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. 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:

• a ∈ S: a belongs to S, or a is an element of S, or S contains a;

• a ∉ S: a does not belong to S.

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

(∀x)(x ∈ A) ⇔ (x ∈ 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}

a ∈ S is true, whereas d ∈ 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}

5 ∈ S is true, whereas 3 ∉ S, 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:

∅ := {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 | x ∈ S and x ≠ 0}

S^{+} := {x | x ∈ S and x ≥ 0}

1.2.3. Relationships between sets.

• A is a subset of B (A ⊆ B) iff any element of A is in B:

(∀x)(x ∈ A) ⇒ (x ∈ B)

• A is a proper subset of B (A ⊂ B) iff A is a subset of B but they are not equal :

(∀x)(x ∈ A) ⇒ (x ∈ B) and (∃x)(x ∈ B) ∧ (x ∉ A)

Exercice 1. [?, Exercise 10 p. 117]

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

T := {1, 3, π}, U := {{1, 3, π}, 1}

(1) S ⊆ R?

(2) 1 ∈ R?

(3) 1 ∈ S?

(4) 1 ⊆ U?

(5) {1} ⊆ T?

(6) {1} ⊆ S?

(7) T ⊂ R?

(8) {1} ∈ S?

(9) ∅ ⊆ S?

(10) T ⊆ U?

(11) T ∈ U?

(12) T ∉ R?

(13) T ⊆ R?

(14) S ⊆ {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. What are the elements of the following powersets ?

(1) ℘({1, 2}) =
(2) ℘({1}) =
(3) ℘(∅) =

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

1.2.5. Binary and unary operations.

Définition. • 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:

(1) is +a binary operation on ℕ ?

(2) is +a binary operation on {0, 1, 2}?

(3) is ×a binary operation on {0, 1}

(4) is −a binary operation on ℕ ?

(5) is /a binary operation on ℤ ?

(6) are +,−, and ×binary operations on ℚ ?

(7) are +,−, ×and /binary operations on ℚ^{∗}?

(8) is \sqrt{x} a unary operation on ℝ^{+} ?

(9) are ∧, ∨, →, ′, … operations on {true, false}? on wff ?

A binary operation can be defined by its multiplication table.

Exemple. ({true, false}, ∧)

\begin{matrix}∧ & false & true \\ false & false & false \\ true & false & true \end{matrix}

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

Exemple. ({2, 5, 9}, •)

\begin{matrix}• & 2 & 5 & 9 \\ 2 & 2 & 5 & 2 \\ 5 & 2 & 9 & 5 \\ 9 & 9 & 2 & 5 \end{matrix}

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 1.3.1. 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. 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. The union of A and B is the set:

A ∪ B := {x | x ∈ A or x ∈ B}

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

A ∩ B := {x | x ∈ A and x ∈ B}

Définition. The complement of A is the set:

A′ := {x | x ∉ A}

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

A − B := {x (x ∈ A) and (x ∉ B)}

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

A × B := {(x, y) | x ∈ A, y ∈ B}

Notation 1.3.2. A^{2} = A × A is the set of all ordered pairs of elements of A; A^{n} = A × ⋯ × A is the set of all n-uples of elements of A. Définition. Venn diagrams.

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

(1) A ∪ B =,

(2) A ∩ B =,

(3) A − B =,

(4) A′ = (assuming that the universe of discourse is {1, 2, 3, 4}),

(5) A × B =,

(6) A^{3} =. Exercice 4. Let A be a set of size n, and B a set of size m.

(1) what is the size of A ∪ B:

(2) what is the size of A ∩ B:

(3) what is the size of A′ ?

(4) what is the size of A × B :

1.3.3. Set identities. Proposition. Let A and B be to sets. Then, A ∩ B = B ∩ A (commutative property)

Proof. x ∈ A ∩ B

⇔ (x ∈ A) ∧ (x ∈ B) by definition of ∩ ⇔ (x ∈ B) ∧ (x ∈ A) by commutativity of ∧ ⇔ x ∈ B ∩ A by definition of ∩ □

The commutativity of ∩derives directly from the commutativity of ∧. All identities on logic operators induce identities on set operators. Proposition. Let A, B, and C be two sets. Then, A ∪ B = B ∪ A (commutative property) A ∩ B = B ∩ A (commutative property) (A ∪ B) ∪ C = A ∪ (B ∪ C) = A ∪ B ∪ C (associative property) (A ∩ B) ∩ C = A ∩ (B ∩ C) = A ∩ B ∩ C (associative property) A ∪ A′ = S (complement, S is the universe of discourse) A ∩ A′ = ∅ (complement) (A ∪ B)′ = A′ ∩ B′ (de Morgan’s) (A ∩ B)′ = A′ ∪ B′ (de Morgan’s) A − B = A ∩ 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:

s_{1}, s_{2}, …, s_{k}

Définition. s_{1}, s_{2}, …, s_{k} is called an enumeration of the set S. Exemple. 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 s_{1}, then a second s_{2}, 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. 0, 1, 2, 3, 4, … is an enumeration of the non-negative integers.

1.4.2. Denumerable and Countable sets.

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

s_{1}, s_{2}, …, s_{n}, …

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

Exemple. Are the following sets countable ?

(1) The set of all students in this class.

(2) 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

(3) The set of the even integers (enumeration: ;

(4) The set of the prime integers (enumeration: ;

(5) The set ℤ of the integers (enumeration: );

(6) The set ℕ × ℕ (e.g. all points in the plane with non-negative integer coordinates):

1.4. COUNTABLE AND UNCOUNTABLE SETS 7

(7) Enumeration: (0, 0), (0, 1), (1, 0), (2, 0), (1, 1), (0, 2), (3, 0), …

(8) The set ℤ × ℤ of all points in the plane with integer coordinates:

8 1. SETS (SECTION 3.1 )

Enumeration: (0, 0), (0, 1), (1, 0), (0, −1), (−1, 0), (2, 0), … (9) The set ℚ of rational numbers:

Théorème. The following sets are countable:

(1) Any subset A ⊂ B of a countable set B.

(2) The union A ∪ B of two countable sets A and B is countable.

(3) The union A_{1} ∪ A_{2} ∪ ⋯ ∪ A_{i} of a finite number i of countable sets A_{1}, …, A_{i}.

(4) The union A_{1} ∪ A_{2} ∪ ⋯ ∪ A_{i} ∪ ⋯ of a countable number of countable sets.

(5) The cross product A × B of two countable sets A and B is countable.

(6) The cross product A_{1} × ⋯ × A_{i} of a finite number of countable sets A_{1}, …, A_{i}.

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

□

(1) Enumerate the elements of B, and ignore those that are not in A.

(2) Enumerate the elements of A and B alternatively (a_{1}, b_{1}, a_{2}, b_{2}, …).

(3) Induction, using 2.

(4) 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}).

(5) As for ℕ × ℕ: (a_{1}, b_{1}), (a_{1}, b_{2}), (a_{2}, b_{1}), (a_{1}, b_{3}), (a_{2}, b_{2}), (a_{3}, b_{1}), . . ..

(6) Induction, using 5.

1.4.3. Uncountable sets ?

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

Problem 1.4.1. Are there any uncountable sets ? Théorème. (Cantor) ℝ is uncountable. Lemme. Any real number x ∈ [0, 1) can be written uniquely in decimal form:

x = 0.d_{1}d_{2}d_{3}d_{4} ⋯ 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. 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)

• \frac{1}{3} = 0.333333 ⋯

• π = 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 s_{1}, s_{2}, …, 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:

\begin{aligned}s_{1} & = & 0, & \underline{d_{1,1}} & d_{1,2} & d_{1,3} & d_{1,4} & d_{1,5} & ⋯ \\ s_{2} & = & 0, & d_{2,1} & \underline{d_{2,2}} & d_{2,3} & d_{2,4} & d_{2,5} & ⋯ \\ s_{3} & = & 0, & d_{3,1} & d_{3,2} & \underline{d_{3,3}} & d_{3,4} & d_{3,5} & ⋯ \\ s_{4} & = & 0, & d_{4,1} & d_{4,2} & d_{4,3} & \underline{d_{4,4}} & d_{4,5} & ⋯ \\ s_{5} & = & 0, & d_{5,1} & d_{5,2} & d_{5,3} & d_{5,4} & \underline{d_{5,5}} & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{aligned}

We want to construct an element x which is not in this enumeration. Let’s look at an example:

\begin{aligned}s_{1} & = & 0, & \underline{1} & 5 & 7 & 3 & 5 ⋯ \\ s_{2} & = & 0, & 0 & \underline{0} & 4 & 7 & 3 ⋯ \\ s_{3} & = & 0, & 4 & 5 & \underline{7} & 0 & 3 ⋯ \\ s_{4} & = & 0, & 9 & 7 & 3 & \underline{5} & 7 ⋯ \\ s_{5} & = & 0, & 4 & 3 & 8 & 1 & \underline{0} ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ ... \\ x & = & 0, & 0 & 1 & 0 & 0 & 1 ⋯ \end{aligned}

The trick : let x_{i} = 1 if d_{i,i} = 0 and x_{i} = 1 else. Then, x ≠ s_{1}. But also, x ≠ s_{2}, x ≠ s_{3}, … Therefore x is indeed never enumerated! Contradiction! Conclusion: [0, 1) is not countable. □

Remarque. 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 1.4.2. We have a hierarchy of bigger and bigger sets:

(1) The empty set

(2) Finite sets (the empty set, sets with 1 element, sets with 2 elements, …)

(3) Denumerable sets: ℕ, ℤ, ℚ, …

(4) Uncountable sets: ℝ, ℂ, …

(5) …

Problem 1.4.3. 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 ?