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Section 16 Inclusion/Exclusion

Section 16 Inclusion/Exclusion. 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.

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Section 16 Inclusion/Exclusion

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  1. Section 16Inclusion/Exclusion MATH 106, Section 16

  2. 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. MATH 106, Section 16

  3. #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} MATH 106, Section 16

  4. 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} 4 6 3 2 {1, 2, 4, 5, 7. 8, 10, 11} {1, 3, 5, 7, 9, 11} {1, 2, 3, 5, 6, 7, 9, 10, 11} {1, 2, 3, 4, 6, 7, 8, 9, 11, 12} MATH 106, Section 16

  5. {6, 12} {4, 8, 12} 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. { } or  {2, 3, 4, 6, 8, 9, 10, 12} {3, 5, 6, 9, 10, 12} {2, 4, 6, 8, 10, 12} 1 5 7 11 12 3 9 4 8 10 6 2 A B MATH 106, Section 16

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

  7. 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 0 MATH 106, Section 16

  8. 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) 35 0 84 0 7 4 6765 6765 262,144 262,142 255,379 2 MATH 106, Section 10

  9. NAME__________________________________________________ A child is playing with large box of blocks that are identical, except that some are red and some are blue. 2. (a) (b) 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? 0 MATH 106, Section 10

  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 MATH 106, Section 10

  11. 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? 0 MATH 106, Section 10

  12. (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 MATH 106, Section 10

  13. 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 MATH 106, Section 10

  14. (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. MATH 106, Section 10

  15. 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 freshmen taking computer science all freshmen MATH 106, Section 10

  16. #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} 15 #A = #B = #C = #(AB) = #(AC) = #(BC) = #(ABC) = 10 6 MATH 106, Section 16

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

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

  19. 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)

  20. #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 not together the set of arrangements where the letters E, F are the set of arrangements where the letters N, O, P are not together MATH 106, Section 16

  21. 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 not together the set of arrangements where the letters E, F are 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 MATH 106, Section 16

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

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

  24. #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 not together set of arrangements where letters O, P, Q, R are MATH 106, Section 16

  25. 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 not together set of arrangements where letters O, P, Q, R are not together set of arrangements where letters U, V, W are MATH 106, Section 16

  26. the set of arrangements where the letters E, F, G, H, I are together the letters O, P, Q, R are together AB AC BC ABC ABC 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 MATH 106, Section 16

  27. (c) (d) Find each of the following: #A = #B = #C = #(AB) = #(AC) = #(BC) = #(ABC) = 9!  4! = 8,709,120 8!  5! = 4,838,400 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. MATH 106, Section 16

  28. 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. (d) #(ABC) = #A + #B + #C – #(AC) – #(BC) + #(ABC) – #(AB) = 4,838,400 + 8,709,120+ 21,772,800 – 345,600– 518,400– 725,760 + 103,680= 33,834,240 MATH 106, Section 16

  29. 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 44 non-ace and non-king cards all 5-card hands from the 48 non-ace cards As time permits, let’s begin work on Problem #5 on the Section 16 Handout.  MATH 106, Section 16

  30. #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 MATH 106, Section 16

  31. 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, 1are together and the digits 5, 6 are together MATH 106, Section 16

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

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

  34. #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 appears at least once the set of integers where the digit 5 appears at least once the set of integers where the digit 6 MATH 106, Section 16

  35. 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 or the set of integers where the digit 5 does not appear the digit 6 does not appear 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 MATH 106, Section 16

  36. (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 MATH 106, Section 16

  37. #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 MATH 106, Section 16

  38. the set of integers divisible by AB AC BC ABC ABC both 10 and 15 15 = 35 10 = 25 Integers divisible by each of 10 and 15 must be divisible by 235 = 30 both 10 and 25 the set of integers divisible by 25 = 55 10 = 25 Integers divisible by each of 10 and 25 must be divisible by 255 = 50 both 15 and 25 the set of integers divisible by 25 = 55 15 = 35 Integers divisible by each of 15 and 25 must be divisible by 355 = 75 each of 10, 15, and 25 the set of integers divisible by 15 = 35 25 = 55 10 = 2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 MATH 106, Section 16

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

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

  41. (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C) – #(BC) + #(ABC) – #(AB) = 100 + 66+ 40 – 33– 20– 13 + 146 6 = MATH 106, Section 16

  42. (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 MATH 106, Section 16

  43. For next class, do the following problems in the Section 16 Homework: integers divisible by 15 integers divisible by 18 integers divisible by 12 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 5 integers with no digit 3 Problem #7: Use #U – #(ABC) = #U – [#A + #B + #C– #(AB) – #(AC) – #(BC) + #(ABC)] MATH 106, Section 16

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