Section 16 Inclusion/Exclusion 6/4/2018 MATH 106, Section 16

Set Notation A set is any well-defined collection of objects. U represents the universal set (i.e., the universe). A  B means that A  B represents A  B represents ~A represents #A represents A is a subset of B, that is, every item in A must also be an item in B. (Every set is a subset of U.) the intersection of A and B, that is, the set of all items that are in both A and B. the union of A and B, that is, the set of all items that are in at least one of A or B. the complement of A, that is, the set of all items that are not in A. the size of A, that is, the number of items that are in A. 6/4/2018 MATH 106, Section 16

#1 Consider a universal set consisting of the positive integers up to (and including) 12, that is, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. We define the subsets A = multiples of 2 B = multiples of 3 C = multiples of 4 D = multiples of 5 Use set notation to display each subset; then find each of the following: #A = #B = #C = #D = ~A = ~B = ~C = ~D = {2, 4, 6, 8, 10, 12} {3, 6, 9, 12} {4, 8, 12} {5, 10} 6/4/2018 MATH 106, Section 16

Consider a universal set consisting of the positive integers up to (and including) 12, that is, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. We define the subsets A = multiples of 2 B = multiples of 3 C = multiples of 4 D = multiples of 5 Use set notation to display each subset; then find each of the following: #A = #B = #C = #D = ~A = ~B = ~C = ~D = {2, 4, 6, 8, 10, 12} {3, 6, 9, 12} {4, 8, 12} {5, 10} 6 4 3 2 {1, 3, 5, 7, 9, 11} {1, 2, 4, 5, 7. 8, 10, 11} {1, 2, 3, 5, 6, 7, 9, 10, 11} {1, 2, 3, 4, 6, 7, 8, 9, 11, 12} 6/4/2018 MATH 106, Section 16

AB = AC = BD = AB = AC = BD = In each section of the Venn diagrams displayed, list the members of the set that section represents. {6, 12} {4, 8, 12} { } or  {2, 3, 4, 6, 8, 9, 10, 12} {2, 4, 6, 8, 10, 12} {3, 5, 6, 9, 10, 12} 1 5 7 11 2 4 8 10 6 12 3 9 A B 6/4/2018 MATH 106, Section 16

1 3 5 7 9 11 2 6 10 4 8 12 A C 1 2 4 7 8 11 3 6 9 12 5 10 B D 6/4/2018 MATH 106, Section 16

Complete each of the following formulas, and then verify that each is true: #(AB) = #A + #B #(AC) = #A + #C #(BD) = #B + #D – #(AB) 8 6 4 2 – #(AC) 6 6 3 3 When we continue with this handout, we shall consider this same type of situation with three sets A, B, and C. Right now, we return to the Subsets, Strings, Equations Handout. – #(BD) 6 4 2 6/4/2018 MATH 106, Section 16

Use the rest of this class period to work on #1 on the Subsets, Strings, Equations Handout, which must be submitted for homework either at the end of this class or in the class indicated on the course schedule. As you work on each problem, check to see if your final answer is correct. We will go over in class any problems there are questions about. There will also be some time this period for questions. CHECK YOUR ANSWERS: 2. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) 84 35 4 7 262,144 6765 6765 2 255,379 262,142 6/4/2018 MATH 106, Section 10

NAME__________________________________________________ 2. (a) (b) A child is playing with large box of blocks that are identical, except that some are red and some are blue. How many different ways can six red blocks and three blue blocks be arranged in a row? 9! ——– = 84 6! 3! How many different ways can six red blocks and three blue blocks be arranged in a row so that none of the red blocks are adjacent? 6/4/2018 MATH 106, Section 10

(c) (d) How many different ways can six red blocks and three blue blocks be arranged in a row so that none of the blue blocks are adjacent? 7! ——– = 35 3! 4! How many different ways can six red blocks and three blue blocks be arranged in a row so that all of the red blocks are adjacent? 4 6/4/2018 MATH 106, Section 10

2.-continued (e) (f) How many different ways can six red blocks and three blue blocks be arranged in a row so that all of the blue blocks are adjacent? 7 How many different ways can six red blocks and three blue blocks be arranged in a row so that blocks of the same color are not adjacent? 6/4/2018 MATH 106, Section 10

(g) (h) How many different ways can 18 blocks, each one either red or blue, be arranged in a row? 218 = 262,144 How many different ways can 18 blocks, each one either red or blue, be arranged in a row so that none of the blue blocks are adjacent? F(18) = 6765 6/4/2018 MATH 106, Section 10

2.-continued (i) (j) How many different ways can 18 blocks, each one either red or blue, be arranged in a row so that none of the red blocks are adjacent? F(18) = 6765 How many different ways can 18 blocks, each one either red or blue, be arranged in a row so that blocks of the same color are not adjacent? 2 6/4/2018 MATH 106, Section 10

(k) (l) How many different ways can 18 blocks, each one either red or blue, be arranged in a row so that at least two of the red blocks are adjacent? 218 – F(18) = 262,144 – 6765 = 255,379 How many different ways can 18 blocks, each one either red or blue, be arranged in a row so that at least two blocks of the same color are adjacent? 218 – 2 = 262,144 – 2 = 262,142 In the class after the exam, we shall begin with #2 on the Section #16 Handout. 6/4/2018 MATH 106, Section 10

Problem #3: Use #U – #(AB) = #U – [#A + #B – #(AB)] In the class after the exam, we shall begin with #2 on the Section #16 Handout. For next class, do the following problems in the Section 16 Homework: Problem #3: Use #U – #(AB) = #U – [#A + #B – #(AB)] freshmen taking mathematics all freshmen freshmen taking computer science 6/4/2018 MATH 106, Section 10

#2 Consider a universal set consisting of the positive integers up to (and including) 30, that is, U = {1, 2, …, 29, 30}. We define the subsets A = multiples of 2 B = multiples of 3 C = multiples of 5 Use set notation to display each subset. In each section of the Venn diagram displayed, list the members of the set that section represents. Then find each of the following: {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30} {3, 6, 9, 12, 15, 18, 21, 24, 27, 30} {5, 10, 15, 20, 25, 30} #A = #B = #C = #(AB) = #(AC) = #(BC) = #(ABC) = 15 10 6 6/4/2018 MATH 106, Section 16

1 7 11 13 17 19 23 29 2 4 8 14 6 12 18 24 3 9 21 27 16 22 26 28 30 A 10 20 15 B 5 25 C 6/4/2018 MATH 106, Section 16

Complete the following formula, and then verify that it is true: #(ABC) = #A + #B + #C – #(AB) – #(AC) – #(BC) + #(ABC) 15 10 6 5 3 2 1 This is 22, which is what we count for ABC from the Venn diagram. #A = #B = #C = #(AB) = #(AC) = #(BC) = #(ABC) = 15 10 6 5 3 2 1 6/4/2018 MATH 106, Section 16

U = {1, 2, …, 29, 30} A = multiples of 2 B = multiples of 3 C = multiples of 5 Note the transition from words to set notation and vice versa: = multiples of each (all) of 2 and 3 and 5 = multiples of (at least) one of 2 or 3 or 5 = multiples of none of 2 and 3 and 5 = not a multiple of (at least) one of 2 or 3 or 5 ABC ABC ~(ABC) OR (~A)(~B)(~C) ~(ABC) OR (~A)(~B)(~C)

A = the set of arrangements where the letters E, F are together #3 Let the set of all different possible arrangements (permutations) of the letters EFNOPZ be the universal set. We define the subsets A = the set of arrangements where the letters E, F are together B = the set of arrangements where the letters N, O, P are together (a) (b) Find the size of the universe. #U = 6! = 720 Describe in words each of the sets listed. ~A ~B AB the set of arrangements where the letters E, F are not together the set of arrangements where the letters N, O, P are not together 6/4/2018 MATH 106, Section 16

A = the set of arrangements where the letters E, F are together Let the set of all different possible arrangements (permutations) of the letters EFNOPZ be the universal set. We define the subsets A = the set of arrangements where the letters E, F are together B = the set of arrangements where the letters N, O, P are together (a) (b) Find the size of the universe. #U = 6! = 720 Describe in words each of the sets listed. ~A ~B AB the set of arrangements where the letters E, F are not together the set of arrangements where the letters N, O, P are not together the set of arrangements where the letters E, F are together the letters N, O, P are together and 6/4/2018 MATH 106, Section 16

the set of arrangements where the letters E, F are together ~(AB) AB ~(AB) the set of arrangements where the letters E, F are not together the letters N, O, P are not together or the set of arrangements where the letters E, F are together the letters N, O, P are together or the set of arrangements where the letters E, F are not together the letters N, O, P are not together and (c) (d) Find each of the following: #A = #B = 5!  2! = 240 4!  3! = 144 Find the number of arrangements where the letters E, F are together and the letters N, O, P are together. #(AB) = 3!  2!  3! = 72 6/4/2018 MATH 106, Section 16

(e) (f) Find the number of arrangements where the letters E, F are together or the letters N, O, P are together. #(AB) = #A + #B – #(AB) = 240 + 144 – 72 = 312 Find the number of arrangements where the letters E, F are not together and the letters N, O, P are not together. #~(AB) = #U – #(AB) = 720 – 312 = 408 6/4/2018 MATH 106, Section 16

B = the set of all arrangements where the letters O,P,Q,R are together #4 Let the set of all different possible arrangements (permutations) of the letters EFGHIOPQRUVW be the universal set. We define the subsets A = the set of all arrangements where the letters E,F,G,H,I are together B = the set of all arrangements where the letters O,P,Q,R are together C = the set of all arrangements where the letters U,V,W are together (a) (b) Find the size of the universe. #U = 12! = 479,001,600 Describe in words each of the sets listed. ~A ~B ~C set of arrangements where letters E, F, G, H, I are not together set of arrangements where letters O, P, Q, R are not together 6/4/2018 MATH 106, Section 16

B = the set of all arrangements where the letters O,P,Q,R are together Let the set of all different possible arrangements (permutations) of the letters EFGHIOPQRUVW be the universal set. We define the subsets A = the set of all arrangements where the letters E,F,G,H,I are together B = the set of all arrangements where the letters O,P,Q,R are together C = the set of all arrangements where the letters U,V,W are together (a) (b) Find the size of the universe. #U = 12! = 479,001,600 Describe in words each of the sets listed. ~A ~B ~C set of arrangements where letters E, F, G, H, I are not together set of arrangements where letters O, P, Q, R are not together set of arrangements where letters U, V, W are not together 6/4/2018 MATH 106, Section 16

AB AC BC ABC ABC the set of arrangements where the letters E, F, G, H, I are together the letters O, P, Q, R are together and the set of arrangements where the letters E, F, G, H, I are together the letters U, V, W are together and the set of arrangements where the letters O, P, Q, R are together the letters U, V, W are together and the set of arrangements where the letters E, F, G, H, I are together the letters O, P, Q, R are together the letters U, V, W are together and and the set of arrangements where the letters E, F, G, H, I are together the letters O, P, Q, R are together the letters U, V, W are together or or 6/4/2018 MATH 106, Section 16

Find each of the following: #A = #B = #C = #(AB) = #(AC) = #(BC) = #(ABC) = 8!  5! = 4,838,400 9!  4! = 8,709,120 10!  3! = 21,772,800 5!  5!  4! = 345,600 6!  5!  3! = 518,400 7!  4!  3! = 725,760 3!  5!  4!  3! = 103,680 Find the number of arrangements where the letters E, F, G, H, I are together or the letters O, P, Q, R are together or the letters U, V, W are together. 6/4/2018 MATH 106, Section 16

(d) Find the number of arrangements where the letters E, F, G, H, I are together or the letters O, P, Q, R are together or the letters U, V, W are together. #(ABC) = #A + #B + #C – #(AB) – #(AC) – #(BC) + #(ABC) = 4,838,400 + 8,709,120 + 21,772,800 – 345,600 – 518,400 – 725,760 + 103,680 = 33,834,240 6/4/2018 MATH 106, Section 16

Problem #4: Use #(ABC) = For next class, do Problem #5 on the Section 16 Handout, and do the following problems in the Section 16 Homework: Problem #4: Use #(ABC) = #A + #B + #C – #(AB) – #(AC) – #(BC) + #(ABC) children who play tennis children who play soccer children who play baseball Problem #5: Use #U – #(AB) = #U – [#A + #B – #(AB)] all 5-card hands from 52 cards all 5-card hands from the 48 non-king cards all 5-card hands from the 48 non-ace cards all 5-card hands from the 44 non-ace and non-king cards As time permits, let’s begin work on Problem #5 on the Section 16 Handout.  6/4/2018 MATH 106, Section 16

A = the set of arrangements where the digits 0, 1 are together #5 Let the set of all different possible arrangements (permutations) of the digits 0123456789 be the universal set. We define the subsets A = the set of arrangements where the digits 0, 1 are together B = the set of arrangements where the digits 5, 6 are together (a) (b) Find the size of the universe. #U = 10! = 3,628,800 Describe in words each of the sets listed. ~A ~B AB the set of arrangements where the digits 0, 1 are not together the set of arrangements where the digits 5, 6 are not together 6/4/2018 MATH 106, Section 16

A = the set of arrangements where the digits 0, 1 are together Let the set of all different possible arrangements (permutations) of the digits 0123456789 be the universal set. We define the subsets A = the set of arrangements where the digits 0, 1 are together B = the set of arrangements where the digits 5, 6 are together (a) (b) Find the size of the universe. #U = 10! = 3,628,800 Describe in words each of the sets listed. ~A ~B AB the set of arrangements where the digits 0, 1 are not together the set of arrangements where the digits 5, 6 are not together the set of arrangements where the digits 0, 1 are together and the digits 5, 6 are together 6/4/2018 MATH 106, Section 16

Find each of the following: #A = #B = 9!  2! = 725,760 9!  2! = the set of arrangements where the digits 0, 1 are not together or the digits 5, 6 are not together the set of arrangements where the digits 0, 1 are together or the digits 5, 6 are together the set of arrangements where the digits 0, 1 are not together and the digits 5, 6 are not together (c) (d) Find each of the following: #A = #B = 9!  2! = 725,760 9!  2! = 725,760 Find the number of arrangements where the digits 0, 1 are together and the digits 5, 6 are together. #(AB) = 8!  2!  2! = 161,280 6/4/2018 MATH 106, Section 16

(e) (f) Find the number of arrangements where the digits 0, 1 are together or the digits 5, 6 are together. #(AB) = #A + #B – #(AB) = 725,760 + 725,760 – 161,280 = 1,290,240 Find the number of arrangements where the digits 0, 1 are not together and the digits 5, 6 are not together. #~(AB) = #U – #(AB) = 3,628,800 – 1,290,240 = 2,338,560 6/4/2018 MATH 106, Section 16

A = the set of integers where the digit 5 does not appear #6 Let the set of all integers from 000000 to 999999 be the universal set, where we use leading zeros as needed, i.e., we write integers such as 538 as 000538. We define the subsets A = the set of integers where the digit 5 does not appear B = the set of integers where the digit 6 does not appear (a) (b) Find the size of the universe. #U = 106 = 1,000,000 Describe in words each of the sets listed. ~A ~B the set of integers where the digit 5 appears at least once the set of integers where the digit 6 appears at least once 6/4/2018 MATH 106, Section 16

the set of integers where the digit 5 does not appear AB ~(AB) AB ~(AB) the set of integers where the digit 5 does not appear the digit 6 does not appear and neither of the digits 5 and 6 appear the set of integers where the digit 5 appears at least once the digit 6 appears at least once or at least one of the digits 5 and 6 appears at least once the set of integers where the digit 5 does not appear the digit 6 does not appear or at least one of the digits 5 and 6 does not appear the set of integers where the digit 5 appears at least once the digit 6 appears at least once and each of the digits 5 and 6 appears at least once (c) Find each of the following: #A = #B = 96 = 531,441 96 = 531,441 6/4/2018 MATH 106, Section 16

#U – #(A  B) = #U – [#A + #B – #(A  B)] = (d) (e) Find the number of integers where neither of the digits 5 and 6 appear. #(AB) = 86 = 262,144 Find the number of integers where the digits 5 and 6 each appears at least once. #~(A  B) = #U – #(A  B) = #U – [#A + #B – #(A  B)] = 106 – [96 + 96 – 86] = 199,262 6/4/2018 MATH 106, Section 16

A = the set of integers divisible by 10 #7 Let the set of all integers from 1 to 1000 be the universal set. We define the subsets A = the set of integers divisible by 10 B = the set of integers divisible by 15 C = the set of integers divisible by 25 (a) (b) {10, 20, 30, …, 990, 1000} {15, 30, 45, …, 975, 990} {25, 50, 75, …, 975, 1000} Find the size of the universe. #U = 1000 Describe in words each of the sets listed. ~A ~B ~C set of integers not divisible by 10 set of integers not divisible by 15 set of integers not divisible by 25 6/4/2018 MATH 106, Section 16

the set of integers divisible by both 10 and 15 10 = 25 15 = 35 AB AC BC ABC ABC the set of integers divisible by both 10 and 15 10 = 25 15 = 35 Integers divisible by each of 10 and 15 must be divisible by 235 = 30 the set of integers divisible by both 10 and 25 10 = 25 25 = 55 Integers divisible by each of 10 and 25 must be divisible by 255 = 50 the set of integers divisible by both 15 and 25 15 = 35 25 = 55 Integers divisible by each of 15 and 25 must be divisible by 355 = 75 the set of integers divisible by each of 10, 15, and 25 10 = 25 15 = 35 25 = 55 Integers divisible by all of 10, 15, and 25 must be divisible by 2355 = 150 the set of integers divisible by at least one of 10, 15, and 25 6/4/2018 MATH 106, Section 16

Find each of the following: #A = #B = #C = #(AB) = 100 110 = 10, 210 = 20, 310 = 30, …, 10010 = 1000 66 115 = 15, 215 = 30, 315 = 45, …, 6615 = 990, 6715 = 1005 40 125 = 25, 225 = 50, 325 = 75, …, 4025 = 1000 33 10 = 25 15 = 35 Integers divisible by each of 10 and 15 must be divisible by 235 = 30 130 = 30, 230 = 60, 330 = 90, …, 3330 = 990, 3430 = 1020 6/4/2018 MATH 106, Section 16

Integers divisible by each of 10 and 25 must be divisible by 255 = #(BC) = #(ABC) = 20 10 = 25 25 = 55 Integers divisible by each of 10 and 25 must be divisible by 255 = 50 150 = 50, 250 = 100, 350 = 150, …, 2050 = 1000 13 15 = 35 25 = 55 Integers divisible by each of 15 and 25 must be divisible by 355 = 75 175 = 75, 275 = 150, 375 = 225, …, 1375 = 975, 1475 = 1050 6 10 = 25 15 = 35 25 = 55 Integers divisible by all of 10, 15, and 25 must be divisible by 2355 = 150 6/4/2018 MATH 106, Section 16

(d) (e) Find the number of integers which are divisible by all of 10, 15, or 25. #(ABC) = 6 Find the number of integers which are divisible by at least one of 10, 15, or 25. #(ABC) = #A + #B + #C – #(AB) – #(AC) – #(BC) + #(ABC) = 100 + 66 + 40 – 33 – 20 – 13 + 6 = 146 6/4/2018 MATH 106, Section 16

(f) Find the number of integers which are divisible by none of 10, 15, or 25. #~(ABC) = #U – #(ABC) = 1000 – 146 = 854 6/4/2018 MATH 106, Section 16

For next class, do the following problems in the Section 16 Homework: integers divisible by 15 integers divisible by 12 integers divisible by 18 Problem #6: Use #U – #(ABC) = #U – [#A + #B + #C – #(AB) – #(AC) – #(BC) + #(ABC)] You will need to know that 12 = 223, 15 = 35, 18 = 233. integers with no digit 4 integers with no digit 3 integers with no digit 5 Problem #7: Use #U – #(ABC) = #U – [#A + #B + #C – #(AB) – #(AC) – #(BC) + #(ABC)] 6/4/2018 MATH 106, Section 16