 8 cid 99 2 Give the truth value of the following propositions concerning the real numbers R a 22 4 32 5 b x x2 x x 3 4 7 c x x2 0 y x y d x x 3 1 2 4 3 Express the negations of the following propositions so that no negation appears outside a quanti er or an expression involving a logical operation Your nal answers should not involve the implication operation a x P x Q x b x y P x y P y x 4 Let f R R and g R R be de ned by f x x2 2 and g x 3x 1 Express each of the following as a function of x Present your answer in the simplest possible form a f g b f g 5 Let p q r be the following propositions p Bob is a statistician q Bob likes numbers r Bob is fun at parties Translate each of the following sentences into a logical expression involving p q and r a Bob likes numbers and he is a statistician b Although Bob likes numbers and he is a statistician he is also fun at parties c Bob likes numbers but he is not a statistician d For Bob to be a statistician it is necessary for him to like numbers and be boring at parties e Bob is a statistician if and only if he likes numbers and is fun at parties 6 Consider the following functions f R R one to one ii onto a iii bijection Brie y explain your answer a f x 2x 1 In each case say whether the function is i 7 Let A a b c B b c d e and C c be subsets of a universal set U a b c d e f b f x 3 sin x c f x 1 x Identify the following a A B b B C c A Indicate whether each of the following relations is true or false d A e b c A f B 2B 8 Consider a group of climbers that climb together on weekends Let the predicate C x denote that x has a climbing certi cate and let A x y denote that climber x has assisted climber y Translate each of the following statements into a logical expression involving C x A x y and quanti ers Let the universal set U be the set of climbers attending the monthly hikes a Sylvia has a certi cate b Amy does not have a certi cate but has assisted Todd c Rob has not assisted Sandra d No one has assisted Joseph e Everyone with a certi cate has assisted another climber 9 Establish the each of the logical equivalences below in a step by step fashion using the rules of propositional equivalence from the text At each step you should either identify the rule used to obtain that step by name if that rule appears on the study list or provide the general form of the rule You need not indicate uses of the commutative or distributive laws a p q q p b p q p q p q q p 10 Show that for integers x 3x 2 is even if and only if x 5 is odd Provide a clear argument using English as appropriate


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UNC-Chapel Hill STOR 215 - Midterm 1

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