Name printed Student ID Section or TA s name and time CMSC 250 Exam 1 Thursday Oct 16 2003 Read this page and all directions carefully 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 exam is 150 and you will have exactly 1 75 hours to complete this exam You may not use calculators textbooks or any other aids during this exam If you need more space for any answer ask for an extra paper these extra papers must be turned in and you must mark so we can find the answer corresponding to a question The cheatsheet which is the last page of the exam can be ripped off and used during the exam the back of the cheat sheet can be used for scratch paper You do not need to turn in the cheat sheet paper at the end of the exam period Even if you do anything written on the cheatsheet or its back will not be graded Please copy the university honor pledge written below and sign your name on the line labeled Signature I pledge on my honor that I have not given nor received unauthorized assistance on this examination Signature This area is for grading purposes points lost per page Do not write below this line 2 3 4 5 6 7 8 1 9 10 11 12 Total 1 15 pnts Use a complete truth table to determine if the following two statements are logically equivalent Use 1 for true and 0 for false to create the complete truth table p q r p q r q p r Yes or No These statements are logically equivalent Why did you answer this way Indicate row and or column which implied the answer above 2 2 6 pnts Convert the following to 8 bit two s complement representation a 2410 b 610 c 1410 3 6 pnts Convert the following from to the bases indicated a 2510 2 b 1A216 2 c BA16 10 3 4 20 pnts For each of the following English sentences translate the meaning into formal notation using the logic symbols and In addition to these you may also use mathematical grouping and set notations symbols as needed On the next line write the negation of the original statement using formal notation Note On the negation no quantifier nor quantitified expression parentheses may be negated in the final answer There are no more than two tall people in my class Domains P all people Predicates T x x is tall C x x is in my class statement negation No person can survive in space without being dressed in a space suit Domains P all people Predicates D x is dressed in a space suit S x x can survive in space statement negation At least 1 young person must go to the meeting Domains U universe of all things Predicates P x x is a person M x x goes to the meeting Y x x is young statement negation Exactly two people like me Domains P all people Predicates L x x likes me statement negation 4 5 10 pnts Use an Euler diagram to determine if each of the following represents a valid argument form Make sure to label the parts of the diagram If it is invalid you must draw a diagram that is not supportive of the conclusion If it is a valid argument draw a diagram that does support the conclusion since you can t draw one that doesn t All pieces of luggage have handles All of my suitcases have handles therefore All of my suitcases are pieces of luggage Circle One Valid Invalid All dogs are cute My pet is cute therefore My pet is a dog Circle One Valid Invalid 5 6 20 pnts Use only the rules provided on the cheatsheet to prove the following It is a Valid Argument you only need to prove that it is You may also assume the Domain is not empty it has at least three members that member is named a b d P1 P2 P3 P4 line 1 x D P x Q x R x S x P a R a y D Q y S y M y z D M z R z therefore w D Q w S w Statement Reason 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 6 Line s 7 43 pnts For each of the following either give a counter example with justification to prove that the statement is false or give a complete proof to show that it is true a For any two perfect squares their product is also a perfect square In other words that the set of perfect squares is closed under multiplication Hint Remember the definition of perfect square is a Z a Z perfect square m Z m2 a 7 b For all positive integers n 3 does not divide n2 2 8 c For all rational numbers x and y x y is rational 9 d The cube root of 5 is irrational If you prove this to be true you may use any of the following lemmas in the proof without needing to prove them if you would like Lemma 1 a b Z m n Z such that m and n have no common factors and ab m n Lemma 2 a Z p Z prime p a2 p a Lemma 3 a Z p Z prime p a3 p a 10 8 15 pnts Given the following logical statement convert it so that it can be built using only OR and NOT gates No AND or other gates may be used The OR gates can only have two inputs and the NOT gates can only have one input P Q R P P Q a Give the logical statement as it would be expressed using only OR and NOT operators giving the proof that they are equivalent in the table below b Show the proof of this translation here to prove that your statement is equivalent to the original line Statement Reason Line s 1 2 3 4 5 6 7 8 9 10 11 12 c Draw the Circuit as it is expressed in the statement you just gave 11 9 15 pnts Use only those rules given on the cheatsheet to prove that the following is a valid argument It is a Valid Argument you only need to prove that it is P1 P2 P3 P4 line 1 P Q R S P Q R P S Q P M N Therefore Q M Statement Reason 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 12 Line s
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