Math Prelims

Lecture 1 Mathematical Preliminaries

A. Banerji

July 21, 2014

(Delhi School of Economics) Introductory Math Econ July 21, 2014 1 / 20

Outline

1 PreliminariesSetsLogicSets and Functions


Preliminaries Sets

Basic Concepts

Take as understood the notion of a set. Usually use upper caseletters for sets and lower case ones for their elements. Notation.a ∈ A, b /∈ A.A ⊆ B if every element of A also belongs to B. A = B if A ⊆ B andB ⊆ A are both true.A ∪ B = {x |x ∈ A or x ∈ B}. The ‘or’ is inclusive of ‘both’.A ∩ B = {x |x ∈ A and x ∈ B}. What if A and B have no commonelements? To retain meaning, we invent the concept of an emptyset, ∅.A ∪ ∅ = A, A ∩ ∅ = ∅. For the latter, note that there’s no element incommon between the sets A and ∅ because the latter does nothave any elements.∅ ⊆ A, for all A. Every element of ∅ belongs to A is vacuouslytrue since ∅ has no elements. This brings us to some logic.


Preliminaries Logic

Logic

Statements or propositions must be either true or false.P ⇒ Q. If the statement P is true, then the statement Q is true.But if P is false, Q may be true or false. For example, on realnumbers, let P = x > 0 and Q = x2 > 0. If P is true, so is Q, butQ may be true even if P is not, e.g. x = −2. We club thesetogether and say P ⇒ Q.Let P = x2 < 0 and Q = x = 5. Then P ⇒ Q is vacuously true.∼ Q ⇒∼ P is the contrapositive of P ⇒ Q.For example: If x3 ≤ 0, then x ≤ 0 is the contrapositive of “ifx > 0, then x3 > 0 ".


Preliminaries Logic

Logic

Claim. A statement and its contrapositive are equivalent.

Proof.Suppose P ⇒ Q is true. Suppose Q is false. Then P must be false, forif not, then via P ⇒ Q, Q would be true, contradicting our assumptionthat Q is false.On the other hand, suppose P ⇒ Q is false. It follows that P is trueand Q is false (for if P is false, then P ⇒ Q is vacuously true). Butthen, ∼ Q ⇒∼ P cannot be true.

DefinitionQ ⇒ P is called the converse of the statement P ⇒ Q.

No relationship between a statement and its converse. e.g., if x > 0then x2 > 0 is true, but its converse is not. On the other hand, if somestatement and its converse are both true, we say P if and only if Q orP ⇔ Q.


Preliminaries Logic

Quantifiers and Negation

2 logical quantifiers: ‘for all’ and ‘there exists’.P : For all a ∈ A, property Π(a) holds.The negation of P (i.e. ∼ P) is the statement: There exists at least onea ∈ A s.t. property Π(a) does not hold.e.g. P : For every x ∈ <, x2 > 0.The negation of P : There exists x ∈ < s.t. x2 ≤ 0.

Q : There exists b ∈ B s.t. property Θ(b) holds.∼ Q : For all b ∈ B, property Θ(b) does not hold.Note. Order of quantifiers matters. e.g. ‘for all x > 0, there existsy > 0 s.t. y2 = x ’, says that every positive real has a positive squareroot. This is not the same as ‘there exists y > 0 s.t. for allx > 0, y2 = x ’, which says some number y is the common square rootof every positive real.


Preliminaries Logic

Logic

Let Π(a,b) be a property defined on elements a and b in sets A and Brespectively. LetP : For every a ∈ A there exists b ∈ B s.t. Π(a,b) holds. Then,∼ P : There exists a ∈ A s.t. for all b ∈ B, Π(a,b) does not hold.

Necessary and Sufficient Conditions

Let P ⇒ Q be true. We say Q is necessary for P. Or P is sufficient forQ. So, if P ⇔ Q, P is necessary and sufficient for Q.


Preliminaries Sets and Functions

Sets

Set Difference: A− B = {x |x ∈ A and x /∈ B}.Also called the complement of B relative to A. More familiar is thenotion of the universal set X , and X − B = Bc .Some set-theoretic ‘laws’Distributive Laws(i) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)(ii) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Proof.(i) Suppose x ∈ LHS. So, x ∈ A and x ∈ (B ∪ C). So x ∈ A and eitherx ∈ B or x ∈ C or both. So, either x ∈ (A ∩ B) or x ∈ (A ∩ C) (or both).Converse?

DeMorgan’s Laws(i) A− (B ∪ C) = (A− B) ∩ (A− C) or more familiarly,(B ∪ C)c = Bc ∩ Cc

(ii) A− (B ∩ C) = (A− B) ∪ (A− C) or (B ∩ C)c = Bc ∪ Cc .(Delhi School of Economics) Introductory Math Econ July 21, 2014 8 / 20


Sets

Arbitrary Unions and IntersectionsLet A be a collection of sets. Then⋃

A∈A = {x |x ∈ A for at least one A ∈ A}⋂A∈A = {x |x ∈ A for every A ∈ A}

Cartesian ProductsWe’ll say (a,b) is an ordered pair if the order of writing these 2objects matters: i.e. if (a,b) and (b,a) are not the same thing. (Thinkof points on the plane). Alternatively, we can derive the notion ofordered pair from the more primitive notion of a set as follows. Define(a,b) = {{a}, {a,b}}. On the right is a set of 2 sets; the first of these isthe singleton that we want to be ‘first’ in the ordered pair. The 2nd isthe set of both objects (obviously, {a,b} = {b,a}, so that alone cannotbe sufficient to define an ordered pair). Thus (b,a) is defined to be theset {{b}, {a,b}}. Let A and B be sets. We then have



Functions

DefinitionThe Cartesian product A× B = {(x , y)|x ∈ A and y ∈ B}.

We can formally define a function using the notion of Cartesianproduct, as follows. Let C,D be 2 sets. A rule of assignment r is asubset of C × D s.t. elements of C appear as first coordinates ofordered pairs belonging to r at most once. The set A of elements of Cappearing as first coordinates in r is called the domain of r . The set ofelements of D comprising 2nd coordinates of r is called the image setof r . Then

DefinitionA function f is a rule of assignment r along with a set B that containsthe image set of r .

A is called the domain of f and the image set of r is called the imageset or range of f .



Functions

We write f : A→ B and think of f as a rule carrying every elementa ∈ A to exactly one element b ∈ B.Examples. 1. f : < → < defined by f (x) = x2,∀x ∈ <.2. Using the notation <2 for the Cartesian product <× < (representingthe plane), let f : <2 → < be defined by f (x1, x2) = x1x2, ∀(x1, x2) ∈ <2.3. g : <2 → <2 defined by g(x1, x2) = (x1x2, x1 + x2),∀(x1, x2) ∈ <2.4. Arbitrary Cartesian Products. We first formally define n-tuples interms of functions. Let X be a set and define the functionx : {1, ...,n} → X . This function is called an n-tuple of elements of X .It’s image at i ∈ {1, ...,n}, x(i), is written as xi . The n-tuple is written as(x1, ..., xn). The order counts. We write X n for the set of all n-tuples ofelements of X . The leading example is <n, n-dimensional Euclideanspace.



Functions

Definition

Let A1, ...An be a collection of sets, and let X =⋃n

1 Ai . The cartesianproduct of this collection of sets, written as A1 × ...× An or Πn

i=1Ai , isthe set of all n-tuples (x1, ..., xn) such that xi ∈ Ai ,∀i ∈ {1, ...,n}.

5. Let X be a set. A sequence or infinite sequence of elements of Xis a function x : Z++ → X . (Z++ is the set of positive integers).Sequences are written by collecting images in order. We writex = (x1, x2, ......). This definition generalizes the notion of infinitesequences of real numbers.



Functions - Images, Preimages

Let f : A→ B.

DefinitionLet A0 ⊆ A. The image of A0 under f , denoted f (A0), is the set{b|b = f (a), for some a ∈ A0}.Let B0 ⊆ B. The preimage of B0 under f , f−1(B0) = {a|f (a) ∈ B0}.

So the image of a set is the collection of images of all its elements, andthe preimage or inverse image of a set is the collection of all elementsin the domain that map into this set.

Example

For the function f (x) = x2, let A0 = [−2,2]. Then, f (A0) = [0,4]. LetB0 = [−2,9]. Then, f−1(B0) = [−3,3].



Functions - Injective, Surjective

Fact. Let f : A→ B, A0,A1 ⊆ A,B0,B1 ⊆ B. Then (i)B0 ⊆ B1 ⇒ f−1(B0) ⊆ f−1(B1).(ii) f−1(B0 ∗ B1) = f−1(B0) ∗ f−1(B1), where ∗ can be ∪,∩,−.i.e., f−1 preserves set inclusion, union, intersection and difference. fonly preserves the first two of these. The 3rd holds with ⊆ and the 4thwith ⊇. To find counterexamples, many-to-one functions (to which wenow move) are helpful. Proofs of the above claims are homework.

Definition

f : A→ B is injective (one-to-one) if [f (a) = f (a′)]⇒ [a = a

′]. It is

surjective (onto) if for every b ∈ B, there exists a ∈ A s.t. b = f (a). Afunction that is both of these is called bijective.

For example, f : < → < defined by f (x) = x2 is many-to-one and notsurjective. If the domain is <+ instead, then f is injective, and further ifthe codomain is <+, then it is bijective.



Functions, Inverse

Fact. Let f : A→ B, A0 ⊆ A, B0 ⊆ B. Then (i)f−1(f (A0)) ⊇ A0; equalityholds if f is injective.(ii) f (f−1(B0)) ⊆ B0; equality holds if f is surjective.You should also show by examples that equality does not in generalhold.Now we use the f−1 to mean something different, namely the inversefunction. If f is bijective, then define a function f−1 : B → A byf−1(b) = a if a is the unique element of A s.t. f (a) = b. Note that f−1 isalso bijective. Indeed, suppose b 6= b

′and f−1(b) = f−1(b

′) = a. Then

f (a) = b and f (a) = b′

which is not possible. So f−1 is injective.Moreover, for every a ∈ A, there is a b ∈ B s.t. b = f (a), since f is afunction. So, there is a b ∈ B s.t. f−1(b) = a. So f−1 is also surjective;hence it is bijective.



Functions

One way to check whether f is bijective uses the following

LemmaLet f : A→ B. If there are 2 functions g,h from B to A s.t.g(f (a)) = a, ∀a ∈ A and f (h(b)) = b∀b ∈ B, then f is bijective andg = h = f−1.

Proof.

f injective. Suppose a 6= a′

and f (a) = f (a′) = b. Then

g(f (a)) = g(b) = g(f (a′)). So g(b) cannot be equal to both a and a

′.

Contradiction. f is also surjective. Indeed, suppose there is a b ∈ Bwith no preimage under f . However, we require f (h(b)) = b. Thisimplies h(b) is a preimage. Contradiction.



Simultaneous Equations

Let f : <n → <m. Fix y ∈ <m. Then the equation f (x) = y represents asystem of m simultaneous equations in n variables. This is clear sincethe equation can be rewritten as

f1(x1, ..., xn) = y1. . . . . . . . . . . . . . . . . .fm(x1, ..., xn) = ym

where y = (y1, ..., ym), x = (x1, ..., xn),f (x) = (f1(x), ..., fm(x)), ∀x ∈ <n, where for each i ∈ {1, ...,m}, thecomponent function fi : <n → <. An x which satisfies f (x) = y is calleda solution to the equation or system of equations. Note that whetherthe system of equations has a solution is the same question aswhether f−1({y}) is a nonempty set. Exercise. f : <2 → <2,f (x1, x2) = (x1x2, x1 + x2). For what values of y in the codomain doesthe equation f (x) = y have a solution?



Simultaneous Equations - contour surfaces

One way to look at a solution: the intersection of (hyper)-surfaces. Forexample, in the exercise above, given y = (y1, y2), the equationsx1x2 = y1 and x1 + x2 = y2. These are 1-dimensional curves in <2, andtheir intersection is the set of solutions. For the more generaln-variable case, fi(x1, ..., xn) = yi describes an (n − 1)-dimensionalsurface, and the solution set is the intersection of the m such surfaces.

DefinitionLet g : <n → < and let y ∈ <. The contour set of g at y ,Cg(y) = {x ∈ <n|g(x) = y}. The upper contour setUg(y) = {x |g(x) ≥ y}. The lower contour set Lg(y) = {x |g(x) ≤ y}.

Observe that Cg(y) = Ug(y) ∩ Lg(y).



Simultaneous Equations in Economics

First order conditions in optimization problems; general equilibrium;Nash equilibrium in some problems. All these can be cast as solutionsto a system of equations F (x) = 0, where F : <n → <n and 0 ∈ <n.General equilbrium is sometimes described as a fixed point of afunction. (i.e., if say f : <n → <n, x is a fixed point if it satisfiesf (x) = x). But x is a fixed point of f if and only if it is a zero ofF (x) ≡ f (x)− x , so the the question is really of finding the zeros of Fi.e. solving F (x) = 0.Continuity of F is an important player in the existence of a solution.(Just as in 1-dimensional case: if f : < → <, is continuous, andf (x1) > 0 > f (x2), then the intermediate value theorem assures asolution x ∈ (x1, x2).) For the more general higher dimensional case, incomputational economics methods like Gauss-Jacobi make use of1-dimensional solutions to the n different equations in an iterative wayto converge to a solution. (see Judd - Numerical Methods inEconomics).



Relations

Recall that we can define a function as a subset of a Cartesian productC × D such that elements of C appear as first coordinates of orderedpairs at most once. Relations are more general in a way.

Definition

A relation � on a set X is a subset of X 2, i.e. �⊆ X × X .

Conventionally, if (x , y) ∈�, we write x � y . As in the case of definingfunctions, the idea of the definition is to not take anything more thanthe meaning of a set to be understood, and to successively definethings in terms of it. (Set -> Cartesian Product -> Relation). But as inthe case of a function, we think of relations in specific ways not directlyrelated to the definition. For example, in consumer theory, if X is theconsumption set and x , y ∈ X , we think of x � y directly as “x ispreferred to y".


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