Name Recitation circle one F 10 30 G 12 30 H 1 30 J 3 30 Math 127 Exam 1 14 February 2018 Turn o and put away your cell phone No notes or books are permitted during this exam No calculators or any other devices are permitted during this exam Read each question carefully answer each question completely and show all of your work Write solutions clearly and legibly credit will not be given for illegible solutions If any question is not clear ask for clari cation Points Score 1 2 3 4 5 20 20 20 20 20 100 1 Consider the following statement x R y R xy 0 a Write this statement as a sentence without using any of the symbols or R b Is this statement true If so prove it If not explain why not c If we reversed the order of the quaniti ers would the statement be true If so prove it If not explain why not 2 a Each of the propositions on the right is logically equivalent to one on the left Match them by writing the appropriate number in the blank You need not justify your answers in any way p i p q p q p q p q p q p q ii p q iii p q iv p p q v p p q b Prove that p q r is logically equivalent to p r q r 3 Find all real valued solutions to the equation x2 3x 2 2x Prove that your answer is correct 4 Let k N with k 1 Prove that for all n N with n 1 cid 18 i k 1 cid 19 k cid 18 n k cid 19 k 1 n cid 88 i 1 5 De ne a sequence ak recursively by a0 1 a1 1 Prove that for all k 1 ak cid 0 3 cid 1 k 2 2 ak ak 1 ak 2 for k 2
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