1 / 16

7.5 Inclusion/Exclusion

7.5 Inclusion/Exclusion. Definition and Example- 2 sets. | A  B| =|A| + |B| - |A ∩ B| Ex1: |A|=9, |B|=11, |A∩B|=5, | A  B | = ?. 3 sets. |A  B C |= ?. Proof for 3 sets. |A1  A2  A3| =∑|Ai| - ∑|Ai ∩ Aj | + |A1∩ A2 ∩ A3| =∑|Ai| =∑|Ai| - ∑|Ai ∩ Aj | .

cid
Download Presentation

7.5 Inclusion/Exclusion

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 7.5 Inclusion/Exclusion

  2. Definition and Example- 2 sets |A  B| =|A| + |B| - |A ∩ B| Ex1: |A|=9, |B|=11, |A∩B|=5, |A  B| = ?

  3. 3 sets |A  B C |= ?

  4. Proof for 3 sets |A1  A2  A3| =∑|Ai| - ∑|Ai ∩ Aj| + |A1∩ A2 ∩ A3| =∑|Ai| =∑|Ai| - ∑|Ai ∩ Aj|

  5. =∑|Ai| - ∑|Ai ∩ Aj| + |A1∩ A2 ∩ A3|

  6. 3 sets |A1  A2  A3| =∑|Ai| - ∑|Ai ∩ Aj| + |A1∩ A2 ∩ A3| Ex. 2: |A|=13, |B|=12, |C|=14, |A∩B|=7, |A∩C|=8, |B∩C|=9, |A∩B∩C|=5, |A  B  C|=?

  7. |A1 A2 A3 A4| =∑|Ai| - ∑|Ai ∩ Aj| + ∑ |Ai∩ Aj ∩ Ak| - |A1∩ A2 ∩ A3∩ A4|

  8. In General: Theorem: |A1 A2  … An| =∑|Ai| - ∑|Ai ∩ Aj| + …+(-1)n+1|A1∩ A2 ∩…∩ An| Proof: …

  9. Proof Proof idea: Show that the right hand side counts each element in the union exactly once. Suppose that a is a member of exactly r of the sets A1, A2, A3,… An where 1≤r ≤n. This element is counted ____ times by ∑|Ai|, ____ times by ∑|Ai ∩ Aj|,… Thus it is counted C(r,1)-C(r,2)+…+(-1)r+1 C(r,r) times by the right side of the equation. By Cor. 2 of Sec. 5.4, C(r,0)-C(r,1)+C(r,2)+…+(-1)r C(r,r)=0 Since C(r,0)=1, Hence, 1=___________________ So each element is counted once on both the right and the left.

  10. Applications- 2 sets Ex : Find the number of positive integers not exceeding 100 that are divisible by 5 or 7.

  11. Ex Find the number of positive integers not exceeding 100 that are NOT divisible by 5 or 7.

  12. Applications- 3 sets

  13. Applications- 3 sets Ex: A survey of 63 students reports that 20 are involved in sports, 23 are involved in social clubs, 29 are involved in academic clubs, 7 are in sports and social clubs, 6 are in social and academic clubs, 8 are in sports and academic clubs, and 5 are in all three. Use a Venn diagram to answer some questions

  14. Venn diagram

  15. questions • How many were in none of these activities? • How many were in sports or social? • How many were in sports or social, but not academic? • How many were in social and academic, but not sports? • How many were in just one activity? • How many were in at least 2 activities?

  16. Assume that |A1|=100, |A2|=1000, and |A3|=10,000 Calculate |A1  A2  A3| if: a) A1A2 and A2 A4 b) The sets are pairwise disjoint c) There are 2 elements common to each pair of sets and 1 element in all 3 sets

More Related