Inclusion- A B Exclusion 2 set proof Albert R Meyer, - - PowerPoint PPT Presentation

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Albert R Meyer, April 24, 2013

Mathematics for Computer Science

MIT 6.042J/18.062J

Inclusion- Exclusion 2 set proof

incexcI.1 Albert R Meyer, April 24, 2013

Inclusion-Exclusion

|A ∪B| =|A|+|B|-|A∩B|

A B

incexcI.2 Albert R Meyer, April 24, 2013

Inc-Exc from Sum Rule

A B

A ∪B = A∪(B−A)

disjoint

incexcI.3

proof:

Albert R Meyer, April 24, 2013

Inc-Exc from Sum Rule

A B proof:

∪ |A B| =|A|+|B−A|

by Sum Rule

incexcI.4

2

Albert R Meyer, April 24, 2013

Inc-Exc from Sum Rule

A B

∪ |A B| =|A|+|B−A|

• |B|

− | A∩B|

incexcI.5 Albert R Meyer, April 24, 2013

Lemma:

A B

|B | −A|=|B|−|A∩B

B = (B∩A) ∪(B−A)

disjoint

incexcI.6

proof:

Albert R Meyer, April 24, 2013

Lemma:

A B

|B−A|=|B|−|A∩B|

QED

proof:

|B|=|B∩A|+|B−A|

by Sum Rule

incexcI.7 Albert R Meyer, April 24, 2013

A B C

Inclusion-Exclusion (3 Sets)

|ABC| = |A|+|B|+|C| |AB| |AC| |BC| + |ABC|

incexcI.8

3

Albert R Meyer, April 24, 2013

Incl-Excl (n sets)

A ∪

1 ∪A2 ∪⋯∪A n =

∑

(−1)

S+1 ∩Ai ∅≠S⊆ 1,2,…,n

{ }

i∈ S

incexcI.10 Albert R Meyer, April 24, 2013 incexcI.13

Incl-Excl Formula: Proofs

by induction on n

• -uninformative

by binomial counting

• -next

by distributivity

• -also

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