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Section 3.3 Principle of Inclusion and Exclusion

Section 3.3 Principle of Inclusion and Exclusion. ( 2 more counting principles ) Principle of Inclusion and Exclusion Given finite sets A 1 , A 2 , …A n , n 2, then |A 1 A 2 …  A n | =. + (-1) n+1 | A 1 A 2  …  A n |. A-B, A B, B-A are mutually disjoint sets.

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Section 3.3 Principle of Inclusion and Exclusion

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  1. Section 3.3 Principle of Inclusion and Exclusion

  2. ( 2 more counting principles ) Principle of Inclusion and Exclusion Given finite sets A1, A2, …An, n 2, then |A1A2 …  An| = + (-1)n+1| A1A2  …  An |

  3. A-B, AB, B-A are mutually disjoint sets. i.e. xA-B, then xB, and therefore x B-A, x AB. AB A-B B-A S: universal set |AB| = |A| + |B| - |AB| |A-B| = |A| - |AB| |B-A| = |B| - |AB|  e.g. |AB| can be computed in several ways depends on the information given.

  4. e.g. In a group of 42 tourists, everyone speaks English or French; there are 35 English speakers and 18 French speakers. How many speak both English & French? 42 |AB| = |A| + |B| - |AB|  |AB| = 11 English French 42 35 18 35 18 ?

  5. e.g. What if we have 3 sets: A For |A| + |B| + |C|: I: counted 1 time II: counted 2 times III: counted 3 times I II II III I II I C B |ABC| = |A| + |B| + |C| - |A  B| - |A  C| - |B  C| + |A  B  C| L.H.S. = |A  (B  C)| = |A| + |B  C|-|A  (B  C)| = |A| + |B| + |C| - |B  C| - |(A  B)  (A  C)| = |A| + |B| + |C| - |B  C| - |A  B| - |A  C| + |ABC| = R.H.S. (can also be seen from the picture)

  6. eg: a survey of 150 college students reveals that: 83 own Cars, 97 own Bikes, 28 own Motorcycles, 53 own a car and a bike, 14 own a car and a motorcycle, 7 own a bike and a motorcycle, 2 own all three.

  7. How many own a bike and nothing else? C 83 M 28 B 97 |B – (C  M)| = |B| - |B  (C  M)| = |B| - |(B  C) (B  M)| = |B| - (|B  C| + |B  M| - |B  C  M|) = 97 – (53 + 7 – 2) = 39

  8. b. How many students do not own any of the three? 150 - |C  B  M| = 150 – (83+97+28-53-14-7+2) = 150 – 136 = 14  The general formula can be derived on P.1 C 14 18 150 51 12 2 39 9 5 B M

  9. Pigeonhole Principle: If more than k items are placed into k bins, then at least one bin contains more than one item. e.g. How many people must be in a room to guarantee that two people have last names that begin with the same letter?  there are 26 letters(or bins). If there are 27 people, then at least 2 people will have last names beginning with the same letter.

  10. e.g. The population of city x is about 35,000. If each resident has three initials, is it true that there must be at least 2 individuals with the same initials? Yes,  26  26  26 = 17,576 < 35,000 Why? Pigeonhole principle & Multiplication rule

  11. eg: How many numbers less than 1 million contain the digit 2? 987234  398173 × 222333  106 - 96 (including 0) (without digit 2) eg: How many bit strings have length 3, 4, or 5? 101  1110  00110  100001× 23 + 24 + 25

  12. eg: Quality control in a factory pulls 40 parts with paint, packaging, or electronics defects from an assembly line. Of these, 28 had a paint defect, 17 had a packaging defect, 13 had an electronics defect, 6 had both paint and packaging defects, 7 had both packaging and electronics defects, and 10 had both paint and electronics defects. Did any part have all three types of defect?

  13. Yes, x = 5 • 40 parts w/paint, packaging, or electronics defects: • 40 = 28 + 17 + 13 – 6 – 7 – 10 + x x = 5 Paint 28 6 10 17 ? 13 Packaging electronics 7

  14. eg: In a group of 25 people, must there be at least 3 who were born in the same month? 25 people, since there are only 12 months(bins)  more than 24 items means that at least one bin has more than 2 items.  Yes

  15. e.g. How many numbers must be selected from the set {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} in order to guarantee that at least one pair adds up to 22? Find all pairs add to 22: (2, 20) (4, 18) (6, 16) 5 bins (8, 14) (10, 12)  6 numbers selected, at least 2 of them will be from the same pair, which gives 22.

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