Combinatorics
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Sets

Chapter 1 Sets (section 3.1 )

1.1 Introduction

Problem p 161

Goal:

Introduce the formalism of set theory
Use it to solve counting problems

1.2 Sets and basic manipulations of sets

1.2.1 Definition

Définition 1 A set A is a collection of objects.

Defining property: you can ask whether an object a belongs to A or not.

Objects or elements: a,b,c,....

Sets: A,B,C,....

Predicates:

ain S: a belongs to S, or a is an element of S, or S contains a;
anot in S: a does not belong to S.

Two sets A and B are equal (A=B) if they contain the same elements:

(for all x)(xin A)<=>(xin B)

1.2.2 How to define a set S?

You have to describe which elements are in the set S and which are not.

List the elements of the set S:

S:={ a,b,c}
ain S is true, whereas din S is false.
The order and the repetition are irrelevant:
{ a,b,c}={ a,b,b,b,c}={ c,b,a}

Note that a and { a} are two different things. The first is the object a, whereas the second one is the set containing a, and no other elements.
Provide a predicate which determines which element are in S:
S:={ x | P(x)}

S:={ x | x is an integer and 3<x<=7}
5in S is true, whereas 3not inS, and no car belongs to A.
Use a recursive definition, as for the set of well formed formulas:
S contains basic propositions A,B,C,...
If P and Q belong to S, then P', P/\ Q, P\/ Q, P-> Q, P<-> Q also belong to S.

Some usual sets:

The empty set:
Ø:={ x | false}
Numbers:
N:={ x | x is a non negative integer }

Z:={ x | x is an integer }

Q:={ x | x is a rational number }

R:={ x | x is a real number }

C:={ x | x is a complex number }

S^*:={ x | xin S and x!=0}

S⁺:={ x | xin S and x>=0}

1.2.3 Relationships between sets

A is a subset of B (Aincluded in or equal to B) iff any element of A is in B:
(for all x)(xin A)=>(xin B)
A is a proper subset of B (Aincluded in B) iff A is a subset of B but they are not equal :
(for all x)(xin A)=>(xin B) and (there exists x)(xin B)/\(xnot in A)

Exercice 1 [, Exercise 10 p. 117]

R:={1,3,pi,4,1,9,10}, S:={{1},3,9,10}

T:={1,3,pi}, U:={{1,3,pi},1}

Sincluded in or equal to R?
1in R?
1in S?
1included in or equal to U?
{1}included in or equal to T?
{1}included in or equal to S?
Tincluded in R?
{1}in S?
Øincluded in or equal to S?
Tincluded in or equal to U?
Tin U?
Tnot in R?
Tincluded in or equal to R?
Sincluded in or equal to{1,3,9,10}?

1.2.4 The powerset of a set

The powerset (S) of a set S is the set of all subsets of S.

Exemple 1 What are the elements of the following powersets ?

({1,2})=
({1})=
(Ø)=

Exercice 2 What is the size of (S) if S has n elements ?

1.2.5 Binary and unary operations

Définition 2 ¤ is a unary operation on a set S if for every element x of S, ¤ x exists, is unique and is a member of S.

¤ is a binary operation on a set S if for every ordered pair (x,y) of elements of S, x¤ y exists, is unique and is a member of S.

Examples:

is +a binary operation on N ?
is +a binary operation on {0,1,2}?
is ×a binary operation on {0,1}
is -a binary operation on N ?
is /a binary operation on Z ?
are +,-, and ×binary operations on Q ?
are +,-, ×and /binary operations on Q^*?
is (x)^1/2 a unary operation on R⁺ ?
are /\, \/, ->, ', ... operations on { true,false}? on wff ?

A binary operation can be defined by its multiplication table.

Exemple 2 ({ true,false},/\)

/\	false	true
false	false	false
true	false	true

An operation does not necessarily need to have a particular meaning.

Exemple 3 ({2,5,9},¤)

¤	2	5	9
2	2	5	2
5	2	9	5
9	9	2	5

5¤9=?

1.3 Operations on sets

1.3.1 The universe of discourse

We have seen operations on numbers, or other objects.

We can also define operations on sets (union, intersection, ...).

We need a set that contains all the sets we want to deal with.

It would be tempting to consider the set of all sets.

Problem 1 Let S be the set of all sets that does not contain themselves.

Does S contain itself ?

if S does not contain itself, then it contains itself !
if S contains itself, then it does not contain itself !

There is a strange loop here which yields a contradiction: S cannot exist.

For a similar reason, the set of all sets cannot exist.

See Gödel Esher Bach[] for a lot of similar fun stuff with strange loops.

That's been a major source of trouble and work a century ago, when mathematicians tried to define a strong basis for set theory.

To be safe, we always work inside a set big enough to contain all the sets we need, but small enough to avoid the strange loop above.

Définition 3 This set is the universal set, or the universe of discourse.

We define operations on the powerset of the universal set.

1.3.2 Union, intersection, complement and Cartesian product

We define operations on subsets of the universe of discourse.

Définition 4 The union of A and B is the set:

Aunion B:={ x | xin A or xin B}

Définition 5 The intersection of A and B is the set:

Aintersect B:={ x | xin A and xin B}

Définition 6 The complement of A is the set:

A':={ x | xnot in A}

Définition 7 The set difference of A and B is the set:

A-B:={ x (xin A) and (xnot in B)}

Définition 8 The Cartesian product (or cross product) of A and B is the set:

A× B:={(x,y) | xin A,yin B}

Notation 2 A²=A× A is the set of all ordered pairs of elements of A;

Aⁿ=A×···× A is the set of all n-uples of elements of A.

Définition 9 Venn diagrams.

Exercice 3 Let A:={1,2} and B:={2,3,4}.

Aunion B=,
Aintersect B=,
A-B=,
A'= (assuming that the universe of discourse is {1,2,3,4}),
A× B=,
A³=.

Exercice 4 Let A be a set of size n, and B a set of size m.

what is the size of Aunion B:
what is the size of Aintersect B:
what is the size of A' ?
what is the size of A× B :

1.3.3 Set identities

Proposition 1 Let A and B be to sets. Then,

Aintersect B=Bintersect A (commutative property)

Proof. xin Aintersect B

<=>(xin A)/\(xin B) by definition of intersect

<=>(xin B)/\(xin A) by commutativity of /\

<=> xin Bintersect A by definition of intersect

The commutativity of intersectderives directly from the commutativity of /\.

All identities on logic operators induce identities on set operators.

Proposition 2 Let A, B, and C be two sets. Then,

Aunion B=Bunion A (commutative property)

Aintersect B=Bintersect A (commutative property)

(Aunion B)union C=Aunion(Bunion C)=Aunion Bunion C (associative property)

(Aintersect B)intersect C=Aintersect(Bintersect C)=Aintersect Bintersect C (associative property)

Aunion A'=S (complement, S is the universe of discourse)

Aintersect A'=Ø (complement)

(Aunion B)'=A'intersect B' (de Morgan's)

(Aintersect B)'=A'union B' (de Morgan's)

A-B=Aintersect B'

1.4 Countable and uncountable sets

1.4.1 Enumerations

If a set S has k elements, these can be listed one after the other:

s₁,s₂,...,s_k

Définition 10 s₁,s₂,...,s_k is called an enumeration of the set S.

Exemple 4 1,5,4,3, 5,1,3,4, and 4,1,3,5 are enumerations of the set {1,5,4,3}.

When a set S is infinite, you can not enumerate all of it's elements at once.

However, some times, you can still pickup a first element s₁, then a second s₂, and so on forever. This process is an enumeration of the elements of S if:

Any element of S gets enumerated eventually;
No element is enumerated twice.

Exemple 5 0,1,2,3,4,... is an enumeration of the non-negative integers.

1.4.2 Denumerable and Countable sets

Définition 11 A infinite set S is denumerable if it has an enumeration

s₁,s₂,...,s_n,...

A set S is countable if it's either finite or denumerable.

Exemple 6 Are the following sets countable ?

The set of all students in this class.
The set N of the non-negative integers (enumeration: 0,1,2,3,4,...);
Actually, the order does not need to be logical.
This is a perfectly legal enumeration: 0,10,5,11,4,2,17,2,1,28,3
The set of the even integers (enumeration: ;
The set of the prime integers (enumeration: ;
The set Z of the integers (enumeration: );
The set N×N

(e.g. all points in the plane with non-negative integer coordinates):
Enumeration: (0,0),(0,1),(1,0),(2,0),(1,1),(0,2),(3,0),...
The set Z×Z of all points in the plane with integer coordinates:

Enumeration: (0,0),(0,1),(1,0),(0,-1),(-1,0),(2,0),...
The set Q of rational numbers:

Théorème 1 The following sets are countable:

Any subset Aincluded in B of a countable set B.
The union Aunion B of two countable sets A and B is countable.
The union A₁union A₂union···union A_i of a finite number i of countable sets A₁,...,A_i.
The union A₁union A₂union···union A_iunion···of a countable number of countable sets.
The cross product A× B of two countable sets A and B is countable.
The cross product A₁×···× A_i of a finite number of countable sets A₁,...,A_i.

Proof. Enumerations of those sets can be constructed in the following way:

Enumerate the elements of B, and ignore those that are not in A.
Enumerate the elements of A and B alternatively (a₁,b₁,a₂,b₂,...).
Induction, using 2.
As for N×N: a_1,1,a_2,1,a_1,2,a_3,1,a_2,2,a_1,3,... (a_i,j is the jth element of A_i).
As for N×N: (a₁,b₁),(a₁,b₂),(a₂,b₁),(a₁,b₃),(a₂,b₂),(a₃,b₁),....
Induction, using 5.

1.4.3 Uncountable sets ?

Remarque 1 All the sets we have seen so far are countable.

Problem 1 Are there any uncountable sets ?

Théorème 2 (Cantor) R is uncountable.

Lemme 1 Any real number xin[0,1) can be written uniquely in decimal form:

x=0.d₁d₂d₃d₄··· d_i···,

where the d_i are digits between 0 and 9, and the sequence does not end by an infinite number of 9.

Exemple 7 Here is how some classical numbers can be written:

0=0.0000···
0.5=0.50000···
0.5=0.49999··· (Same number as above, we don't use this form)
1/3=0.333333···
pi=3.141592653···

Let's jump to the proof of the theorem, without detailing the proof of this lemma.

Proof. (Of the theorem): Cantor diagonalization method.

It's sufficient to prove that [0,1) is not countable.

Let's do this by contradiction: assume [0,1) is countable.

Let s₁,s₂,...,s_i,... be an enumeration of [0,1).

Goal: construct an element which is not in this enumeration.

Using the lemma, we can write the s_i as follow:

s₁

d_1,1

d_1,2

d_1,3

d_1,4

d_1,5

···

s₂

d_2,1

d_2,2

d_2,3

d_2,4

d_2,5

···

s₃

d_3,1

d_3,2

d_3,3

d_3,4

d_3,5

···

s₄

d_4,1

d_4,2

d_4,3

d_4,4

d_4,5

···

s₅

d_5,1

d_5,2

d_5,3

d_5,4

d_5,5

···

·
·
·

We want to construct an element x which is not in this enumeration.

Let's look at an example:

s₁

5···

s₂

3···

s₃

3···

s₄

7···

s₅

···

·
·
·

·
·
··
·
·

1···

The trick : let x_i=1 if d_i,i=0 and x_i=1 else.

Then, x!= s₁. But also, x!= s₂, x!= s₃, ...

Therefore x is indeed never enumerated! Contradiction!

Conclusion: [0,1) is not countable.

Remarque 2 We can deduce that many sets like C, or [5.3,10) are uncountable.

A similar proof can be used to prove that many other sets are uncountable.

This includes the set of all infinite strings, or the set (N) of all subsets of N.

Résumé 2 We have a hierarchy of bigger and bigger sets:

The empty set
Finite sets (the empty set, sets with 1 element, sets with 2 elements, ...)
Denumerable sets: N, Z, Q, ...
Uncountable sets: R, C, ...
...

Problem 3 Is there any set bigger than N and smaller that R? Nobody knows!

1.5 Conclusion

We are now done with basic properties of sets. We have seen enough formalism

to go on to the next section, Combinatorics:

How to count the objects in a finite set ?

Analysis Of Algorithms

Class notes

Combinatorics
Sets / CSM MACS-358 / Nicolas M. Thiéry
Last modified: Fri Jan 19 13:12:17 2007