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7.5 Inclusion/Exclusion

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

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3 sets |A B C |= ? A CB

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Proof for 3 sets |A1 A2 A3| =∑|Ai| - ∑|Ai ∩ Aj| + |A1∩ A2 ∩ A3| =∑|Ai| =∑|Ai| - ∑|Ai ∩ Aj| A1 A3A2 A1 A3A2

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=∑|Ai| - ∑|Ai ∩ Aj| + |A1∩ A2 ∩ A3| A1 A3A2

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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|=?

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

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

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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.

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

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

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Applications- 3 sets Female sophomore Math major

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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

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Venn diagram Sports socialacademic

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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?

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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

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