Ho to write pro ofs quic guide Eugenia Cheng Departmen of Mathematics Univ ersit of Chicago Email eugeniamath
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Ho to write pro ofs quic guide Eugenia Cheng Departmen of Mathematics Univ ersit of Chicago Email eugeniamath

uc hicagoedu eb ttpwwwmathuc hicagoedu eugenia Octob er 2004 pr of is like em or ainting or building or bridge or novel or symphony Help dont kno ho to write pro of ell did an one ev er tell ou what pro of is and how to go ab out writing one Ma yb n

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Ho to write pro ofs quic guide Eugenia Cheng Departmen of Mathematics Univ ersit of Chicago Email eugeniamath

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Presentation on theme: "Ho to write pro ofs quic guide Eugenia Cheng Departmen of Mathematics Univ ersit of Chicago Email eugeniamath"‚ÄĒ Presentation transcript:

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Ho to write pro ofs: quic guide Eugenia Cheng Departmen of Mathematics, Univ ersit of Chicago E-mail: eugenia@math.uc hicago.edu eb: ttp://www.math.uc hicago.edu/ eugenia Octob er 2004 pr of is like em, or ainting, or building, or bridge, or novel, or symphony. \Help! donít kno ho to write pro of !" ell, did an one ev er tell ou what pro of is and how to go ab out writing one? Ma yb not. In whic case itís no onder ouíre erplexed. riting go pro of is not supp osed to something can just sit do wn and do. Itís lik writing em in foreign language. First ou ha to learn the language.

And then ou ha to kno it ell enough to write etry in it, not just sa \Whic is it to the train station please?" Ev en when ou kno ho to do it, writing pro of tak es planning, eort and inspira- tion. Great artists do mak sk etc hes efore starting pain ting for real; great arc hitects mak plans efore building building; great engineers mak plans efore building bridge; great authors plan their no els efore writing them; great usicians plan their symphonies efore comp osing them. And es, great mathematicians plan their pro ofs in adv ance as ell.
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Con ten ts What do es pro of

lo ok lik e? Wh is writing pro of hard? What sort of things do try and pro e? The general shap of pro of What do esnít pro of lo ok lik e? Practicalities: ho to think up pro of Some more sp ecic shap es of pro ofs 10 Pro of con tradiction 15 Exercises: What is wrong with the follo wing \pro ofs"? 16
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What do es pro of lo ok lik e? pr of is series of statements ach of which fol lows lo gic al ly fr om what has gone efor e. It starts with things we ar assuming to true. It ends with the thing we ar trying to pr ove. So, lik go story pro of has eginning, middle and an end.

Beginning: things are assuming to true, including the denitions of the things eíre talking ab out Middle: statemen ts, eac follo wing logically from the stu efore it End: the thing eíre trying to pro The oin is that eíre given the eginning and the end, and someho ha to ll in the middle. But canít just ll it in randomly ha to ll it in in that \gets us to the end". Itís lik putting in stepping stones to cross riv er. If put them to far apart, eíre in danger of falling in when try to cross. It migh ok but it migh not and itís probably etter to safe than sorry Wh is

writing pro of hard? One of the dicult things ab out writing pro of is that the order in whic write it is often not the order in whic though it up. In fact, often think up the pro of ackwar ds Imagine you want to atch movie at the Music Box. How ar you going to get ther e? ou se that the Br own Line wil take you ther fr om the op. ou know that you an get the #6 Bus to the op, and you know that you an walk to the #6 Bus stop fr om Campus. But when you actual ly make the journey, you start by walking to the #6, and you end by getting the Br own Line. nd if some one asks you for dir

ctions, it wil not very helpful if you explain it to them ackwar ds Or to put it another to build bridge across riv er, migh ell start at oth ends and ork our to ards the middle. migh ev en put some preliminary supp orts at arious oin ts in the middle and ll in all the gaps afterw ards. But when actually go across the bridge, start at one end and nish at the other. One of the easiest mistak es to mak in pro of is to write it down in the or der you thought of it This ma con tain all the righ steps, but if theyíre in the wrong order itís no use. Itís lik taking piece of usic and

pla ying all the notes in dieren order. Or writing ord with all the letters in the wrong order. This means that for all but the simplest pro ofs, ouíll probably need to plan it out in adv ance of actually writing it do wn. Lik building long bridge or large building it needs some planning, ev en though building small bridge or tin ut migh not.
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What sort of things do try and pro e? Here is classication of the sorts of things pro (this list is not exhaustiv e, and itís also not clear cut there is some erlap, dep ending on ho ou lo ok at it): 1. i.e. \something

equals something else" 2. 3. 4. is purple (or has some other in teresting and relev an prop ert y) 5. is true i.e. \all animals of certain kind eha in certain )" 6. s.t. is true i.e. \there is an animal that eha es in certain )" 7. Supp ose that a; b; and are true. Then is true. [Note that this is just ersion of in disguise.] The general shap of pro of Letís no ha lo ok at the general shap of pro of, efore taking closer lo ok at what it migh lo ok lik for eac of the cases ab e. ust alw ys remem er that there is eginning, middle and an end. Example 1. Using the eld axioms, pr ove that

ab ac for any al numb ers a; b; ou may use the fact that x: for any al numb er beginning eld axioms denition given middle )) denition ab distributive la ac )) distributive la additive inverse given ac denition of additive inverse ab ab ac end line 2, ab ac as required
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Example 2. et and functions Show that if and ar inje ctive then is inje ctive beginning denition of injective denition )( )) assumption that and re injective i.e. middle )( )( )) )) denition since is injective since is injective )( )( end i.e. is injective,

as required Example 3. Pr ove by induction that 1) beginning Principle of Induction middle fo LHS RHS result is true fo If result is true fo then )( i.e. result true fo result true fo result true fo end the Principle of Induction, the result is true fo all
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Of course, when write go story donít actually lab el the eginning, the middle and the end with BEGINNING, MIDDLE, and END itís supp osed to sort of ob vious where they are. The same is true of pro of. So hereís the thing eep going on ab out but whic is apparen tly not as ob vious as it migh sound: The end of pro of should

come at the end, not at the eginning. Of course, Iív delib erately made it sound really ob vious there. But hereís more illuminating of putting it: The pro of should end with the thing ouíre trying to pro e. The pro of should not gin with the thing ouíre trying to pro e. This is not to confused with the fact that often egin announcing what the end is going to e. This is bit lik story that starts at the end and then the en tire story is ash bac k. Lik The Go-Betwe en or Brideshe ad evisite or eb Or, itís lik taking someone on journey ou migh ell tell them where ouíre going righ at the

start. But once ouív told them what the destination is you stil start the journey fr om the ginning The same is true of pro ofs. Ev en if egin announcing what the end is going to e, we then have to start at the ginning and ork our to the pre-announced end. What do esnít pro of lo ok lik e? Ther ar mor plastic amingo es in meric than al ones. There are more bad no els in the orld than go ones, and there are more bad pro ofs in the orld than go ones. Here are some of the most opular ys to write bad pro of. 1. Begin at the end and end at the eginning This is really really terrible thing

to do. This migh ev en orse than lea ving out gaps in the middle. Because if ou egin at the end and end at the eginning ou monumental ly ha enít got where ouíre trying to go. Hereís an example of this for Example from Section 4: ab ac ab ab ac ac ac )) ry comparing this with the go pro of giv en in Section ouíll see that all the correct steps are there, but theyíre all in the wrong order.
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Sense any make do esn ít it ackwar ds but things right the write you if. This is terrible thing to do but not terminal catastrophe if ou ha all the righ ideas but in the wrong order, all ou

need to do is ork out ho to put them in the righ order 2. ak ying leaps instead of earth ound steps. This category includes leaping from one statemen to another without justifying the leap lea ving out to man steps in et een using profound theorem without pro ving it (w orse) using profound theorem without ev en men tioning it or example, sp ot the ying leap in the follo wing \pro of": ab ab ac 3. ak ying leaps and land at on our face in the ud By whic mean making steps that are actually wrong. The end ma ell justify the means in some orlds, but in mathematics

if ou use the wrong means to get to the righ end, ou ha enít actually got to the end at all. ou just think ou ha e. But itís gmen of our imagination. Hereís an example of ery imaginitiv \pro of" that is denitely at on its face in the ud: ab ab ab ab ac Of course, itís ev en orse if ou do something illegal and thereb reac conclusion that isnít ev en true. Lik or What is wrong with these \deductions"?
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4. Handw ving Handw ving is when ou arriv at statemen some not-v ery-mathematical means. The step isnít necessarily wrong, but ou ha enít arriv ed at it in

go logical manner. erhaps ou had to resort to writing few sen tences of prose in English rather than Mathematics-sp eak. This is often sign that ouív got the righ idea but ou ha enít ork ed out ho to express it. Sp ot the handw ving here ou can see it from mile o: ab ac ecause if ou add ac to oth sides then oth sides vanish which means they re inverse ab ab ac Handw ving is bad but is not ultimately catastrophic ou just need to learn ho to translate from English in to Mathematics. This is probably easier to learn than the problem of coming up with the righ idea in the rst

place. 5. Incorrect logic This includes the great classics negating statemen incorrectly pro ving the con erse of something instead of the thing itself What is the negation of the follo wing statemen t: s.t. satisfying The correct answ er is at the ottom of the page If ou get it wrong, ou go directly to Jail. Do not pass Go. Do not collect $200. 6. Incorrect assumption ou could ha all our logic righ t, ou could mak series of erfectly go and sensible steps, but if ou start in the wrong place then ouíre not going to ha go pro of. Or, if ou use an assumption along the that simply isnít true, then

itís all going to go horribly ear-shap ed 7. Incorrect use of denitions or use of incorrect denitions This is ery ery oidable error. Esp ecially if itís not test and so ou ha all our notes and all the oks in the orld to consult: getting the denitions wrong is really s.t. satisfying s.t.
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oin tless of going wrong. Whatís wrong with the follo wing \pro of" for the second example from Section 4? )( )( is injective. 8. Assuming to uc This is tric ky one, esp ecially when ouíre studen at the eginning of course. What are ou allo ed to assume? Ho uc do ou ha to

justify eac step? go rule of th um is: ou ne to justify everything enough for your ers to understand it. This is not hard and fast rule, but itís guideline that will alw ys remain true ho ev er far ou progress in mathematics, ev en if ou ecome an in ternationally acclaimed Fields- medal-winning mathematician. The oin is that as ou ecome more adv anced our eers do to o, so ou are ev en tually going to taking bigger steps in our pro ofs than ou do no w. i.e. donít orry ou onít required to write do wn ev ery use of the distributiv la forev er! If in doubt, justify things mor ather than less. ery

few eople giv to uc explanation of things. In fact, ha only ev er encoun- tered one studen who consisten tly explained things to uc h. Practicalities: ho to think up pro of The harsh realit is that when ou sit do wn to pro something ou usually ha to start just staring at it really hard and hoping for some inspiration to hit ou. Ho ev er, ou can put ourself in the est ossible place to nd that inspiration doing some of the follo wing things, probably on piece of rough pap er. rite out the beginning ery carefully rite do wn the denitions ery explicitly write do wn the things ou

are allo ed to assume, and write it all do wn in careful mathematical language. rite out the end ery carefully That is, write do wn the thing ouíre trying to pro e, in careful mathematical language.
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ry and manipulate oth the eginning and the end to try and mak them lo ok lik one another. This is lik building from oth ends of the bridge un til they meet in the middle, and itís ok as long as ou write the whole thing out pr op erly in the right or der afterwar ds ak big leaps to see what happ ens, and then mak the big leaps in to smaller leaps afterw ards. See if the situation

reminds ou of an situations ouív ev er seen efore. If so, erhaps ou can cop the metho d. ou should always read er our pro of after ouív written it to mak sure ev ery single step mak es sense. When ouíre writing pro of the rst time through, ou migh get carried in frenzy of inspiration and ecome blind to the orld around ou whic mean that ou migh do something wrong without noticing it. Itís imp ortan to go bac in calm state and pr etend to mor stupid than you ar Or more sceptical. Or un trusting. When ou nish pro of ou should feel lik ou understand whatís going on, but ou should

go bac er it pretending that ou don ít understand, and see if our pro of explains it to ou. Some more sp ecic shap es of pro ofs No letís lo ok at the arious yp es of things that try to pro (as listed in Section 3), and think ab out ho can pro them. 1. or \something equals something else" The pro of migh tak the follo wing general shap e: Or: 10
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Note that this is very dieren from: 2. No the pro of migh lo ok lik this: Or: kno that Also and 3. No the pro of migh lo ok lik this: Conversely Hence 11
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Or Ho ev er, ew are that this can dangerous of

taking short cut ou migh nd that ouíre going the wrong up one street. Do those implications al ly ork bac kw ards? Itís alw ys safer to do the forw ards and the bac kw ards separately and write \con ersely" at the oin where ouíre ab out to start doing the con erse direction. 4. is purple go to start is to write do wn the denition. What do es it mean for to purple? \x is purple" means kno and is purple as required 5. x; is true In practice this will usually of certain kind", lik \for an rational um er or \for an con tin uous function or \for an braided monoidal category ". Then

the oin is probably to use the assume pr op erties of to pro ). So go to start is to write do wn the denition of those assumed prop erties, carefully in mathematical language. e.g. Pr ove that any ational numb er an expr esse as wher and ar inte gers that ar not oth even. So start sa ying: Let rational numb er. So can exp ressed as where and re integers and 12
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Note that ha pic ed an arbitr ary and then just can just pro that this has the desired prop ert and eíre done. ou migh sa \But eív only pro ed it for this and not every ". But the oin is that this is andom not

one sp ecic one, itís sort of generic that sho ws the pro of will ork for an sp ecic one that substituted in. Itís not lik pro ving the result for one particular um er, sa 23. 6. s.t. is true Here, all ha to do is nd one for whic is true. So can just sa \let 23" and then sho that 23 has the desired prop ert or example, pro that: s.t. 100 Put 10 No so have 10 100 This is ne; of course could also ha pic ed 1000 or 476002 The latter esp ecially ould little eccen tric but ould still erfectly alid (if violen t) hoice of to nish the problem o. Of

course, sometimes itís bit hard to just pluc alid out of thin air. Itís bit lik pulling rabbit out of hat it lo oks lik magic, but of course you are the one who put the rabbit there in the rst place. So if itís complicated example probably ork out (on rough piece of pap er) whic is going to do the tric k, and once ha it all ork ed out can pull it out of the hat. 7. If a; b; c; are true then is true When ou ha whole lot of things a; b; c; d; ouíre allo ed to assume, it gets more complicated. ou migh ha to dev elop sev eral parts of it sort of at the same time efore pro ceeding to the

end, lik no el where there are sev eral strands of plot happ ening at the same time efore they all come together at the end for the nal enouement The pro of migh lo ok lik this: and and and 13
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In fact if dra little picture of what happ ened, itís uc easier to see whatís going on (and see where the eginning, the middle and the end ha got to): Hereís an example of this phenomenon at ork: Pr ove that if then ou may assume that for al x; )( xy and x: No means i.e. Therefo re )( closure of under multiplication. No )( by the rst given assumption (( )( ))

commutativit of multiplication )) irst given second given )( as required 14
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And hereís the picture of it: rdered eld axioms )( xy )( )) )( )( Pro of con tradiction Pro of con tradiction is ery useful tec hnique whic itís imp ortan to understand. The idea is that statemen is either true or false. So if it isnít false then it must true So instead of pro ving that is true can pro that isn ít false ecause if it isnít false then it ust true. then sho that eing false ontr adicts something kno to true, and this means that canít ossibly fase, so it ust true. summarise: are

trying to pro that some statemen is true. sa \supp ose ere not true", and nd con tradiction Since eing false giv es con tradiction, deduce that ust true. usually write big at the oin where reac hed the con tradiction, to dra atten tion to it. 15
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Hereís an example: Using the eld axioms, pr ove that has no multiplic ative inverse in Supp ose that do es have multiplicative inverse. This means s.t. But kno that 0x and :x has no multiplicative inverse. Exercises: What is wrong with the follo wing \pro ofs"? Example 1. Pr ove that so so 16
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2. et and functions Show that if and ar inje ctive then is inje ctive )( )( )) )) But injective is injective as required. Example 3. Pr ove that s.t. 2a 4a put 2a Example 4. Using only the eld axioms, pr ove that x; )( )( distributive la yx distributive la xy commutativit of multiplication xy additive inverse xy xy commutativit of multiplication xy distributivit additive inverse xy additive inverse denition of additive identit 17