SETS, LOGIC, PROOFS, AND RELATIONS

Transcription

1 VII. SETS, LOGIC, PROOFS, AND RELATIONS Durig your first year especially, you will do a lot of proofs. If you have t had a proof-based math class before, these will seem weird ad awkward. They are simply a way of makig a argumet, just as essays are. Like essays, good proofs use a certai laguage ad follow a certai formula. You will see these logical symbols: Q sice therefore ad or ot implies is equivalet to cotradictio For the most part, I discourage the sice ad therefore symbols because they re cofusig (at least for me). I ve ever bee able to fid a way to keep them straight, which oe meas what. There are other symbols to express all of these (icludig about fourtee differet ways to express a cotradictio). I also dislike usig the symbols ad for or ad ad, sice they are also used to represet the miimum ad the maximum of two thigs, or the joi ad meet of two partitioigs. I logic, their meaig is clear, but outside of that, there may be some ambiguity. A useless piece of trivia is that the symbol comes from the first letter of the Lati cojuctio vel, which is a iclusive or. I logic, Or is always used i the iclusive sese: sayig that all first year grad studets take prob/stats or metrics does ot rule out that some might take both. It oly rules out people takig either. I fact, the egatio of ad ad or are: ( P Q) ( P) ( Q) ( P Q) ( P) ( Q) A propositio is a statemet that is either true or false. It might be like the moo is i the seveth house. You ca go cosult a astrology chart ad verify whether it is. A coditioal statemet like if the moo is i the seveth house ad Jupiter aligs with Mars, the peace will rule the plaets cosists of a hypothesis ad a coclusio. This is ofte writte P Q, ad read P implies Q or P is sufficiet for Q. The coverse of a coditioal is Q P or P Q, ad does ot eed to be true eve if the origial statemet is. The coverse is ofte read P oly if Q or P is ecessary for Q. The cotrapositive ( Q) ( P ) is. If a coditioal ad its coverse are both true, the statemet is kow as a bicoditioal statemet, P Q. The bicoditioal is sometimes writte P iff Q ad ofte read P if ad oly if Q or P is ecessary ad sufficiet for Q It hard to come up with a better defiitio of a set tha just a collectio of thigs. These are called elemets or members. Commo set otatio icludes: Summer 00 math class otes, page 47

2 the empty set for all or for ay or for each or for every there exists or for some is a elemet of the set is a subset of the set is a (proper) subset of the set (May people ited the last of these to allow weak iclusio as well.) For oe set A to be a subset of aother set B meas that every elemet of A is also a elemet of B. If every elemet i A is also i B, ad B elemets that are ot i A, the A is a proper subset of B. Alteratively, we ca say that A is cotaied i B whe it is a subset ad properly cotaied or strictly cotaied whe it is a proper subset. Whe two sets are each a (weak) subset of each other, the they are equivalet, deoted with a equal sig. The complemet of a set A is everythig i the uiverse that is ot i A, usually deoted by A or A C. The cardiality of a set is the umber of elemets i it, usually deoted by # A or A. (Thik of the set of all people i the classroom; the cardiality of this set is the umber of people i the classroom.) The set without ay elemets is the empty set, deoted with the ifty Norwegia letter Ø. Two sets are disjoit whe their itersectio is the empty set; that is, whe they share o elemets. Elemets of a set ca ofte be characterized by the commo features that they share, at the exclusio of all elemets ot i the set, like the set of all squirrels i the classroom or the set of all vegetables that have a purple exterior. There is a simple otatio for expressig this: { x R+ :0x N} This is much easier tha actually amig all oegative real umbers that are some iteger divided by te, a list which starts with 0, 0., 0., 0.3, 0.4, ad goes o for quite a while. Recall the defiitio of a fuctio cotiuous at some x: for ay ε > 0 there exists a δ > 0 such that if a poit y is less tha distace δ from x, the f( y) f( x) is less ε. I mathematical shorthad, this becomes: ( ε > 0) ( δ > 0 ) s.t. y x < δ f y f x ε [( ) ( ( ) ( ) < )] This is much more compact though harder to read. Example: The utility fuctio Ui: Xi R is locally o-satiated if xi Xi ad ε > 0, xi Xi s.t. xi xi ε ad Ui xi Ui xi ( )> ( ). Example: A idividual s demad correspodece is defied as: xi( p, wi) { x X : U ( x) U ( x ) x B( p, w )} where his budget correspodece is i i i i Summer 00 math class otes, page 48

3 L L Bpw (, i) x Xi, Xi R + : xp il l wi. l= Ca you traslate this ito words, ad the, ca you traslate it ito somethig that makes sese? There is some coectio betwee the logic above ad sets. If A is the set of evets for which statemet P is true (for example, the set of days o which the moo is i the seveth house), ad B is the set of evets for which statemet Q is true, the we have the followig relatioships: ( P Q) ( A B) ( P Q) ( A B) ( P Q) ( A = B) ( P) ( A ) C ( P Q) ( A B) ( P Q) ( A B) Ofte, these relatioships are expressed usig a Ve diagram. I ecoomics, we are ofte asked to provide a formal proof of some assertio. Writig a proof is like writig a essay. There is a proper way to do each, there are several stadard formats, ad there are stylistic details which make your argumet easier to follow. Geerally, this is the format I like to follow for a proof. O the first lie, I state exactly what I will be provig. O the ext lie, I begi my proof. I the body of the proof, I first aouce the method of the proof (if othig is stated, direct proof is uderstood). The I state my suppositios, followed by relevat defiitios. This is like a itroductio i a essay. Havig clearly spelled out the backgroud, I give the argumet. Ultimately, the proof comes to some puchlie. I a direct proof, this is the coclusio; i a proof by cotradictio, it is the critical cotradictio ad its implicatio. This is like the coclusio of the essay. Traditioally, this is followed by a black box or the letters QED to aouce that the proof is complete. A direct proof is the most basic way to show if P, the Q. You start off by supposig that P is true, ad the you show the logical sequece that implies Q must also be true. Prove: If is a odd iteger, the is odd. Proof: Suppose that is a odd iteger. By defiitio of odd, this meas that k Z s.t. = k+. The = ( k+ ) = 4k + 4k+. Set m= k + k. We kow that m is a iteger, sice Z is closed uder multiplicatio ad additio. Thus, = k+ 4k 4k m ( ) = + + = + for m Z, ad so by defiitio, is odd. QED. Summer 00 math class otes, page 49

4 A proof by cotradictio or a proof by cotrapositive is sometimes the easiest way to show how P implies Q. You start by supposig that P is true but Q is ot, ad show how this leads to two coclusios which are mutually exclusive. These proofs ofte start with the statemet, The proof is by cotradictio, or, Suppose ot. Whe you arrive at the two cotradictory statemets, deote that this is a impossibility with oe of several commo symbols or the words which is a cotradictio. The you state that this cotradictio meas that P must imply Q. Prove: If r is a real umber such that r =, the r is irratioal. Proof: Suppose ot: suppose r is ratioal ad r =. By defiitio, r is ratioal if m, Z such that r= m/. We ca assume without loss of geerality that m ad have o commo divisor greater tha oe (if they did, we could simply fid m Z : m = m/ gcd ( m, ), similarly, to reduce the fractio). The r= m/ r = m / m = r =. Sice m is eve, we kow that m is eve (as show i the previous proof). The by defiitio, k Z s.t. m= k. The m = 4k = k =. Thus, is eve ad as a cosequece, is eve. The two divides both m ad, but they have o commo divisor greater tha oe.. This is a cotradictio. Whe r =, it must be the case that r is irratioal. QED. A third method is proof by iductio. This works extremely well i certai cases, geerally, whe the assertio looks somethig like P implies Q for all greater tha or equal to N. There are two steps here. First, show that P implies Q for N. The you just assume that (P Q) is true for some N ad show that this leads to (P Q) must be true for +. It works just like kockig over domioes: if the - th domio falls, it pushes over the +-th domio, provided that you started the chai at the N-th oe. These proofs ofte start with the words, The proof is by iductio. They aouce the two steps. Fially, they state that the assertio has bee show for the first case, ad that oe case implies the ext, ad thus is must be true for all cases greater tha the first. k x Prove: Let x R, x. The Z ++, x = x. Proof: The proof is by iductio. Let us take ay x R, x. I the first step, we k wat to show that ( x )= ( x ) ( x ) for =. The left-had side of this is k 0 x = x =. The right-had side is ( x ) ( x )=. Thus, it is true whe =. k Now suppose that ( x )= ( x ) ( x ) is true for some arbitary Z ++. We wat to show that this implies that the relatioship also holds for +. For +, + k k ( x )= x + ( x )= x + ( x ( x ) = xx ( ) ( x ) + ( x ) ( x ) = ( ( + x + ) ) ( x x ) + ( x ) = ( x ) ( x ). This is what we eeded to show: k + k + k x = x x ( x )= x x x ( ) ( ) ( ). Sice ( ) Summer 00 math class otes, page 50

5 k + k = x x for =, ad sice ( x )= x x ( x )= + k x x, we ca coclude that Z ++ ( x )= x x. QED. This is a relatioship which show up i discoutig future payoffs ow you ll kow the derivatio whe you see it. Here is the sort of thig that you ll be asked to prove i micro classes this ext year, ad what a good proof looks like: ( ) is homogeeous of degree oe with respect to w ad satisfies Walras Prove: If xpw, law (igore this for ow!), the l L, ε ( p, w)=. lw ( ) is homogeous of degree oe with respect to Proof: Suppose that the demad fuctio xpw, xl( pw, ) xl( pw, ) w ad satisfies Walras law. Icome elasticity for good l is defied as ε lw( pw, ) ww. α 0, xp, αw αxpw., Differetiatig with respect to α, By defiitio of homgeeity, > ( )= ( ) α xp (, αw)= α αxpw (, ) wdw xp (, αw)= xpw (, ). Evaluatig this at α =, for each good xl( pw, ) xl( pw, ) x l L xl( p, w) w= xl( p, w) w ww =. Thus, ε l( pw, ) xl( pw, ) lw pw, ww. QED. ( ) = I your micro class this year, you will work with preferece relatios before you are allowed to play with utility fuctios. For some set X, a biary relatio gives you a true or false statemet about ay pair ( xy, ), with x X ad y Y. For example, thik about a group of five kids (Aabel, Beladoa, Clarabel, Isabel, ad Gargamel). Perso x likig perso y is a biary relatio: you ca pull out ay pair, such as (Aabel, Clarabel), ad ask: does Aabel like Clarabel? The aswer is yes or o. The order of the pair ( xy, ) matters: this might ot be the same as the aswer to does Clarabel like Aabel? Similarly, you ca pull out the pair (Clarabel, Gargamel) or (Isabel, Isabel) or ay others, ad the relatio gives you a aswer to all of these. Because each of these statemets is a true or false propositio, sometimes a relatio will be called a subset of X X. The set X X is all possible pairs from X, ad the relatio is the subset o which this is true. O a set of ames, a possible relatio is x comes earlier tha or at the same place as y i alphabetic order. O a set of potetial budles of goods, a possible relatio is I like x at least as much as I like as y. The last of these is a preferece relatio. Here is almost the right defiitio of a ice property of some orderigs: Defiitio: A biary relatio R o a set X is (close to) a partial orderig if, xyz,, X, the followig three properties hold: Summer 00 math class otes, page 5

6 . reflexive: ( xx, ) R. complete: xy, R yx, R ( ) ( ) [(, ) (, ) ] (, ). 3. trasitive: xy R yz R xz R You ca verify that the alphabetic order rule above does i fact fit the defiitio of a partial orderig. Somethig very similar will arise later uder the ame of the lexicographic orderig. Whe we have a partial orderig, it is more ituitive to write xf y for ( xy, ) R. This makes sese whe you compare it to a certai partial orderig o the real umbers, the greater tha or equal to orderig. Defiitio: A partial orderig R o a set X is a well-orderig if every oempty subset S has a first elemet s i X; that is, ( xs, ) R x A. What this gets at is the problem that i some sets it s very hard to ame exactly what the largest or smallest elemet is. Uder a few circumstaces it is: Priciple: Ay partial orderig o a fiite set is a well-orderig. This relates to a topic metioed previously. The relatios ad are partial orderigs o the real umbers. The real umbers are ot well-ordered uder these relatios, though. As metioed, ot all sets have a maximum ad a miimum. However, all bouded sets have upper bouds (who would have thought?); that is, umbers that are above everythig i the set. They also have lower bouds. The least upper boud of a set, also kow as a supremum, is the smallest umber with the property that it is greater tha or equal to ay elemet i the set. Ulike a maximum, it does ot eed to be i the set however, if the set has a maximum, it is also the supremum. O the other had, whe a set does t have a maximum, like the iterval ( 0, ), the supremum is the ext best thig. The greatest upper boud or ifimum of a set is the largest umber such that it is less tha or equal to aythig i the set. If a miimum exists, it is the ifimum. Suprema of sets are deoted i oe of these ways: sup xyz,, yxz { } or: lub {,, } while ifima are represeted as: if xyz,, { } or: glb { xyz,, } Summer 00 math class otes, page 5

7 Refereces: Kolmogorov ad Fomi, Chapter. Roselicht, Chapter. Fletcher ad Patter. Summer 00 math class otes, page 53