Name printed Student ID Section or TA s name and time CMSC 250 Exam 1 Thursday Mar 11 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 exam is 100 and you will have exactly 1 hour and 10 minutes 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 and the back of the cheat sheet can be used for scratch paper 1 15 pnts Use a COMPLETE truth table to determine if the following argument is valid or not Use 1 for true and 0 for false to create the complete truth table If it is not valid indicate all rows columns indicate that it is not valid if it is valid mark all rows columns which prove that it is valid P1 A W R P2 A W P3 R A Therefore W R Yes or No These statements do represent a valid argument Explain why you selected this answer for validity invalidity indicate how specific rows columns indicated this answer to you This area is for grading purposes points lost per page Do not write below this line 1 2 3 4 5 6 7 1 8 9 10 Total 2 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 x D P x Q x R x x D P x M x x D R x Z x x D Z x P x Therefore x D Z x Q x M x Statement Reason 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 2 Line s 3 36 pnts For each of the following either give a complete proof to demonstrate that the statement is true or a counter example with validation to show that it is false For these problems you may use any of the formal definitions given in class or the textbook and you may use the fact that every integer is either even or odd but not both a For all integers a b m n greater than 1 if m n and a m b then a n b 3 b n Z n Z odd n 3 2 12 n2 3 12 n2 1 4 c a Z prime b Z 1 a b3 a b 5 4 14 pnts Each item in the alphabetized list below completely describes a situation if the situation says Jan grooms Butch Lori grooms Butch for example this means that Jan only grooms Butch and neither of the other two dogs and Lori grooms Butch and neither of the other two dogs Each item in the number list below gives a formal logic statement You must determine which of the formal logic statements would be true in each of the lettered real world situations For each situation there may be one more than one or none of the listed logic statements that are true in that situation If there are none you must write the word NONE if there is more than one you must write all of the logic statements that are true in that described situation Defined are the following sets you can assume everything in that set is listed and that these are all of the sets you have to work with D dogs Alphie Butch Carlos P people Jan Lori The predicate is G x y meaning person x grooms dog y Note it is possible for more than one person to work on the same dog it is possible for a dog not to be groomed at all it is possible a person can work on more than one dog etc If the situation does not say that a certain person grooms a certain dog you must assume that that person DOES NOT groom that dog Logic Statements 1 x P y D G x y 2 x P y D G x y 3 y D x P G x y 4 y D x P G x y Next to each of the situations descriptions of that world s real life configuration write all numbers which correspond to logic statements that are true in that situation a Jan grooms Alphie and Butch Lori grooms Butch and Carlos b Jan does not groom any dog Lori grooms Alphie Butch and Carlos c Jan grooms Alphie Lori grooms Butch d Jan grooms Alphie Butch and Carlos Lori grooms Alphie Butch and Carlos e Jan grooms Alphie and Carlos Lori grooms Butch f Jan grooms Alphie Butch and Carlos Lori grooms Butch g Jan does not groom any dog Lori does not groom any dog 6 5 10 pnts Given the following truth table do each of the following tasks The operators and the gates they represent must only be those we have discussed in class the AND gate that has exactly 2 inputs and 1 output the OR gate that has exactly 2 inputs and 1 output and the NOT gate that has exactly 1 input and 1 output p 1 1 1 1 0 0 0 0 q 1 1 0 0 1 1 0 0 r 1 0 1 0 1 0 1 0 output 0 0 0 1 0 0 1 1 a Give the three situations values for p q and r that will give a 1 as the output in a single long logic statement of the form situation1 situation2 situation3 b Show the reduction of that line to an equivalent statement that has the minimum number of logical operators This reduction is in the standard form of a proof Equivalent Statement Rule 1 2 3 4 5 6 7 8 9 11 c Draw the circuit represented by this truth table using as few gates as possible 7 6 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 hunters are animals that like meat Some animals that like meat like to sleep therefore Some hunters like to sleep Circle One Valid Invalid No spiders are animals with 6 legs All spiders spin webs therefore No 6 legged animals spin webs Circle One Valid Invalid 8
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