Mathematics for Economists Chapter 1: Basic DeÖnitions 1. Operations on sets Intersection X \ Y; union X [ Y; complement Z ! Z c ; di§erence X Y = X \ Y c : Distributivity X \ ( Y [ Z ) = ( X \ Y ) [ ( X \ Z ) X [ ( Y \ Z ) = ( X [ Y ) \ ( X [ Z ) ( X \ Y ) c = X c [ Y c ; ( X [ Y ) c = X c \ Y c union, intersection are commutative and associative Cartesian product: X Y is the set of ordered pairs ( x; y ) with x 2 X; y 2 Y: The product X X is not the same as the set of subsets of X with two elements. 2. Functions, mappings f : X ! Y has domain X and values/image in Y; it associates with every x 2 X an element y = f ( x ) in Y: The range of f is the set f ( X ) = f y 2 Y j 9 x 2 X : y = f ( x ) g The graph of f is the subset of X Y : graph ( f ) = f ( x; f ( x )) j x 2 X g It is sometime convenient to identify a function with its graph: then a func- tion is deÖned as a subset of X Y such that every x -section contains exactly one y 2 Y: The set of functions from X into Y is denoted Y X : Example: the set of subsets of X is denoted f 0 ; 1 g X or 2 X : The function f 2 Y X is onto Y if f ( X ) = Y ; it is one-to-one if x 6 = x 0 ) f ( x ) 6 = f ( x 0 ); it is a bijection if it is one to one and onto Y: In the later case, we can speak of the inverse of f; the function f 1 from Y into X deÖned by y = f ( x ) , x = f 1 ( y ) : Note: if f 2 Y X is not a bijection, its inverse as a function from Y to X is not deÖned. However, if f is one-to-one, we may identify it with a function g from X into f ( X ) ; and that function has an inverse g 1 from f ( X ) into X: Hence the need to always specify the domain of a function. Composition of f 2 Y X ; g 2 Z Y : g f ( x ) = g ( f ( x )) : For instance, if id X denotes the identity mapping of X; i.e., id X ( x ) = x for all x; and f 2 Y X is a bijection, we have f f 1 = id Y ; f 1 f = id X : The composition is an associative operation, it is not commutative. If f 2 Y X and g 2 X Y are such that g f = id X ; we cannot deduce g = f 1 ; or f = g 1 : If g f = id X and f g = id Y ; we can. Exercise 1 What can we deduce from g f = id X , or f g = id Y ? 1 Exercise 2 If f 2 Y X and g 2 Z Y are onto, so is g f: If they are one to one, so is g f: A bijection from X into itself is called a permutation . Their set is denoted A ( X ) : The permutations of X form a group for the composition: every f 2 A ( X ) has an inverse f 1 2 A ( X ) : Exercise 3 If X contains three or more distinct elements, construct f; g 2 A ( X ) such that f g 6 = g f: Exercise 4 If X is Önite, j X j = n; then A ( X ) is Önite and A ( X ) = n ! = n ( n 1) ::: 2 : 1 : Exercise 5 A permutation h is called elementary if it exchanges two elements x; y of A and leaves every other element Öxed: h ( x ) = y; h ( y ) = x and h ( z ) = z for all z 6 = x; y: If j X j is Önite, every permutation f 2 A ( X ) can be written as a product of elementary permutations. Exercise 6 If X = f 1 ; 2 ; :::; n g and a permutation f of X is written as f i 1 ; i 2 ; :::; i n g ; where i k = f ( k ) for k = 1 ; :::; n; we can check the parity of f as follows. Denote sign ( z ) = +1 ( resp: 1 ; resp: 0 ) if z is positive ( resp: negative, resp: zero). Then deÖne the signature of a permutation f as ( f ) = 1 k;l n k y 1 g or f x 1 = y 1 ; and x 2 > y 2 g or ... or f x i = y i for i = 1 ; :::; n 1 and x n > y n g It is a linear ordering of R n : Exercise 8 Check that the recursive formula to count the number r n of di§ erent preorderings over a set X with n elements is : r n ( k ) = k ( r n 1 ( k )+ r n 1 ( k 1)) Compute r 2 ; r 3 ; r 4 ; r 5 : 4. Cardinals The binary relation over the set of all sets ( X; Y ) 2 R , def there exists f 2 Y X and onto Y , there exists g 2 X Y and one-to-one on Y is a preordering. Its indi§erence classes are written j X j and R is written j X j j Y j : We call j X j the cardinal of X: Also j X j > j Y j means j X j j Y j and Not j Y j j X j : The equivalence relation \ X and Y have the same cardinality " is character- ized as follows: j X j = j Y j , there is a one-to-one f 2 Y X and a one-to-one g 2 X Y , there is an onto f 2 Y X and an onto g 2 X Y , there is a bijection h from X into Y: Note: the latter equivalence requires a non trivial proof, given in the Appen- dix. If X Y; then j X j j Y j : A set X is Önite i§ for any proper subset X 0 of X , we have j X 0 j < j X j : It is inÖnite i§ for some proper subset X 0 of X; we have j X 0 j = j X j (note that j X 0 j j X j is clear, if X 0 is a subset of X ) : 4 The set N of natural integers is the set of Önite cardinals. It is an inÖnite set. Conventional notation N = f 0 ; 1 ; 2 ; :::; n; ::: g : The preordering of Önite cardinals is called the natural order of N : Note that j N j is the smallest inÖnite cardinal. A set with cardinality j N j is called enumerable Examples : Q , the set of rational numbers, is enumerable. R ; the set of real numbers, is not: j R j > j N j : (this important fact is an easy consequence of the decimal development of real numbers: see Chapter 2). N = N ::: N | {z } k tim es is enumerable: N k = j N j : Continuum hypothesis: there is no cardinal between j R j and j N j : Exercise 9 R k = j R j Exercise 10 If X; Y are Önite: j X [ Y j = j X j + j Y j j X \ Y j and Y X = j Y j j X j Exercise 11 Show the following properties for intervals of real numbers: j [0 ; 1] j = j [0 ; 1[ j = j ]0 ; 1[ j Exercise 12 For any set, Önite or not, 2 X > j X j Exercise 13 The lexicographic ordering of R 2 cannot be represented by a utility function. Suppose such a function u ( x 1 ; x 2 ) exists, then we can pick for all x 1 2 R a rational number r ( x 1 ) such that u ( x 1 ; 2) > r ( x 1 ) > u ( x 1 ; 1) : Deduce a contradiction. Appendix: Two proofs on Cardinals 1. The relation j X j j Y j is complete. Given two sets X; Y; consider the set Z of triples z = ( A; B; f ) where A X; B Y; and f is a bijection from A into B The binary relation % on Z : ( A; B; f ) % ( A 0 ; B 0 ; f 0 ) , def A 0 A; B 0 B; f = f 0 on A 0 ; is clearly transitive (but not complete). Given any subset Z 0 of Z on which % is complete, we construct an upper bound e z of Z 0 (i.e., e z % z for all z 2 Z 0 ) as follows: e z = ( e A; e B; e f ) where e A = [ A; e B = [ B and e f = f on A for all z = ( A; B; f ) in Z 0 : An important result in set theory, known as Zorní s lemma, says that the relation % must have a maximal element z in Z; namely such that z z holds for no z 2 Z: Let z = ( A ; B ; f ) : If X A and Y B are both non empty, we can ìaugmentî z by adding a 2 X A to A ; b 2 Y B to B and setting f ( a ) = b: This contradicts the maximality of z : Therefore A = X or B = Y or both. The former implies j X j j Y j and the latter j Y j j X j : 5 2. If j X j = j Y j ; there exist a bijection from X into Y: a. We prove an auxiliary result Örst. Let A; A 1 ; A 2 be 3 nested sets, A 2 A 1 A; and f a bijection from A into A 2 : Then there exists a bijection g from A into A 1 : We construct a nested sequence A 2 A 3 A 4 ::: as follows: A 3 = f ( A 1 ) ( ) A 3 f ( A ) = A 2 ) A 4 = f ( A 2 ) ( ) A 4 f ( A 1 ) = A 3 ) A 5 = f ( A 3 ) ( ) A 5 f ( A 2 ) = A 4 ) etc. Note that f ( A A 1 ) = A 2 A 3 (because f is a bijection) and similarly f ( A 2 A 3 ) = A 4 A 5 (for the same reason) f ( A 4 A 5 ) = A 6 A 7 etc. DeÖne g 2 A A g = f on A A 1 ; A 2 A 3 ; A 4 A 5 ; ::: g = identity on A 1 A 2 ; A 3 A 4 ; A 5 A 6 ; ::: Write B = ( A A 1 ) [ ( A 2 A 3 ) [ ( A 4 A 5 ) [ ::: C = ( A 1 A 2 ) [ ( A 3 A 4 ) [ ( A 5 A 6 ) [ ::: and note that f ( B ) B ; id ( C ) = C: Therefore g is one-to-one on A: More- over f ( B ) [ C = A 1 ; therefore g is a bijection from A into A 1 : b. We Öx X; Y such that j X j j Y j and j Y j j X j : Thus there are two one- to-one mappings f; g; where f is from X into Y; and g from Y into X: DeÖne A = X; A 1 = g ( Y ) ; A 2 = ( g f )( X ) : Note that g f is a bijection from A into A 2 ; and moreover A 2 A 1 A: By the auxiliary result, there is a bijection h from A into A 1 : Thus h 1 g is the desired bijection from Y into X: 6 Chapter 2: Real Analysis 1. Sequences Z = f :::; n; :::; 2 ; 1 ; 0 ; 1 ; 2 ; :::; n; ::: g is the set of relative integers. Q = f p q ; p; q 2 Z g is the set of rational numbers. R =] 1 ; + 1 [ is the set of real numbers R + = [0 ; + 1 [ is the set of non negative real numbers R ++ =]0 ; + 1 [ is the set of positive real numbers Integer part: for all x 2 R ; this is the unique relative integer p; p 2 Z ; such that p x < p + 1; it is written b x c R is archimedian: 8 b 2 R 9 p 2 N p > a Exercise 14 8 a > 0 ; a 2 R ; 8 b 2 R 9 p 2 N p:a > b Exercise 15 8 " > 0 ; " 2 R ; 9 2 Q : 0 < < " A sequence of real numbers is an element of R N ; and the conventional no- tation is ( x n ) n 2 N or ( x 0 ; x 1 ; :::; x n ; ::: ) or ( x 1 ; :::; x n ; ::: ) : Note: a sequence in an arbitrary set X is an element of X N : The sequence ( x n ) is bounded if there is a 2 N such that j x n j a for all n: The sequence ( x n ) of real numbers converges to x if we have 8 " > 0 9 m 2 N 8 n m j x n x j " a property that we write lim n x n = x Note: in the deÖnition above, we can limit the choice of " to the numbers 1 p ; p = 1 ; 2 ::: Thus, for instance, lim n !1 x n = x is equivalent to 8 p 2 N 9 m 2 N 8 n m j x n x j 1 p Decimal development of a real number. Given x 2 R ; we construct inductively a sequence ( a n ) of integers such that 0 a n 9 for all n b 10 x c = 10 b x c + a 1 b 10 2 x c = 10 2 b x c + 10 a 1 + a 2 ... b 10 n x c = 10 n b x c + 10 n 1 a 1 + 10 n 2 a 2 + ::: 10 a n 1 + a n i.e., for all n the decimal development of the integer b 10 n x c 10 n b x c is a 1 ; a 2 ; :::; a n : The sequence x n = b x c + n P k =1 10 k a k converges to x Conversely if the sequence ( a n ) of integers between 0 on 9 is such that: lim f n b x c + n P k =1 10 k a k g = x then this sequence is the decimal development of x; unless a n = 9 for all n large enough. A number is rational if andf only if its decimal development is periodic. Exercise 16 Prove the two statements above. 7 Two important applications of the decimal development : Between any two distinct real numbers, there is an inÖnityof rational num- bers and an inÖnity of irrational numbers. j R j > j N j : Operations on limits: lim n x n + lim n y n = lim n ( x n + y n ) (lim n x n ) (lim n y n ) = lim n ( x n y n ) f lim n x n 6 = 0 g ) lim n ( 1 x n ) = 1 lim n x n where we assume that the left-hand limits exist, and deduce that the right- hand limit exists, too. f ( x n ) bounded and lim n y n = 0 g ) lim n x n y n = 0 Exercise 17 The sum (or product) of two non convergent sequences may be convergent. Exercise 18 The product of a bounded sequence and a convergent (resp. a non convergent) sequence, may or may not be convergent. Cauchy sequences: A Cauchy sequence is a sequence (x n ) such that : 8 n x n y n ) lim n x n lim n y n Cauchy sequences: A Cauchy sequence is a sequence (x n ) such that : 8 " > 0 9 m 2 N 8 n; n 0 m j x n x n 0 j " Fundamental result: a sequence of real numbers is convergent if and only if it is a Cauchy sequence. Note that the result is in a way tautological, because the set of real numbers is deÖned as the set of Cauchy sequences in Q , up to a quotient operation. In Q , a Cauchy sequence may not be convergent. Example: a non periodic decimal development. Exercise 19 Prove this claim. Upper and lower bounds: the subset X of R is bounded above (resp. below ) if there exists an integer a such that 8 x 2 X; x a (resp. x a ) : In this case a is called an upper bound of X (resp. a lower bound ). A key application of the fundamental result : every set X bounded above has a smallest upperbound denoted sup( X ) and called the supremum of X . Every set X bounded below has a largest lowerbound inf( X ) called the inÖmum of X 8 Exercise 20 The supremum a = sup( X ) is characterized by the two following properties: a is an upper bound of X 8 " > 0 9 x 2 X : x > a " (a similar statement holds for the inÖmum). The sequence ( x n ) is called non-decreasing (resp. non increasing) if x n x n +1 for all n ( resp. x n x n +1 ) : A non decreasing (resp. non increasing) sequence that is bounded above (resp. below) converges to the supremum (resp. inÖmum) of its values. If I n is a nested sequence of closed intervals I n = [ a n ; b n ] ; i.e., I n +1 I n for all n; then the intersection \ n I n is non empty. If, moreover, lim n ( b n a n ) = 0 ; then \ n I n contains exactly one real number. The lim inf and lim sup of a bounded sequence ( x n ) are deÖned as follows: lim sup( x n ) = inf f sup m n x m j n = 1 ; 2 ; ::: g = lim n ( sup m n x m ) lim inf( x n ) = sup f inf m n x m j n = 1 ; 2 ; ::: g = lim n ( inf m n x m ) Fact: lim inf( x n ) lim sup( x n ) with equality if and only if ( x n ) converges, in which case its limit is the common value. Exercise 21 The lim sup and lim inf operators do not commute with addition, or with multiplication. Exercise 22 Prove : lim sup( x n + y n ) lim sup x n + lim sup y n If x n ; y n are positive prove: lim sup( x n y n ) (lim sup x n ) (lim sup y n ) Similar properties hold for lim inf : What is the composition of lim sup / lim inf with the inverse operator: x n ! 1 x n ? Limit points of a sequence ( x n ) : a number y such that 8 " > 0 8 m 2 N 9 n m j x n y j ": A subsequence of the sequence ( x n ) 2 R N is a sequence ( x ' ( n ) ) = ( x ' (0) ; x ' (1) ; :::; x ' ( n ) ; ::: ) where ' is a strictly increasing function of N into itself. Every subsequence of a converging sequence converges to the same limit. The number y is a limit point of the sequence ( x n ) if and only if there exists a subsequence ( x ' ( n ) ) of ( x n ) converging to y: If ( x n ) is a bounded sequence, lim n sup x n and lim n inf x n are, respectively, its largest and its smallest limit point. Corollary: a bounded sequence is convergent if and only if it has a unique limit point. Corollary: every bounded sequence has at least one limit point. Note: The set of limit points of a sequence ( x n ) can be any enumerable subset of R : Given a Önite set f 0 ; :::; K 1 g the sequence x n = k if k is the remainder of ( n : K ) has the limit points 0 ; :::; K 1 and only those. 9 Exercise 23 Given an arbitrary sequence ( n ) of real numbers, construct a sequence ( x n ) of which the limit points are exactly the values of the sequence ( n ) and its limit points. 2. Continuity Interval: an interval of R is a subset I such that 8 x; y = f x 2 I and y 2 I g ) [ x; y ] I There are nine types of interval depending upon their endpoints being real numbers or 1 ; and on whether an endpoint belongs to I or not. Limit of a function over a subset: Let A be a subset of R and a be in the closure A of A (that is, a is the limit of at least one sequence in A ), we say that f 2 R A converges to b 2 R when x converges to a in A (or simply, the limit of f in A at a is b ) and we write lim x ! a x 2 A f ( x ) = b; the following property: 8 " > 0 9 > 0 8 x 2 A : j x a j ) j f ( x ) b j " We say that the limit of f in A at a is + 1 if we have: 8 n 2 N 9 > 0 8 x 2 A : j x a j ) f ( x ) n We say that the limit of f in A at + 1 is b if we have: 8 " > 0 9 n 2 N 8 x 2 A x n ) j f ( x ) b j " Exercise 24 DeÖne similarly lim x ! a x 2 A f ( x ) = 1 ; lim x !1 x 2 A f ( x ) = + 1 ; etc. Equivalent formulation: for any sequence ( x n ) in A : lim n x n = a ) lim n f ( x n ) = b Examples: lim x ! 0 x> 0 Logx = 1 ; lim x ! 2 x< 2 b x c = 1; lim x ! 2 x> 2 b x c = 2 If f is monotone (non increasing or non decreasing) then f has a right-limit and a left-limit (in A ) everywhere: lim x ! a xa x 2 A f ( x ) exist for all a 2 A (in the above statement, the limit may be 1 ; and a may be replaced by 1 ) : Continuity of a function f 2 R A where A is a bounded interval: [ ; ] ; ] ; ] ; [ ; [ ; ] ; [ : If a is interior to A : f is continuous at a def , lim x ! a f ( x ) = f ( a ) , lim n f ( x n ) = f ( a ) for every ( x n ) s.t. lim n x n = a if a = 2 A : f is continuous at a def , lim x ! x> f ( x ) = f ( a ) , lim n f ( x n ) = f ( a ) for every ( x n ) s.t. x n and lim n x n = a 10 Exercise 25 If a = 2 A write the similar deÖnition of lim n x n = a: Right continuity, left continuity Same premises as above. If a is interior to A : f is right-continuous at a def , lim x ! x> f ( x ) = f ( a ) f is left-continuous at a def , lim x ! x< f ( x ) = f ( a ) If a = : f is right-continuous at a , f is continuous at a left continuity is not deÖned. Operations on continuous functions: the sum, the product, the quotient respect the property of continuity at a given point, that of right continuity, and that of left continuity. The composition of functions respects the continuity property: assume f 2 R A ; g 2 R B ; f ( A ) B; then { f continuous at a; and g continuous at b = f ( a ) g ) g f continuous at a: Exercise 26 The composition of right-continuous functions may not be right- continuous. The supremum (resp. inÖmum) of Önitely many functions respects the prop- erties of continuity, right-continuity, left-continuity. f k 2 R A ; k = 1 ; :::; K ; g ( x ) = sup k =1 ;:::;K f k ( x ) f for all k; f k is continuous at a g ) f g is continuous at a g Examples: f ( x ) = b x c ; A = R ; is right-continuous everywhere and continuous for non integer values of x: f ( x ) = sin x if x 6 = 0; f (0) = 0; A = R is neither right nor left-continuous at 0; is continuous everywhere else. f ( x ) = 0 if x 2 Q ; f ( x ) = 1 if x 2 RQ ; is neither right nor left-continuous anywhere. If f is monotone on R ; it is continuous at all points, with the exception of at most an enumerable set of points. Exercise 27 Construct a monotone function on R that is discontinuous at every rational number and continuous at every irrational number. What about a function discontinuous at every irrational number and continuous at every rational number ? A closed bounded interval of R is a set [ a; b ] = f x j a x b g where a; b 2 R : If f is continuous on A (i.e., continuous at every point of A ) ; and [ a; b ] A; then f ([ a; b ]) is a closed bounded interval. This crucial observation has two important corollaries. 11 Intermediate value theorem: if f is continuous on the closed, bounded inter- val [ a; b ] ; f takes all the values between f ( a ) and f ( b ) : 8 x 2 ] f ( a ) ; f ( b )[ 9 c 2 ] a; b [ f ( c ) = x Maximum theorem: if f is continuous on the closed, bounded interval [ a; b ] ; f reaches its maximum and its minimum: 9 c; d 2 [ a; b ] : f ( c ) = sup a x b f ( x ); f ( d ) = inf a x b f ( x ) Variant of the IVT: if f is continuous on the interval I; then f ( I ) is an interval of R : Exercise 28 We may have I =] a; b [ and f ( I ) = [ c; d ] ; or I = [ a; + 1 [ and f ( I ) =] c; d [ ; etc. An application of the IVT: a polynomial of odd degree has at least one real root. Homeomorphism Let I be an interval and f 2 R I : The four following properties are equivalent: i) f is a bijection from I into f ( I ) , f is continuous on I and f 1 is continuous on f ( I ) ; ii) f is continuous and one-to-one on I; iii) f is continuous and strictly monotone (strictly increasing or strictly de- creasing), iv) f is strictly monotone and f ( I ) is an interval. Uniform Continuity: f 2 R A is uniformly continuous on A if we have: 8 " > 0 9 > 0 8 x; y 2 A : j x y j ) j f ( x ) f ( y ) j " Let I be a closed bounded interval and f 2 R I Then f f is continuous on I g , f f is uniformly continuous on I g : On the other hand, if A is not bounded, uniform continuity is a more de- manding property than continuity: f ( x ) = x 2 is c: but not u:c: on R + f ( x ) = sin x is u:c: on R Exercise 29 Show that f ( x ) = 1 x is uniformly continuous on [1 ; + 1 [ : Show that f ( x ) = sin 1 x is not uniformly continuous on ]0,1]. Is f ( x ) = sin( x 2 ) u.c. on R ? What about f ( x ) = sin( x 2 ) x for x 6 = 0 ; f (0) = 0? 3. Derivatives Given f 2 R A ; where A is an interval, and a an interior point of A; we say that f is di§ erentiable (or derivable ) at a if lim x ! a x 6 = a f ( x ) f ( a ) x a exists. This limit is called the derivative of f at a and denoted f 0 ( a ) : 12 Right-derivative of f at a : it is the limit lim x ! a x>a f ( x ) f ( a ) x a = f 0 + ( a ) ; when this limit exists Left-derivative of f at a : it is the limit lim x ! a x From the above discussion we get: A 2 I ( n ) i§ for some B 2 M ( n ) : AB = BA = I In this case the matrix B is unique and denoted B = A 1 : A matrix A 2 M ( n ) is invertible i§ it is onto ( A R n = R n ) , and i§ it is one-to-one ( Ax = 0 ) x = 0) : Computation rules on inverse matrices: A 2 I ( n ) ) A 1 2 I ( n ) AB 2 I ( n ) ) A; B 2 I ( n ) ( :A ) 1 = 1 A 1 when 2 R f 0 g ; ( A 1 ) 1 = A ; ( AB ) 1 = B 1 A 1 Determinants. The transposed of the m n matrix A = [ a ij ] is the n m matrix t A = [ a ji ] : We can write any n n matrix as the list of its column vectors A = [ a 1 ; ::::a n ] ; all in R n : Thus t A = [ b 1 ; ::; b n ] where b i is the i th row vector of A: Denote S n = A ( f 1 ; :::; n g ) the set of permutations of f 1 ; :::; n g and recall from Chapter 1

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