Mathematical Foundations -1- Sets and Sequences. Sets and Sequences

Transcription

1 Mathematical Foudatios -1- Sets ad Sequeces Sets ad Sequeces Methods of proof 2 Sets ad vectors 13 Plaes ad hyperplaes 18 Liearly idepedet vectors, vector spaces 2 Covex combiatios of vectors 21 eighborhoods, boudary poit ad iterior poit of a set 24 Closed ad ope sets, covex sets, compact sets 28 Sequeces ad coverget sub-sequeces 33 Readig: EM Appedix A.1, B.1 Joh Riley October 8, 213

2 Mathematical Foudatios -2- Sets ad Sequeces Methods of proof (i) (ii) (iii) (iv) direct proof, proof by iductio, proof of the cotrapositive, proof by cotradictio Joh Riley October 8, 213

3 Mathematical Foudatios -3- Sets ad Sequeces Direct proof 1 Propositio: Let be the PV of 1 dollar a period ito the future. The the preset value of a 1 r ivestmet yieldig a costat paymet of 1 dollar after each of periods is V [1... ]. 1 The bracketed expressio is a example of a geometric sum 2 1 S 1 a a... a. The well kow formula for this sum is S 1 a 1 a, a1. Proof: as a a... a Hece, 1 as (1 a a... a ) a. But the term i the paretheses is S. Therefore 1 as S a. As log as a 1, we ca solve for S rearrage to obtai S Q.E.D. 1 a. 1 a Joh Riley October 8, 213

4 Mathematical Foudatios -4- Sets ad Sequeces Applicatio: Prisoer s Dilemma Each firm chooses either a High or Low price. Payoffs are as show (row player idicated first) Both firm follow the strategy of settig a high price I the first period. However if cheated a firm will follow the motto of my Graham acestors. Never forget a fried. Never forgive a eemy. That is they will evermore play low. Suppose firm 2 cheats i the first period. Its payoff stream the has a PV of 2 2 PV{5,1,1,1,...} 5 (...) 5 (1...)(1) 5 (1). 1 If firm 2 does ot cheat his payoff is 2 2 PV{2,2,2,2,...} 25 (...) 5 (1...)(2) 2 (2). 1 Therefore cheatig is uprofitable if 5 (1) 2 (2), that is if , that 3 / 4 Firm 1 High Firm 2 Low High 2,2,5 Low 5, 1,1 Joh Riley October 8, 213

5 Mathematical Foudatios -5- Sets ad Sequeces Proof by iductio Proof by iductio is ofte helpful if the goal is to establish that some propositio P is true for all itegers = 1,2,. Step 1: establish that the propositio is true for 1. Step 2: show that if the propositio holds for all itegers up to k, it must hold for k 1. If both ca be proved the we are fiished because if P 1 is true the, by the secod step P 2 is true. The, by the secod step, sice P 1 ad P 2 are both true so is P 3. Sice this argumet ca be repeated over ad over agai it follows that for all P is true. Joh Riley October 8, 213

6 Mathematical Foudatios -6- Sets ad Sequeces Geometric sum example Defie S 1 a Claim: 1 a 1 1 a... a S. Note that S1 1 a 1 a 1 so the claim is true for =1. Suppose true for =k. The k k k k1 k1 k k 1 a k 1 a a (1 a) 1 a k 1 a... a a S a a S 1 a 1 a 1 a Therefore true for k 1 k1. Joh Riley October 8, 213

7 Mathematical Foudatios -7- Sets ad Sequeces Proof of the cotrapositive Suppose we would like to prove that if the statemet A is true the the statemet B must also be true. That is, ay evet i which A is true is also a evet i which B is true. ( B is ecessary for A.) A B Cosider the three Ve diagrams. I each case the box is the set of all possible evets. I the left diagram the heavily shaded regio is the set of evets i which A is true. The set of evets i which A is false is the set A. Similarly for B. Note that A B is equivalet to the statemet B A. if A is true the B is true equivaletly if B is true the A is true This statemet is the cotra positive of the origial statemet. (If B is false the A is false) Joh Riley October 8, 213

8 Mathematical Foudatios -8- Sets ad Sequeces Thus istead of attemptig a direct proof that A implies B, we ca appeal to the cotrapositive ad attempt to prove that if B is false, the A must also be false. A good example of this approach ca be foud i sectio A.5 where we show how to costruct a proof of the followig statemet. If the differetiable fuctio f takes o a maximum at x, the the slope is zero at x. The cotrapositive of this statemet is that if the slope of a fuctio is ot zero at x, the the fuctio does ot take o a maximum at x. The proof the follows from a examiatio of the formal defiitio of the slope of a fuctio. Joh Riley October 8, 213

9 Mathematical Foudatios -9- Sets ad Sequeces Proof by cotradictio Last but by o meas least, we ofte prove that a statemet is true by derivig the implicatios that follow if the statemet is false. Suppose, by a combiatio of luck ad cuig, we fid a implicatio that is impossible. The we kow that the statemet caot be false ad so it must be true! To illustrate, we prove that the sum of two odd umbers must be eve. Ay eve umber ca be writte as 2 where is a iteger ad so ay odd umber ca be writte as 2+1. Suppose that the statemet is false so that for some itegers, abc,, (2a 1) (2b 1) 2c 1. (*) Rearragig this equatio, 2( c a b) 1. The umber 2( c a b) is divisible by 2 so it caot be equal to 1. Hece equatio (*) leads to a cotradictio. Thus there caot be ay such umbers a,b,c ad so the statemet must be true. Joh Riley October 8, 213

10 Mathematical Foudatios -1- Sets ad Sequeces Propositio: If the demad fuctio q 2 ( p) is strictly more elastic tha q 1 ( p ) the the demad curves have at most oe itersectio. p dq p dq ( q ( p), p) ( q ( p), p) q2 dp q1 dp ** Joh Riley October 8, 213

11 Mathematical Foudatios -11- Sets ad Sequeces Propositio: If the demad fuctio q 2 ( p) is strictly more elastic tha q 1 ( p ) the the demad curves have at most oe itersectio. p dq p dq ( q ( p), p) ( q ( p), p) q2 dp q1 dp If the demad curves itersect at p. dq2 dq1 The q1( p) q2( p) ad so ( p) ( p). (*) dp dp Therefore for some prices greater the p, q2( p) q1( p). q 1 ( p ) * Joh Riley October 8, 213

12 Mathematical Foudatios -12- Sets ad Sequeces Propositio: If the demad fuctio q 1 ( p) is strictly less elastic tha q 2 ( p ) the the demad curves have at most oe itersectio. p dq p dq ( q ( p), p) ( q ( p), p) q2 dp q1 dp If the demad curves itersect at p. dq2 dq1 The q1( p) q2( p) ad so ( p) ( p). (*) dp dp Therefore for some prices greater the p, q2( p) q1( p). Let p 1 be the lowest price at which the curves itersect Ad let p 2 be the secod lowest such price. dq2 dq1 The, as depicted ( p2) ( p2) dp dp But at ay itersectio (*) must hold so we have a cotradictio. Joh Riley October 8, 213

13 Mathematical Foudatios -13- Sets ad Sequeces Sets ad Vectors Set =collectio of objects If x is i the set we write x S x belogs to S If x is ot i the set we write x S x does ot belog to S Subset, S T superset S T Uio, S T x S or x T (shaded) Itersectio S T x S ad x T Set differece S \ T { xs x T} Joh Riley October 8, 213

14 Mathematical Foudatios -14- Sets ad Sequeces Vector a ordered -tuple x ( x1, x2,..., x ) where xi (a real umber) The set of all such vectors is the Euclidea vector space of dimesio. x Ecoomists typically cosider subsets of ( S ) Legth of a vector x 2 2 xj j1 Joh Riley October 8, 213

15 Mathematical Foudatios -15- Sets ad Sequeces Distace betwee 2 vectors If 1, a atural measure is the absolute value of the differece so that d( y, z) y z. Similarly, with 2, a atural measure is the Euclidea distace y z betwee the two vectors. Appealig to Pythagoras Theorem d( x, y) ( y z ) ( y z ) y z 2 d( y, z) y z Note that the distace betwee y ad z is the legth of the vector y-z. Extedig this to -dimesios, the square of Euclidea distace betwee ordered -tuples is defied as follows. 2 2 ( j j) j1 y z y z Joh Riley October 8, 213

16 Mathematical Foudatios -16- Sets ad Sequeces Vector Product The product ( ier product or sumproduct ) of two -dimesioal vectors y ad z is. y z i1 y z i i The square of Euclidea distace betwee two vectors ca the be rewritte as follows. 2 2 d( y, z) y z ( y z) ( y z). (*) Properties of vector products a b a b b a b a i i i i i1 i1 a ( b c) a b a c ad hece a ( b d) a b a d Joh Riley October 8, 213

17 Mathematical Foudatios -17- Sets ad Sequeces Orthogoal Vectors Two vectors, x ad p are depicted i 3-dimesioal space. x lies i the plae perpedicular to p. Questio: What is the equatio of the plae of vectors perpedicular to p? O Pythagoras Theorem x p p x Joh Riley October 8, 213

18 Mathematical Foudatios -18- Sets ad Sequeces Plaes ad hyperplaes p x p x From (*) o page 16, we ca rewrite this as follows. ( p x) ( p x) p p x x Applyig the rules of vector multiplicatio, ( p x) ( p x) ( p x) p ( p x) x p p x p p x x x p p 2p x x x Appealig to Pythagoras Theorem, p p 2p x x x p p x x Hece px. We geeralize perpedicularity to higher dimesios as follows. Defiitio: Orthogoal Vectors The vectors x ad p are orthogoal if their product p x pixi. i1 Joh Riley October 8, 213

19 Mathematical Foudatios -19- Sets ad Sequeces (Hyper)plae through Set of vectors x such that x x is orthogoal to p. x orthogoal to p Example: Budget set Price vector ad icome p, I Cosumptio vector x Suppose that p x I For feasibility p x I p x O Equivaletly, p x x ( ) Geometry of the budget set Set of positive cosumptio vectors bouded by the hyperplae through x orthogoal to p Joh Riley October 8, 213

20 Mathematical Foudatios -2- Sets ad Sequeces Liear combiatio of vectors Liearly idepedet vectors 1 m 1 1 { v,..., v } such that v 1 1 m m y v... v where v, i 1,..., m m m... v if ad oly if... Euclidea Vector Space Set of vectors such that each vector ca be represeted as a combiatio of liearly idepedet vectors 1 m { v,..., v } kow as a basis for the vector space Dimesio of a vector space umber of vectors i a basis 1 i m Class Exercise: v (1, 2,1), v (5,2, 3), v (6,, 2). (a) What is the dimesio of the vector space of all liear combiatios of these vectors? (b) 1 The vector space of liear combiatios of v ad plae.) What is the equatio of this plae? HINT: Fid a vector p that is orthogoal (perpedicular) to both 2 v is a two dimesioal vector space (a 1 v ad 2 v. Joh Riley October 8, 213

21 Mathematical Foudatios -21- Sets ad Sequeces Covex combiatio of vectors For ay 1 x, x X the vector y is a covex combiatio of x ad 1 x if for some (,1) 1 y (1 ) x x. ** Joh Riley October 8, 213

22 Mathematical Foudatios -22- Sets ad Sequeces Covex combiatio of vectors For ay 1 x, x X the vector y is a covex combiatio of x ad 1 x if for some (,1) 1 y (1 ) x x. It is ofte helpful to write a covex combiatio as 1 x (1 ) x x. Note that 1 x x ( x x ). The 1 x x ( x x ) Hece 1 x x x x The covex combiatio with weight is a fractio of the way dow the lie segmet begiig at x ad edig at 1 x. * Joh Riley October 8, 213

23 Mathematical Foudatios -23- Sets ad Sequeces Covex combiatio of vectors For ay 1 x, x X the vector y is a covex combiatio of x ad 1 x if for some (,1) 1 y (1 ) x x. It is ofte helpful to write a covex combiatio as 1 x (1 ) x x. Note that 1 x x ( x x ). The 1 x x ( x x ) Hece 1 x x x x The covex combiatio with weight is a fractio of the way dow the lie segmet begiig at x ad edig at 1 x. Exercise: x ad x are both covex combiatios of to x (a) Depict the 4 vectors i a diagram. x ad 1 x. Suppose that (so x is closer (b) Prove that x is a covex combiatio of x ad x. Joh Riley October 8, 213

24 Mathematical Foudatios -24- Sets ad Sequeces Neighborhood ad deleted (puctured) eighborhood of x eighborhood of x N( x, ) { x d( x, x ) } Deleted Neighborhood of a vector D N ( x, ) { x d( x, x ), x x } Class Exercise: Defie a Local Maximum for the fuctio (i) f : (ii) f : Joh Riley October 8, 213

25 Mathematical Foudatios -25- Sets ad Sequeces Heceforth focus o sets that are subsets of Euclidea space Boudary poit of a set x is o the boudary of y S ad z S S if for ay the eighborhood Nx (, ) cotais some ** Joh Riley October 8, 213

26 Mathematical Foudatios -26- Sets ad Sequeces Heceforth focus o sets that are subsets of Euclidea space Boudary poit of a set x is o the boudary of y S ad z S S if for ay the eighborhood Nx (, ) cotais some Cosider a sequece of eighborhoods t t Nx (, ) where. The there must be a correspodig sequece of poits { y t } i S ad a secod sequece { z } ot i S that both coverge to t x * Joh Riley October 8, 213

27 Mathematical Foudatios -27- Sets ad Sequeces Heceforth focus o sets that are subsets of Euclidea space Boudary poit of a set x is o the boudary of y S ad z S S if for ay the eighborhood Nx (, ) cotais some Cosider a sequece of eighborhoods t t Nx (, ) where. The there must be a correspodig sequece of poits { y t } i S ad a secod sequece { z } ot i S that both coverge to t x Iterior poit of S. Ay poit i S that is ot a boudary poit The set of iterior poits is writte as it S Joh Riley October 8, 213

28 Mathematical Foudatios -28- Sets ad Sequeces Closed Set i Euclidea Space S is closed if it cotais all its boudary poits. Example: [ a, b] { x a x b} Note that the set of real umbers is closed as it has o boudary poits. I ecoomics we are ofte iterested i the set of positive real vectors { x x } This is also a closed set. * Joh Riley October 8, 213

29 Mathematical Foudatios -29- Sets ad Sequeces Closed Set i Euclidea Space S is closed if it cotais all its boudary poits. Example: [ a, b] { x a x b} I ecoomics we are ofte iterested i the set of positive real vectors { x x } This is also a closed set. Ope Set i Euclidea Space S is ope if every poit is S is a iterior poit. That is, for ay that the eighborhood N( x, ) S Example: ( a, b) { x a x b} Class Exercise: Which statemets are correct? The set of real umbers is (a) ope (b) closed (c) either ope or closed. x S there exists such Joh Riley October 8, 213

30 Mathematical Foudatios -3- Sets ad Sequeces Bouded set: The set X is bouded if for some a, a x a for all x X Compact set: A set that is both closed ad bouded ** Joh Riley October 8, 213

31 Mathematical Foudatios -31- Sets ad Sequeces Bouded set: The set X is bouded if for some a, a x a for all x X Compact set: A set that is both closed ad bouded Covex set: S is covex if, for ay 1 x, x S ad ay (,1), the covex combiatio 1 x (1 ) x x S Strictly Covex Set S is strictly covex if, for ay 1 x, x S ad ay (,1), 1 x (1 ) x x it S * Joh Riley October 8, 213

32 Mathematical Foudatios -32- Sets ad Sequeces Bouded set: The set X is bouded if for some a, a x a for all x X Compact set: A set that is both closed ad bouded Covex set: S is covex if, for ay 1 x, x S ad ay (,1), the covex combiatio 1 x (1 ) x x S Strictly Covex Set S is strictly covex if, for ay 1 x, x S ad ay (,1), 1 x (1 ) x x it S Examples of strictly covex sets S { x x x 25}, 2 S { x x x 25}, S { x x x 25} 1 2 Exercise (a) Suppose that ab c. What are the boudary poits of S { x a x b or b x c}? (b) Is S ope? Joh Riley October 8, 213

33 Mathematical Foudatios -33- Sets ad Sequeces t t Ifiite sequece of vectors { x } 1 Coverget sequece t t1 { x } x For ay there exists some T( ) such that for all t t T( ), x N( x, ) The vector x is the limit poit of this sequece. Questio: Does every ifiite sequece have a limit poit? Questio: Does every bouded ifiite sequece have a limit poit? Joh Riley October 8, 213

34 Mathematical Foudatios -34- Sets ad Sequeces Coverget subsequece Bolazo-Weierstrass Theorem: Ay bouded sequece t t of vectors { x } 1 has a coverget subsequece. Proof: x 2 Sequece is bouded so for some a t a x a ifiite sequece i a square of side 2a Step 1: must be a ifiite subsequece i a square of side a Step 2: must be a ifiite subsequece i a square of side a /2. Step : must be a ifiite subsequece i a square of side a/. Joh Riley October 8, 213