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1 Logic and Set Theory2.Pdf February 1, 2018 Any calculator permitted on N.Y.S. Regents examinations may be used. The word \compute" calls for an exact answer in simplest form. 1 - Logic and Set Theory 1. Let P be the set of single-digit prime numbers and let O be the set of single-digit odd numbers. Compute the number of elements in O [ P . 2. In a survey of 120 third-graders, 80 of them liked the color blue and 60 of them liked the color red. If 50 of them liked blue and red, how many of the third-graders didn't like red or blue? 3. In Metropolis, there are Truthies (who always tell the truth) and Fibbies (who always lie). Three citizens of Metropolis make the following statements. Jenny says, \Kenny or Lenny is a Truthie (but not both)." Kenny says, \Jenny or Lenny is a Truthie (but not both)." Lenny says, \Both Jenny and Kenny are Fibbies." Which of the three people are Fibbies? Note: List any member who is a Fibbie or write NONE if all three are Truthies. Monroe County Math League Contest #5. SOLUTIONS February 2018 1 - Logic and Set Theory 1. Let P be the set of single-digit prime numbers and let O be the set of single-digit odd numbers. Compute the number of elements in O [ P . SOLUTION: 6 The set O [ P is f1; 2; 3; 5; 7; 9g, which has 6 elements. 2. In a survey of 120 third-graders, 80 of them liked the color blue and 60 of them liked the color red. If 50 of them liked blue and red, how many of the third-graders didn't like red or blue? SOLUTION: 30 Together, there are 140 third-graders who liked blue and red, but 50 of those were double-counted, so there are only 140 − 50 = 90 of them who liked either red or blue (or both). Subtracting from 120, there are 120 − 90 = 30 third-graders who don't like red or blue. 3. In Metropolis, there are Truthies (who always tell the truth) and Fibbies (who always lie). Three citizens of Metropolis make the following statements. Jenny says, \Kenny or Lenny is a Truthie (but not both)." Kenny says, \Jenny or Lenny is a Truthie (but not both)." Lenny says, \Both Jenny and Kenny are Fibbies." Which of the three people are Fibbies? Note: List any member who is a Fibbie or write NONE if all three are Truthies. SOLUTION: Lenny (only) Suppose Lenny is a Truthie. Then, both Jenny and Kenny are Fibbies. Then, only Lenny is a Truthie, which makes Jenny a Truthie, and that's a contradiction. So, Lenny is a Fibbie. Therefore, at least one of Jenny and Kenny is telling the truth. If Jenny is a Truthie, then exactly one of Jenny or Lenny is a Truthie, so Kenny is a Truthie also. Similarly, if Kenny is a Truthie, so is Jenny. Therefore, there is only one Fibbie, and that's Lenny. February 2, 2017 Any calculator permitted on N.Y.S. Regents examinations may be used. The word \compute" calls for an exact answer in simplest form. 1 - Logic and Set Theory 1. Suppose the following four statements are true: If you study and you get lucky, then you will succeed. If you succeed, then you will be famous. Jimmy is not famous. Jimmy studied. Draw a valid conclusion that uses all of this information. 2. In Mcmland, there are four kinds of people: Amoricans, Braths, Crews, and Dwurfs. All Amoricans are Braths, all Crews are Dwurfs, all Dwurfs are Amoricans, and all Crews are Braths. If none of the people groups are equal in number, which group is the smallest? 3. Students in the ninth grade at Springfield High were polled to see if they were fans of the Mets and/or the Yankees. There are 70 students who do not cheer for the Mets. There are 95 students who do not cheer for the Yankees. There are 127 students who cheer for the Mets or for the Yankees (or both). There are 90 students who cheer for the Mets. Compute the number of students who cheer for the Mets but not the Yankees. Monroe County Math League Contest #5. SOLUTIONS February 2017 1 - Logic and Set Theory 1. Suppose the following four statements are true: If you study and you get lucky, then you will succeed. If you succeed, then you will be famous. Jimmy is not famous. Jimmy studied. Draw a valid conclusion that uses all of this information. SOLUTION: Jimmy did not get lucky. From the first two sentences, one can conclude \If you study and you get lucky, then you will be famous." Incorporating the third given sentence, one can conclude that \You study and you get lucky" is false. Because Jimmy did study, the only way that \You study and you get lucky" is false is if Jimmy did not get lucky. 2. In Mcmland, there are four kinds of people: Amoricans, Braths, Crews, and Dwurfs. All Amoricans are Braths, all Crews are Dwurfs, all Dwurfs are Amoricans, and all Crews are Braths. If none of the people groups are equal in number, which group is the smallest? SOLUTION: Crews All Amoricans are Braths, so the Braths are not the smallest. All Dwurfs are Amoricans, so the Amoricans are not the smallest. All Crews are Dwurfs, so the Dwurfs are not the smallest. The Crews are the smallest. 3. Students in the ninth grade at Springfield High were polled to see if they were fans of the Mets and/or the Yankees. There are 70 students who do not cheer for the Mets. There are 95 students who do not cheer for the Yankees. There are 127 students who cheer for the Mets or for the Yankees (or both). There are 90 students who cheer for the Mets. Compute the number of students who cheer for the Mets but not the Yankees. SOLUTION: 62 Because there are 90 students who cheer for the Mets and 70 who do not, there are 160 students in the ninth grade at Springfield High. Subtracting the 127 who cheer for either team, there are 33 students who do not cheer for either team. There are 95 students who do not cheer for the Yankees, so there are 95 − 33 = 62 who cheer for the Mets. February 4, 2016 Any calculator permitted on N.Y.S. Regents examinations may be used. The word \compute" calls for an exact answer in simplest form. 1 - Logic and Set Theory 1. Let P be the set of prime numbers less than 10. Let Q be the set of single-digit odd numbers. Compute the number of numbers in P [ Q. 2. At a party, there are 40 people. Of the 40 people at the party, 22 people are girls, 5 people are left-handed, and 2 people are left-handed girls. How many right-handed boys attended the party? 3. In the trunk of Juan's car are four bags: a bag of apples, a bag of bananas, a bag of carrots, and a bag of dates. Together, the bag of apples and the bag of carrots weigh the same as the bag of bananas and the bag of dates do together. Together, the bag of apples and the bag of dates weigh more than the bag of bananas and the bag of carrots do together. The bag of bananas weighs as much as the bags of carrots and dates do together. Using A, B, C, and D to abbreviate the bags of apples, bananas, carrots, and dates, arrange their weights in order from greatest to least. Note: If you think the bag of apples weighs more than the bag of bananas, which weighs more than the bag of carrots, which weighs more than the bag of dates, write ABCD as your answer. Monroe County Math League Contest #5. SOLUTIONS February 2016 1 - Logic and Set Theory 1. Let P be the set of prime numbers less than 10. Let Q be the set of single-digit odd numbers. Compute the number of numbers in P [ Q. SOLUTION: 6 The set P = f2; 3; 5; 7g. The set Q = f1; 3; 5; 7; 9g. Therefore, P [ Q = f1; 2; 3; 5; 7; 9g, which has cardinality 6. 2. At a party, there are 40 people. Of the 40 people at the party, 22 people are girls, 5 people are left-handed, and 2 people are left-handed girls. How many right-handed boys attended the party? SOLUTION: 15 At the party, G is the set of girls and L is the set of left-handed people. We know that the cardinality of G [ L is 22 + 5 − 2 = 25. The right-handed boys are the people not in G [ L, so there are 40 − 25 = 15 right-handed boys at the party. 3. In the trunk of Juan's car are four bags: a bag of apples, a bag of bananas, a bag of carrots, and a bag of dates. Together, the bag of apples and the bag of carrots weigh the same as the bag of bananas and the bag of dates do together. Together, the bag of apples and the bag of dates weigh more than the bag of bananas and the bag of carrots do together. The bag of bananas weighs as much as the bags of carrots and dates do together. Using A, B, C, and D to abbreviate the bags of apples, bananas, carrots, and dates, arrange their weights in order from greatest to least.
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