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
Principle of inclusion and exclusion
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
Principle of inclusion and exclusion
For three sets we have,
|ABC|=|A|+|B|+|C|-|AB|-|AC|-|BC|+|ABC|
A B
C
1
2 2 2
|A|+|B|+|C|
-|AB| 1 1
1
Principle of inclusion and exclusion
For three sets we have,
|ABC|=|A|+|B|+|C|-|AB|-|AC|-|BC|+|ABC|
A B
C
1
1 2 1
|A|+|B|+|C|
-|AB|
-|AC|
1 1
1
Principle of inclusion and exclusion
For three sets we have,
|ABC|=|A|+|B|+|C|-|AB|-|AC|-|BC|+|ABC|
A B
C
1
1 1 0
|A|+|B|+|C|
-|AB|
-|AC|
-|BC|
1 1
1
Principle of inclusion and exclusion
For three sets we have,
|ABC|=|A|+|B|+|C|-|AB|-|AC|-|BC|+|ABC|
A B
C
1
1 1 1
|A|+|B|+|C|
-|AB|
-|AC|
-|BC|
+|ABC|
1 1
1
Principle of inclusion and exclusion
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
Principle of inclusion and exclusion
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
Principle of inclusion and exclusion
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
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?
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(||
Illustration of the Principle of inclusion and exclusion
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?
Properties of relations
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:
Properties of relations
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
Properties of relations
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