principle of inclusion and exclusion - warwick...

Principle of inclusion and exclusion

If A and B are disjoint sets, then

|AB|=|A|+|B| Rule of Sum

If A and B are not disjoint, then

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

A B


For three sets we have,

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

A B

C

2

2 2 3

|A|+|B|+|C| 1 1

1




A B

C

1

2 2 2

|A|+|B|+|C|

-|AB| 1 1

1




A B

C

1

1 2 1

|A|+|B|+|C|

-|AB|

-|AC|

1 1

1




A B

C

1

1 1 0

|A|+|B|+|C|

-|AB|

-|AC|

-|BC|

1 1

1




A B

C

1

1 1 1

|A|+|B|+|C|

-|AB|

-|AC|

-|BC|

+|ABC|

1 1

1


For A1,A2,…,An and a natural kn, denote

||2

21

1 k

k

ii

iii

ik AAAS

|||| 11 nAAS ||2 ji

ji

AAS

n

n

n SSSSSAAA 1

432121 )1(||

Theorem


n

n

n SSSSSAAA 1

432121 )1(||

Theorem

Proof.

xA1 … An

Assume x is contained in m sets,

then x contributes m to S1

x contributes to S2,

2

m

k

mx contributes to Sk

m

k

k

m

k

k

m

k

m

k

m

0

1

1

1

1100

)1(

)1(

0)1(0

m

k

k

k

m0)1(

0

1

m

k

k

k

m


n

n

n SSSSSAAA 1

432121 )1(||

Theorem

If A1,A2,…,AnA, then denoting |A|=S0, we have

n

n

n SSSSSAAAA )1(|||| 321021

Illustration of the Principle of inclusion and exclusion

How many ways are there to place k identical balls into n different boxes so that no box contains more than p balls?

kxxx n 21 (1)

How many non-negative integer solutions of (1) are there in which no xi exceeds p?

Before answering this question, recall

How many ways are there to place k identical balls into n different boxes?

How many ways are there to place k identical balls into n different boxes so that each box contains at least p+1 balls?

1

11

n

kn

k

kn

kn(p+1)

1

1)1(

n

pnkn


How many ways are there to place k identical balls into n different boxes so that no box contains more than p balls?

kxxx n 21 (1)

How many non-negative integer solutions of (1) are there in which no xi exceeds p?

Let B be the set of non-negative integer solutions of (1) s.t. xip i

and A be the set of all non-negative integer solutions of (1)

k

knA

1|| |B|=?

Ai the set of non-negative integer solutions of (1) in which xi>p

|B|=|A|-|A1A2 …An|

||2

21

1 l

l

ii

iii

il AAAS

n

n

n SSSSSAAA 1

432121 )1(||


liii AAA 21

pxpxlii ,...,

1

1

1)1(||

21 n

plknAAA

liii

1

1)1(

n

plkn

l

nSl

n

l

l

n

plkn

l

nB

0 1

1)1()1(||

= the set of solutions in which

Ai the set of non-negative integer solutions of (1) in which xi>p

|B|=|A|-|A1A2 …An|

Exercise

Call a 7-digit telephone number d1,d2,d3 d4,d5,d6,d7 memorable if the prefix sequence d1,d2,d3 is exactly the same as either the sequence d4,d5,d6 or d5,d6,d7 (possibly both). Assuming that each di can be any of the ten decimal digits 0,1,2,…,9, find the number of different memorable numbers.

Let A be the set of memorable numbers with d1,d2,d3= d4,d5,d6 and B the set of memorable numbers with d1,d2,d3= d5,d6,d7

Then the number of different memorable numbers is |AB|=|A|+|B|-|AB|=104+104-10

Exercise

In a group of 15 people, 6 people speak English, 4 people speak French, 5 people speak German, 3 speak English and French, 2 speak English and German, 2 speak French and German, 1 speaks all three languages. Determine how many people in the group speak none of the three languages.

|U|=15, |E|=6, |F|=4, |G|=5, |EF|=3, |EG|=2, |FG|=2, |EFG|=1

|None|=|U|-|EFG| |EFG|=|E|+|F|+|G|-|EF|-|EG|-|FG|+|EFG|

|None|=6

Relations

Definition. A binary relation on a set A is a subset of A2

Examples: If A={1,2,3}, then {(1,2),(2,3)} is a relation on A

If A is the set of students taking Combinatorics, then R1={(a,b) | a likes b} is binary relation and R2={(a,b) | a and b have the same birthday} is a binary relation

How many binary relations on a set A are there?

Answer: 2||2 A

Representation of relations

1. By listing the elements: {(1,2),(2,3)} is a relation on A ={1,2,3}

2. By a binary matrix: 1

2

3

1 2 3

0 1 0

1 0 0

0 0 0

2. By a graph: 1

2 3

Properties of relations

Definition. A relation R is symmetric if (a,b) R implies (b,a) R

In terms of graphs: a b

Is R1={(a,b) | a likes b} symmetric?

Is R2={(a,b) | a and b have the same birthday} symmetric?

Is {(1,2),(2,3)} symmetric?


Definition. A relation R on A is reflexive if (a,a) R for each a A

In terms of graphs:

Is R1={(a,b) | a likes b} reflexive?

Is R2={(a,b) | a and b have the same birthday} reflexive?

Is {(1,2),(2,3)} reflexive?

a loop

1

1

1

In terms of matrices:


Definition. A relation R on A is transitive if (a,b) R and (b,c) R implies (a,c) R

In terms of graphs:

Is R1={(a,b) | a likes b} transitive?

Is R2={(a,b) | a and b have the same birthday} transitive?

Is {(1,2),(2,3)} transitive?

a

b

c


Definition. A relation R is an equivalence relation if R is symmetric, reflexive and transitive

R2={(a,b) | a and b have the same birthday} is an equivalence relation

Partitions

Definition. Partition of a set A is an equivalence relation on A

Definition. A collection of subsets A1,…,Ak of a set A is called a partition of A if the subsets are pairwise disjoint and the union of the subsets coincides with A

R={(a,b) | there is Ai such that aAi and bAi}

Each subset Ai is called an equivalence class of the relation

Partitions

How many partitions of an n-set are there?

Bn Bell number

How many ways to partition an n-set into k subsets are there?

}{k

n Stirling number of the second kind

How many ordered partitions of an n-set into k subsets are there?

A partition (A1,…,Ak) is ordered if the order of the subsets matters

Example. there are 2 ordered partitions of the set {1,2} into two subsets: ({1},{2}) and ({2},{1})

Partitions

Theorem. The number of ordered partitions of an n-set into k subsets of cardinalities n1,…,nk is

!!!

!

21 knnn

n

)!(!

!

11 nnn

n

Proof.

)!(!

)!(

212

1

nnnn

nn

)!(!

)!(

1

11

kk

k

nnnn

nnn

…

!!!

!

21 knnn

n

Multinomial coefficient

knnn

n

,,, 21

Partitions

Exercise. Let A={a1,…,ak} be an alphabet and n a natural number, and n=n1+…+nk a partition of n.

How many words of length n in alphabet A in which letter ai appears exactly ni times are there?

!!!

!

21 knnn

n

1 2 3 4 5 ...... n-1 n

…… a1 a1 a1

Multinomial Theorem

k

k

n

k

nn

nnn k

n

k xxxnnn

nxxx

21

1

21

... 21

21,,,

)...(

For any integer n0 and k1,

Proof.

Represent

and expand the brackets

n

kk

n

k xxxxxxxxx )...()...()...( 212121

332313322212312111

321321

2

321 ))(()(

xxxxxxxxxxxxxxxxxx

xxxxxxxxx

For instance, for n=2 and k=3 we have

The set of all words of length n in the alphabet {x1,…,xk}

kn

k

nnxxx 21

21The coefficient of is the number of words containing letter xi

exactly ni times, which is the multinomial coefficient

principle of inclusion and exclusion - warwick...

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