lecture 1 preliminaries - cs.e.ecust.edu.cn

Lecture 1 Preliminaries

Yajing CHEN

Department of Economics, ECUST

2021 Autumn (Week 3)

Contents

1 Notations

2 LogicNecessary and sufficient conditionsMethods of proof

3 Elementary set theory

4 Relations and functionsRelationsFunctions

2 / 32

Contents

1 Notations




Notations 3 / 32

Notations

∀: for all∃: there exits . . .

⇒: A implies B

⇔: A is equivalent to B; A holds if and only if B holds

∈: be the element of A

⊆: set A is included in B

⊂ or ⊊: “A is the proper subset of B”

¬ or ∼: not

∧: conjunction∨: disjunction≡: triple bar

Exercise: Please tell the difference between ax ≡ bx and ax = bx .

Notations 4 / 32

Numbers

N = {1, 2, 3, 4, . . .}: positive integers

Z = {. . . ,−3,−2,−1, 0, 1, 2, 3, . . .}: integersQ ≡ { a

b : a, b ∈ Z; b ̸= 0}: rational numbers

▶√2 is not a rational number. (We will come back to this later.)

Irrational numbers cannot be written as ratios or quotients of integers.

R := {x | −∞ < x < +∞}: real numbers

C: complex numbers a+ ib with a, b ∈ R and i =√−1

Exercise: Try to identify the logical relationship among N,Z,Q,R,C.

Notations 5 / 32

Contents

1 Notations




Logic 6 / 32

Logic: Necessity and Sufficiency

A is necessary for B Whenever B is true, A must also be true.

A is true if B is true.

A is implied by B (A ⇐ B).

A is sufficient for B Whenever A is true, B must also be true.

A is true only if B is true.

A implies B (A ⇒ B).

A is necessary and sufficient for B Whenever A ⇐ B and A ⇒ B.

A is true if and only if (often written as iff) B is true.

A and B are equivalent (A ⇔ B).

Logic Necessary and sufficient conditions 7 / 32

Logic: Necessity and Sufficiency

Illustration of logic:

1 A = /x is an integer less than 4 0;B = /x is an integer less than 10 0

2 A = /y is a number which is multiple of 40;B = /y is an even number 0

3 A = /X is yellow0;B = /X is a lemon 0

4 A = /A person is a father0;B = /A person is a male 0

Logic Necessary and sufficient conditions 8 / 32

Methods of proof

Mathematical theorems usually have the form of implication (⇒) oran equivalence (⇔), where one or more statements are alleged to berelated in particular ways.

Suppose we have a theorem, say /A ⇒ B0. Here A is theassumption and B is the conclusion.

To prove a theorem is to establish the validity of its conclusion giventhe truth of its assumption.

Several methods can be used to do that.

Logic Methods of proof 9 / 32

Methods of proof

How to prove a theorem, say /A ⇒ B0?

Direct Proof (constructive proof)Assume that A is true, deduce various consequences of that, and usethem to show that B must also hold.eg. A ⇒ A1 ⇒ A2 ⇒ . . . ⇒ An−1 ⇒ An ⇒ B

Indirect Proof▶ Contrapositive Proof

Assume that B does not hold, then show that A cannot hold. (Use thelogical equivalence of ¬B ⇒ ¬A and A ⇒ B.)

▶ Proof by ContradictionAssume that A is true, assume that B is not true, and attempt toderive a logical contradiction. (This approach relies on the fact that ifA ⇒ ¬B is false, then A ⇒ B must be true.)

Proof by Mathematical Induction


Direct proof (constructive proof)

Example: Consider the propositions

A : 3x − x2 ≥ 0

B : x ≥ 0

Let us prove the implication A ⇒ B by constructive proof.

Proof.

Assume A, that is, suppose that 3x − x2 ≥ 0.Then, 3x ≥ x2 ≥ 0.Therefore, x ≥ 0. (By transitivity)


Contrapositive proof

Consider a statement A, write ¬A for the statement ”it is not truethat A holds”. ¬A is also called the negation of A.

Consider a proposition P of the form A ⇒ B: if hypothesis A holds,conclusion B holds. The converse of the proposition P is thestatement B ⇒ A, which reverses the hypothesis and conclusion of P.

Consider a proposition P of the form A ⇒ B: if hypothesis A holds,conclusion B holds. The inverse of the proposition P is the statement¬A ⇒ ¬B.Consider a proposition P of the form A ⇒ B: if hypothesis A holds,conclusion B holds. The contrapositive of the proposition P is thestatement ¬B ⇒ ¬A.The contrapositive is equivalent to the original statement.

Exercise: What is the converse of the inverse of an implication? What isthe inverse of the contrapositive? What is the inverse of the inverse of animplication?


Proof by contradiction

Theorem 2.1√2 is not a rational number.

Proof.√2 is either rational or irrational

Suppose√2 is rational.√

2 = p/q, where p and q are not both even. (3)2 = p2/q2 and then 2q2 = p2.p2 is even⇒ p is even. (1)p = 2m for some integer m.p.p = 2m.2m and p2 = 2.(2m2).Then, 2.q2 = 2.(2m2), and subsequently q2 = 2m2.q2 is even ⇒ q is even. (2)(1) and (2) mean p and q are both even, which contradicts ourassumption (3).Therefore,

√2 is irrational.


Proof by mathematical induction

The simplest and most common form of mathematical inductioninfers that a statement involving a natural number n (that is, aninteger n ≥ 0 or 1) holds for all values of n.The proof consists of two steps:

▶ The initial or base case: prove that the statement holds for 0, or 1.▶ The induction step, inductive step, or step case: prove that for every n,

if the statement holds for n, then it holds for n + 1.

The hypothesis in the inductive step, that the statement holds for aparticular n, is called the induction hypothesis or inductive hypothesis.

Exercise: Prove thatk∑

n=1

n =k(k + 1)

2

.


Methods of proof

How to prove a theorem, say /A ⇔ B0?

If /A ⇔ B0, we must give a proof in both directions.

That is, both /A ⇒ B0and /B ⇒ A0must be established beforea complete proof of the assertion has been achieved.

Remark:

Proof by example is no proof.▶ Suppose the following two statements are given:

1 x is a Chinese;2 x has black hair.

Citing a hundred examples can never prove that a certain propertyalways holds, citing one counterexample can disprove that propertyalways holds.


Contents

1 Notations




Elementary set theory 16 / 32

Elementary set theory

How to describe sets

A set is any collection of elements, e.g., S = {2, 4, 6, 8} orS = {x |x is an even integer greater than 0 and less than 10}.To denote inclusion in a set, we use the symbol ∈, e.g., 4 ∈ S .

A set is empty (denoted ∅) if it contains no element at all.

The number of elements in a set S, its cardinality, is denoted |S |.Relationship between sets

A set S is a subset of another set T if every element of S is also anelement of T.

▶ S is contained in T (or, T contains S), and write S ⊂ T .▶ That is, if S ⊂ T , then x ∈ S ⇒ x ∈ T .▶ The set S is a proper subset of T, if S ⊂ T and S ̸= T▶ Empty set ∅ is a subset of every set.



Relationship between sets

Two sets are equal sets (S = T) if they each contain the sameelements.

The set difference, S\T or S − T , is a set of elements in the set Sthat are not elements of T.

The complement of a set S denoted by Sc in a universal set U is theset of all elements in U that are not in S , i.e., U\S(Sc = {x ∈ U|x /∈ S}).The symmetric difference S △ T = (S\T ) ∪ (T\S).

Exercise: If S = T , then S △ T =?


Elementary set theoryThe basic operations on sets are union and intersection.

Union of S and T: S ∪ T := {x |x ∈ S or x ∈ T}Intersection of S and T: S ∩ T := {x |x ∈ S and x ∈ T}

Figure: Venn diagrams

Source: Figure A1.1 (see Jehle and Reny, Advanced Microeconomic Theory, 3rd, pp.498)



Properties of unions and intersections of sets. Let A,B, and C bethree subsets of X . Then the following properties hold:(i) Commutative law: A ∪ B = B ∪ A and A ∩ B = B ∩ A.(ii) Associative law: (A ∪ B) ∪ C = A ∪ (B ∪ C ) = A ∪ B ∪ C and

(A ∩ B) ∩ C = A ∩ (B ∩ C ) = A ∩ B ∩ C .(iii) Distributive law: (A ∪ B) ∩ C = (A ∩ C ) ∪ (B ∩ C ) and

(A ∩ B) ∪ C = (A ∪ C ) ∩ (B ∪ C ).

The collection of all subsets of a set A is also a set, called the powerset of A and denoted P(A) or 2A

Exercise: Let A = {x , y , z}. Then, P(A) =? |P(A)| =?



The (Cartesian) product of two sets S and T, denoted S × T , is theset of (all) ordered pairs in the form (s, t).

▶ First element in the pair is a member of S and the second is T.▶ S × T := {(s, t)|s ∈ S , t ∈ T}

The set of all real numbers is denoted R, and is defined by

▶ R := {x | −∞ < x < +∞}.If we form the set product of real numbers R× R, we get thetwo-dimensional Euclidean space denoted R2 (Cartesian plane).



More generally, any n-tuple, or vector, is just an n-dimensionalordered tuple and can be thought of as a “point” in n-dimensionalEuclidean space denoted by Rn is defined as the set product

▶ Rn :≡ R× R× . . .× R︸︷︷︸n times

≡ {(x1, x2, . . . , xn)|xi ∈ R, i = 1, 2, . . . , n}

We often restrict our attention to a subset of Rn, called thenonnegative orthant and denoted Rn

+ where

▶ Rn+ := {(x1, x2, . . . , xn)|xi ≥ 0, i = 1, 2, . . . , n} ⊂ Rn

We sometimes talk about the strictly /positive orthant0 of Rn,denoted Rn

++ where

▶ Rn++ := {(x1, x2, . . . , xn)|xi > 0, i = 1, 2, . . . , n} ⊂ Rn

+ ⊂ Rn

For any two vectors x and y in Rn, we say that▶ x ≥ y iff xi ≥ yi ,∀i ∈ 1, 2, . . . , n▶ x ≫ y iff xi > yi ,∀i ∈ 1, 2, . . . , n


Contents

1 Notations




Relations and functions 23 / 32

Relations

Recall that the (Cartesian) product of two sets X and Y , denoted X × Y ,is the set of (all) ordered pairs in the form (x , y).

Definition 4.1

A (binary) relation from X to Y is a subset R of X × Y .

When a binary relation is a subset of the product of one set X withitself, i.e., R ⊂ X × X , we say that it is a relation on the set X.

If (x , y) ∈ R,we often write xRy or y ∈ R(x) and say that y is aimage of x .

The domain of R: DR = {x ∈ X |R(x) ̸= ∅}The range of R: RR = {y ∈ Y |R−1(y) ̸= ∅}

Exercise: Let X = {1, 2, 3}. Illustrate the relation ≥ on X .

Relations and functions Relations 24 / 32

Relations

A relation R on X is reflective if xRx for all x ∈ X .

A relation R on X is complete if xRy or yRx for all x , y ∈ X .

A relation R on X is transitive if xRy and yRz implies xRz for allx , y , z ∈ X .

A relation R on X is symmetric if xRy implies yRx .

A relation R on X is anti-symmetric if xRy and yRx implies x = y .

Order relations:

A binary relation ≥ defined in a set X is a preorder or quasiorder if itis reflexive and transitive.

A binary relation ≥ defined in a set X is a partial order if it isreflexive, transitive, and antisymmetric.

A binary relation ≥ defined in a set X is a total order if it iscomplete, transitive, and antisymmetric.

Relations and functions Relations 25 / 32

Functions

Definition 4.2

A function from one set D to another set R is a relation that associateseach element of D with a single, unique element of R.

Function f is mapping from one set D to another set R in which D iscalled the domain and R is the range.

We write such function as f : D → R .

The image of f is that set of points in the range into which somepoint in the domain is mapped, i.e.,I ≡ {y |y = f (x) for some x ∈ D}.The inverse image of a set of points S ⊂ I is defined asf −1(S) ≡ {x |x ∈ D, f (x) ∈ S}.The graph of function f is the set of ordered pairs:

G := {(x , y)|x ∈ D, y = f (x)}

Relations and functions Functions 26 / 32

Figure: Functions and Non-functions

Source: Figure A1.8 (see Jehle and Reny, Advanced Microeconomic Theory, 3rd, pp.504)


Figure: Domain, range, and image

(a)f : R→ R, y = sinx(b)f : [0, 1] → [0, 1], y = 1

2xSource: Figure A1.8 (see Jehle and Reny, Advanced Microeconomic Theory, 3rd, pp.505)


Several Mappings

If every point in range is assigned to at most a single point in thedomain, the function is called one-to-one. That is, f : D → R is aone-to-one function (or injection) if f (x) = f (y) implies that x = y .

If every point in the range is mapped into by some point in thedomain, the function is said to be onto. That is, f : D → R is anonto function (or surjection) if for every y ∈ R there is an x ∈ D suchthat f (x) = y .

If the function f : D → R is a one-to-one and onto function (orbijection, or one-to-one correspondence) if for every y ∈ R there is aunique x ∈ D such that f (x) = y .


Correspondences and sequences

Definition 4.3

A function from one set D to another set R is a relation that associateseach element of D with a single, unique element of R. If more than onepoint is assigned, then such a relation is called a correspondence. Asequence can be defined as a function whose domain is either the set ofthe natural numbers (for infinite sequences), or the set of the first nnatural numbers (for a sequence of finite length n).

Exercise: Show the logical relationship among relations, functions,correspondences, and sequences.


Problem sets I

1 Show that f (x) = x2 is not a one-to-one mapping.

2 Is the function f : R2 → R defined by f (x1, x2) = x1 + x2 surjective?Why?

3 Show the logical relationship between B ∈ P(A) and B ⊂ A.

4 |P(∅)| =?

Problem sets II

5 A binary relation ≥ defined in a set X is an equivalence relation if itis reflexive, transitive, and symmetric. Let X = {a, b, c , d} andR = {(a, a), (a, b), (b, a), (b, b), (c , c), (c , d), (d , c), (d , d)}.Is R an equivalence relation defined on X?

6 Prove that A(n) : 2n > n by mathematical induction.

7 Let A = {0, 1, 2}. Write down the < relation defined on A. Thisrelation is a subset of A× A.

8 Judge whether the following relations defined on R satisfy theproperties defined above.

1 ≥2 >3 =4 ̸=

lecture 1 preliminaries - cs.e.ecust.edu.cn

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