Predicates Sets 110223055946 Phpapp01

8/10/2019 Predicates Sets 110223055946 Phpapp01

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Discrete Mathematics

Predicates and Sets

H. Turgut Uyar Ayseg ul Gencata Yayml Emre Harmanc

2001-2013


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c 2001-2013 T. Uyar, A. Yayml, E. Harmanc

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Topics

1 PredicatesIntroductionQuantiersMultiple Quantiers

2 SetsIntroduction

SubsetSet OperationsInclusion-Exclusion


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Topics


2 SetsIntroduction



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Predicate

Denition

predicate (or open statement ): a declarative sentence whichcontains one or more variables, andis not a proposition, butbecomes a proposition when the variables in it

are replaced by certain allowable choices


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Universe of Discourse

Denitionuniverse of discourse: U set of allowable choices

examples:Z: integersN: natural numbersZ+ : positive integersQ: rational numbersR: real numbersC: complex numbers


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Universe of Discourse

Denitionuniverse of discourse: U set of allowable choices

examples:Z: integersN: natural numbersZ+ : positive integersQ: rational numbersR: real numbersC: complex numbers


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Predicate Examples

Example U = Np (x ): x + 2 is an even integer

p (5): F p (8): T

p (x ): x + 2 is not an even integer

Example U = Nq (x , y ): x + y and x 2y are even integers

q (11, 3): F , q (14, 4): T


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Predicate Examples


p (5): F p (8): T



q (11, 3): F , q (14, 4): T


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Predicate Examples


p (5): F p (8): T



q (11, 3): F , q (14, 4): T


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Topics


2 SetsIntroduction



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Quantiers

Denitionexistential quantier:predicate is true for some values

symbol: read: there exists

symbol: !read: there exists only one

Denitionuniversal quantier:predicate is true for all values

symbol: read: for all


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Quantiers







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Quantiers







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Quantiers

existential quantier

U = {x 1, x 2, . . . , x n }x p (x ) p (x 1) p (x 2) p (x n )

p (x ) is true for some x

universal quantier

U = {x 1, x

2, . . . , x n }

x p (x ) p (x 1) p (x 2) p (x n )

p (x ) is true for all x


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Quantiers

existential quantier


p (x ) is true for some x

universal quantier


p (x ) is true for all x


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Quantier Examples

Example

U = Rp (x ) : x 0q (x ) : x 2 0r (x ) : (x 4)(x + 1) = 0s (x ) : x 2 3 > 0

are the following expressions true?

x [p (x ) r (x )]x [p (x ) q (x )]x [q (x ) s (x )]x [r (x ) s (x )]x [r (x ) p (x )]


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Quantier Examples

Example





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Quantier Examples

Example





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Quantier Examples

Example





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Quantier Examples

Example



x [p (x ) r (x )]x [p (x ) q (x )]x [q (x ) s (x )]x [r (x ) s (x )]

x [r (x ) p (x )]


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Quantier Examples

Example



x [p (x ) r (x )]x [p (x ) q (x )]x [q (x ) s (x )]x [r (x ) s (x )]

x [r (x ) p (x )]


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Negating Quantiers

replace with , and with negate the predicate

x p (x ) x p (x )x p (x ) x p (x )

x p (x ) x p (x )x p (x ) x p (x )


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Negating Quantiers

replace with , and with negate the predicate

x p (x ) x p (x )x p (x ) x p (x )

x p (x ) x p (x )x p (x ) x p (x )


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Negating Quantiers

Theoremx p (x ) x p (x )

Proof.

x p (x ) [p (x 1) p (x 2) p (x n )] p (x 1) p (x 2) p (x n ) x p (x )


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Negating Quantiers


Proof.



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Negating Quantiers


Proof.


N i Q i


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Negating Quantiers


Proof.


P di E i l


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Predicate Equivalences

Theorem

x [p (x ) q (x )] x p (x ) x q (x )

Theoremx [p (x ) q (x )] x p (x ) x q (x )

P di t E i l


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Predicate Equivalences

Theorem

x [p (x ) q (x )] x p (x ) x q (x )


Predicate Implications


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Theoremx p (x ) x q (x ) x [p (x ) q (x )]



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Topics


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Topics

1 PredicatesIntroductionQuantiers

Multiple Quantiers

2 SetsIntroduction


Multiple Quantiers


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Multiple Quantiers

x y p (x , y )x y p (x , y )x y p (x , y )x y p (x , y )

Multiple Quantier Examples


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Example

U = Zp (x , y ) : x + y = 17

x y p (x , y ):for every x there exists a y such that x + y = 17y x p (x , y ):there exists a y so that for all x , x + y = 17

what if U = N?



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Example

U = Zp (x , y ) : x + y = 17


what if U = N?



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Example

U = Zp (x , y ) : x + y = 17


what if U = N?



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Example

U = Zp (x , y ) : x + y = 17


what if U = N?

Multiple Quantiers


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u t p e Qua t e s

Example

U x = {1, 2}U y = {A, B }

x y p (x , y ) [p (1, A) p (1, B )] [p (2, A) p (2, B )]x y p (x , y ) [p (1, A) p (1, B )] [p (2, A) p (2, B )]x y p (x , y ) [p (1, A) p (1, B )] [p (2, A) p (2, B )]

x y p (x , y ) [p (1, A) p (1, B )] [p (2, A) p (2, B )]

Multiple Quantiers


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p Q

Example

U x = {1, 2}U y = {A, B }


x y p (x , y ) [p (1, A) p (1, B )] [p (2, A) p (2, B )]

Multiple Quantiers


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p Q

Example

U x = {1, 2}U y = {A, B }


x y p (x , y ) [p (1, A) p (1, B )] [p (2, A) p (2, B )]

Multiple Quantiers


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p Q

Example

U x = {1, 2}U y = {A, B }


x y p (x , y ) [p (1, A) p (1, B )] [p (2, A) p (2, B )]

Multiple Quantiers


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p

Example

U x = {1, 2}U y = {A, B }


x y p (x , y ) [p (1, A) p (1, B )] [p (2, A) p (2, B )]

References


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Required Reading: Grimaldi

Chapter 2: Fundamentals of Logic2.4. The Use of Quantiers

Supplementary Reading: ODonnell, Hall, Page

Chapter 7: Predicate Logic

Topics


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1 PredicatesIntroductionQuantiers

Multiple Quantiers

2 SetsIntroductionSubsetSet OperationsInclusion-Exclusion

Set


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Denition

set: a collection of elements that aredistinctunorderednon-repeating

Set Representation


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explicit representationelements are listed within braces: {a1, a2, . . . , an }

implicit representation

elements that validate a predicate: {x |x G , p (x )}: empty set

let S be a set, and a be an elementa S : a is an element of set S a / S : a is not an element of set S

|S |: number of elements (cardinality)

Set Representation


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Set Representation


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Set Representation


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Set Representation


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Explicit Representation Example


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Example

{3, 8, 2, 11, 5}11 {3, 8, 2, 11, 5}|{3, 8, 2, 11, 5}| = 5

Implicit Representation Examples


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Example

{x |x Z+ , 20 < x 3 < 100} {3, 4}{2x 1|x Z+ , 20 < x 3 < 100} {5, 7}

Example

A = {x |x R, 1 x 5}

Example

E = {n |n N, k N [n = 2k ]}A = {x |x E , 1 x 5}



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Example

{x |x Z+ , 20 < x 3 < 100} {3, 4}{2x 1|x Z+ , 20 < x 3 < 100} {5, 7}

Example

A = {x |x R, 1 x 5}

Example

E = {n |n N, k N [n = 2k ]}A = {x |x E , 1 x 5}



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Example

{x |x Z+ , 20 < x 3 < 100} {3, 4}{2x 1|x Z+ , 20 < x 3 < 100} {5, 7}

Example

A = {x |x R, 1 x 5}

Example

E = {n |n N, k N [n = 2k ]}A = {x |x E , 1 x 5}

Set Dilemma


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There is a barber who lives in a small town.He shaves all those men who dont shave themselves.He doesnt shave those men who shave themselves.

Does the barber shave himself?

yes but he doesnt shave men who shave themselves nono but he shaves all men who dont shave themselves yes

Set Dilemma


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Set Dilemma


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Set Dilemma


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let S be the set of sets that are not an element of themselvesS = {A|A / A}

Is S an element of itself?

yes but the predicate is not valid nono but the predicate is valid yes

Set Dilemma


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Set Dilemma


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Set Dilemma


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Topics


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2 SetsIntroductionSubsetSet OperationsInclusion-Exclusion

Subset


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DenitionA B x [x A x B ]

set equality:A = B (A B ) (B A)proper subset:A B (A B ) (A = B )

S [ S ]

Subset


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S [ S ]

Subset


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S [ S ]

Subset


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S [ S ]

Subset


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not a subset

A B x [x A x B ] x [x A x B ] x [(x A) (x B )] x [(x A) (x B )]

x [(x A) (x / B )]

Subset


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not a subset


x [(x A) (x / B )]

Subset


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not a subset


x [(x A) (x / B )]

Subset


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not a subset


x [(x A) (x / B )]

Subset


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not a subset


x [(x A) (x / B )]

Power Set


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Denitionpower set: P (S )the set of all subsets of a set, including the empty setand the set itself

if a set has n elements, its power set has 2n elements

Power Set


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Denitionpower set: P (S )the set of all subsets of a set, including the empty setand the set itself

if a set has n elements, its power set has 2n elements

Example of Power Set


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Example

P ({1, 2, 3}) = {,{1}, {2}, {3},{1, 2}, {1, 3}, {2, 3},{1, 2, 3}

}

Topics


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2 SetsIntroductionSubset

Set OperationsInclusion-Exclusion

Set Operations


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complement

A = {x |x / A}

intersectionA B = {x |(x A) (x B )}

if A B = then A and B are disjoint

unionAB = {x |(x A) (x B )}

Set Operations


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complement

A = {x |x / A}




Set Operations


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complement

A = {x |x / A}




Set Operations


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differenceA B = {x |(x A) (x / B )}

A B = A B symmetric difference :A B = {x |(x AB ) (x / A B )}

Set Operations


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Set Operations


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Cartesian Product


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DenitionCartesian product :

A B = {(a, b )|a A, b B }A B C N = {(a, b , . . . , n)|a A, b B , . . . , n N }

|A B C N | = |A| | B | | C | | N |

Cartesian Product


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DenitionCartesian product :

A B = {(a, b )|a A, b B }A B C N = {(a, b , . . . , n)|a A, b B , . . . , n N }

|A B C N | = |A| | B | | C | | N |

Cartesian Product Example


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ExampleA = {a1.a2, a3, a4}B = {b 1, b 2, b 3}

A B = {(a1, b 1), (a1, b 2), (a1, b 3),(a2, b 1), (a2, b 2), (a2, b 3),(a3, b 1), (a3, b 2), (a3, b 3),(a4, b 1), (a4, b 2), (a4, b 3)

}

Equivalences

D bl C l t


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Double Complement

A = A

CommutativityA B = B A AB = B A

Associativity(A B ) C = A (B C ) (AB ) C = A (B C )

IdempotenceA A = A AA = A

InverseA A = AA = U

Equivalences

D bl C l t


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Double Complement

A = A




InverseA A = AA = U

Equivalences

Double Complement


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Double Complement

A = A




InverseA A = AA = U

Equivalences

Double Complement


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Double Complement

A = A




InverseA A = AA = U

Equivalences

Double Complement


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Double Complement

A = A




InverseA A = AA = U

Equivalences

Identity


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Identity

A U = A A = A

DominationA = AU = U

DistributivityA (B C ) = ( A B ) (A C ) A (B C ) = ( AB ) (AC )

AbsorptionA (AB ) = A A (A B ) = A

DeMorgans LawsA B = AB AB = A B

Equivalences

Identity


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Identity

A U = A A = A





Equivalences

Identity


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Identity

A U = A A = A





Equivalences

Identity


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Identity

A U = A A = A





Equivalences

Identity


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Identity

A U = A A = A





DeMorgans Laws

P f


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Proof.

A B = {x |x / (A B )}= {x | (x (A B ))}

= {x | ((x A) (x B ))}= {x | (x A) (x B )}= {x |(x / A) (x / B )}= {x |(x A) (x B )}

= {x |x AB }= AB

DeMorgans Laws

P f


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Proof.

A B = {x |x / (A B )}= {x | (x (A B ))}


= {x |x AB }= AB

DeMorgans Laws

P f


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Proof.

A B = {x |x / (A B )}= {x | (x (A B ))}


= {x |x AB }= AB

DeMorgans Laws

Proof


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Proof.

A B = {x |x / (A B )}= {x | (x (A B ))}


= {x |x AB }= AB

DeMorgans Laws

Proof


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Proof.

A B = {x |x / (A B )}= {x | (x (A B ))}


= {x |x AB }= AB

DeMorgans Laws

Proof


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Proof.

A B = {x |x / (A B )}= {x | (x (A B ))}


= {x |x AB }= AB

DeMorgans Laws

Proof


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Proof.

A B = {x |x / (A B )}= {x | (x (A B ))}


= {x |x AB }= AB

DeMorgans Laws

Proof


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Proof.

A B = {x |x / (A B )}= {x | (x (A B ))}


= {x |x AB }= AB

Example of Equivalence


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TheoremA (B C ) = ( A B ) (A C )

Equivalence Example


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Proof.

(A B ) (A C ) = ( A B ) (A C )= ( A B ) (AC )= (( A B ) A) ((A B ) C ))= ((A B ) C ))= ( A B ) C = A (B C )= A (B C )

Equivalence Example


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Proof.


Equivalence Example

P f


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Proof.


Equivalence Example

P f


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Proof.


Equivalence Example

Proof


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Proof.


Equivalence Example

Proof


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Proof.


Equivalence Example

Proof


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Proof.


Topics


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2 SetsIntroductionSubset

Set OperationsInclusion-Exclusion

Principle of Inclusion-Exclusion

|AB | = |A| + |B | | A B |


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| | | | | | | ||AB C | =|A| + |B | + |C | (|A B | + |A C | + |B C |) + |A B C |

Theorem

|A1 A2 An | =i

|Ai | i , j

|Ai A j |

+i , j , k

|Ai A j Ak |

+ 1n 1 |Ai A j An |


|AB | = |A| + |B | | A B |


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Theorem

|A1 A2 An | =i

|Ai | i , j

|Ai A j |

+i , j , k

|Ai A j Ak |

+ 1n 1 |Ai A j An |


|AB | = |A| + |B | | A B |


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Theorem

|A1 A2 An | =i

|Ai | i , j

|Ai A j |

+i , j , k

|Ai A j Ak |

+ 1n 1 |Ai A j An |

Inclusion-Exclusion Example


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Example (sieve of Eratosthenes)

a method to identify prime numbers2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25 26 27 28 29 302 3 5 7 9 11 13 15 17

19 21 23 25 27 29

2 3 5 7 11 13 1719 23 25 29

2 3 5 7 11 13 17

19 23 29



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18 19 20 21 22 23 24 25 26 27 28 29 302 3 5 7 9 11 13 15 17

19 21 23 25 27 29

2 3 5 7 11 13 1719 23 25 29

2 3 5 7 11 13 17

19 23 29



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18 19 20 21 22 23 24 25 26 27 28 29 302 3 5 7 9 11 13 15 17

19 21 23 25 27 29

2 3 5 7 11 13 1719 23 25 29

2 3 5 7 11 13 17

19 23 29



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18 19 20 21 22 23 24 25 26 27 28 29 302 3 5 7 9 11 13 15 17

19 21 23 25 27 29

2 3 5 7 11 13 1719 23 25 29

2 3 5 7 11 13 17

19 23 29



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18 19 20 21 22 23 24 25 26 27 28 29 302 3 5 7 9 11 13 15 17

19 21 23 25 27 29

2 3 5 7 11 13 1719 23 25 29

2 3 5 7 11 13 17

19 23 29



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number of primes between 1 and 100

numbers that are not divisible by 2, 3, 5 and 7A2: set of numbers divisible by 2A3: set of numbers divisible by 3A5: set of numbers divisible by 5A7: set of numbers divisible by 7

|A2 A3 A5 A7 |



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|A2 A3 A5 A7 |



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|A

2 A

3 A

5 A

7 |



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|A2| = 100/ 2 = 50

|A3| = 100/ 3 = 33|A5| = 100/ 5 = 20|A7| = 100/ 7 = 14

|A2 A3 | = 100/ 6 = 16

|A2 A5 | = 100/ 10 = 10|A2 A7 | = 100/ 14 = 7|A3 A5 | = 100/ 15 = 6|A3 A7 | = 100/ 21 = 4

|A5 A7 | = 100/ 35 = 2



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|A2| = 100/ 2 = 50

|A3| = 100/ 3 = 33|A5| = 100/ 5 = 20|A7| = 100/ 7 = 14

|A2 A3 | = 100/ 6 = 16

|A2 A5 | = 100/ 10 = 10|A2 A7 | = 100/ 14 = 7|A3 A5 | = 100/ 15 = 6|A3 A7 | = 100/ 21 = 4

|A5 A7 | = 100/ 35 = 2



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|A2 A3 A5| = 100/ 30 = 3|A2 A3 A7| = 100/ 42 = 2|A2 A5 A7| = 100/ 70 = 1|A3 A5 A7| = 100/ 105 = 0

|A2 A3 A5 A7 | = 100/ 210 = 0



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|A2 A3 A5| = 100/ 30 = 3|A2 A3 A7| = 100/ 42 = 2|A2 A5 A7| = 100/ 70 = 1|A3 A5 A7| = 100/ 105 = 0

|A2 A3 A5 A7 | = 100/ 210 = 0


E l ( i f E t th )


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|A2 A3 A5 A7 | = (50 + 33 + 20 + 14)

(16 + 10 + 7 + 6 + 4 + 2)

+ (3 + 2 + 1 + 0) (0)= 78

number of primes: (100 78) + 4 1 = 25


E l ( i f E t th )


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|A2 A3 A5 A7 | = (50 + 33 + 20 + 14) (16 + 10 + 7 + 6 + 4 + 2)+ (3 + 2 + 1 + 0) (0)= 78

number of primes: (100 78) + 4 1 = 25

References

Required Reading: Grimaldi


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Required Reading: GrimaldiChapter 3: Set Theory

3.1. Sets and Subsets3.2. Set Operations and the Laws of Set Theory

Chapter 8: The Principle of Inclusion and Exclusion8.1. The Principle of Inclusion and Exclusion

Supplementary Reading: ODonnell, Hall, Page

Chapter 8: Set Theory

Predicates Sets 110223055946 Phpapp01

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