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# Calculus MATH 1120

UNCC

GPA 3.73

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This 4 page Class Notes was uploaded by Mrs. Dangelo Fahey on Sunday October 25, 2015. The Class Notes belongs to MATH 1120 at University of North Carolina - Charlotte taught by Staff in Fall. Since its upload, it has received 38 views. For similar materials see /class/228919/math-1120-university-of-north-carolina-charlotte in Mathematics (M) at University of North Carolina - Charlotte.

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Date Created: 10/25/15

The Inclusion Exclusion Principle 1 Among 18 students in a room 7 study mathematics 10 study science and 10 study computer programming Also 3 study mathematics and science 4 study mathematics and computer programming and 5 study science and computer programming We know that 1 student studies all three subjects How many of these students study none of the three subjects Solution Solution Let M S and 0 denote the sets of students who study math science and computing respectively and let U be the entire set of 18 students Then lMl 7151 10 and 0 10 Also we have lMSl 31MC 4 and 180 5 where denotes the number of elements of the set x and juxtoposition of sets means intersection Finally lMCSl 1 Then lUli MlHSHlCVlilMSlilMOlilSC HlMSC D MSOl 187277121 2 2 Let A B and C be sets with the following properties 0 lAl100 1B1 50 and 01 48 o The number of elements that belong to exactly one of the three sets is twice the number that belong to exactly two of the sets 0 The number of elements that belong to exactly one of the three sets is three times the number that belong to all of the sets How many elements belong to all three sets 3 Three sets AB and C have the following properties NA 63 NB 91NO 44NA B 25NA O 23NO B 21 Also NAUBUO 139 What is NA B O7 Solution Let x denote the cardinality of NA B C Then z satis es z 15 25 45 21 23 139 using a Venn diagram Thus z 10 4 Two circles and a triangle are given in the plane What is the largest number of points that can belong to at least two of the three gures Solution Solution The number is maximized when no point belongs to all three sets Since lTCll S 6 lTOgl S 6 and 101021 3 2 the maximum possible number is 6 6 2 14 It is not hard to draw a picture to show that 14 such points are possible The Inclusion Exclusion Principle 5 03 5 00 How many equilateral triangles have at least two vertices in the hexagonal lat tice shown Solution Solution There are 6 area 177 triangles that have the center point as a vertex and 6 more area 1 triangles that don7t There are 4 triangle with area 3 two of whose vertices are in the set BDF and two others with vertices in the set A C Finally there are 6 triangles of area 4 two each with edges ADBE and CF Thus the total is 12 8 6 26 How many integers in the set 1 2 3 4 360 have at least one prime divisor in common with 3607 Solution Solution 360 23 32 5 Let A2A3 and A5 denote the multiples of 2 3 and 5 respectively among 1 2 3 360 To nd 1A2 UAg UA5l apply the principal of inclusion exclusion for three sets lAz U A3 UAsl lAzl lAsl lAsl lAzAal A2A5l lA3A5l1LlA2A3A5l Note that the number of multiples of 71 less than or equal to N is Thus 142UAsUAsl 1H11l171171171111 1801207276073672412264 Let U 123 de ned as follows A2nl1 n 1000 andn iseven A3 nl1g 71 1000 and n is a multiple of 3 A5 nl1g 71 1000 and n is a multiple of 5 All complements are taken with respect to U Find the number of ele ments of each of the sets listed below a A2 A3 A5 b Az Ag Aia 0 Aim A3 0 A5 Cl A2 Ag e A2 Q As H A5 1000 and let A2A3 and A5 denote the subsets of U 4204730475 f 4720430475 g 4720473045 and h In a math contest three problems A B and C were posed Among the partic ipants there were 25 who solved at least one problem Of all the participants who did not solve problem A the number who solved problem B was twice 2 The Inclusion Exclusion Principle Q the number who solved C The number who solved only problem A was one more than the number who solved A and at least one other problem Of all participants who solved just one problem half did not solve problem A How many solved just problem B How many numbers can be obtained as the product of two or more of the numbers 34455 6 777 Solution Solution Take G as the multiset 34455 6 7 77 and P as the process P1 with the modi cation that we must use at least two members of G and we multiply instead of add Note that each member 71 of R uniquely determines the subset Sn of G whose product it is We claim that each product in R uniquely determines its factors among the multiset Factor the product 71 of members of G into primes to get something of the form n 213751171 The exponent 239 is odd if and only if the 6 appears in the product The number of 57s and 77s in Sn is just k and l respectively and the number of 47s is The number of 67s is 2397 2l j and the number of 37s is j minus the number of 67s Thus the number of members of R is the number alternative ways to treat the various values We can include the 3 or not include the 6 or not include 0 1 or 2 of the 47s 01 or 2 of the 57s and 012 or 3 of the 77s This number is 223347175138 1994 UNC Charlotte Comprehensive Exam How many of the rst 100 pos itive integers are expressible as a sum of three or fewer members of the set 30 31 32 33 34 if we are allowed to use the same power more than once For example 5 can be represented but 8 cannot Solution Solution The number of powers of 3 used is just the sum of the ternary digits It is useful therefore to consider the numbers from 1 to 26 27 to 53 54 to 80 and 81 to 100 Numbers in the range 1 to 26 have ternary representation of the form a2a1a03 How many of these satisfy 12 a1 a0 3 3 There are 16 such numbers Those in the range 27 to 53 all have the form 1a2a1a03 There are 10 for which 12 a1 a0 3 2 The number in the range 54 to 80 have the form 2a2a1a03 Only 4 of these satisfy 12 a1 a0 3 1 The numbers from 81 to 100 all have the form 1a3a2a1a03 We want to know how many of that form are less than 100 and satisfy 1 a3 a2 a1 a0 3 3 There are 10 numbers in this range which satisfy the conditions Hence there are 16 10 4 10 40 such numbers altogether 3 The Inclusion Exclusion Principle H CH How many integers can be expressed as a sum of two or more different members of the set 0 1248 1631 In a survey of the chewing gum tastes of a group of baseball players it was found that 22 liked juicy fruit 25 liked spearmint 39 like bubble gum 9 like both spearmint and juicy fruit 17 liked juicy fruit and bubble gum 20 liked spearmint and bubble gum 6 liked all three 4 liked none of these How many baseball players were surveyed Of 28 students taking at least one subject the number taking Math and En glish but not History equals the number taking Math but not History or En glish No student takes English only or History only and six students take Math and History but not English The number taking English and History but not Math is 5 times the number taking all three subjects If the number taking all three subjects is even and non zero how many are taking English and Math but not History Mr Brown raises chickens Each can be described as thin or fat brown or red hen or rooster Four are thin brown hens 17 are hens 14 are thin chickens 4 are thin hens 11 are thin brown chickens 5 are brown hens 3 are fat red roosters 17 are thin or brown chickens How many chickens does Mr Brown have Consider the following information regarding three sets A B and 0 all of which are subsets of a set U lf NS denotes the number of members of S suppose that NA 14 NB 10 NAU B U C 24 and NA B 6 Consider the following assertions 1 C has at most 24 members 2 C has at least 6 members 3 A U B has exactly 18 members Which ones are true

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