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1、PrefaceFormoststudents,thefirstandoftenonlycourseincollegemathematicsiscalculus.Itistruethatcalculusisthesinglemostimportantfieldofmathematics,whoseemergenceintheseventeenthcenturysignaledthebirthofmodernmathematicsandwasthekeytothesuccessfulapplicationsofmathematicsinthesciencesandengineering.
2、Butcalculus(oranalysis)isalsoverytechnical.Ittakesalotofworkeventointroduceitsfundamentalnotionslikecontinuityandthederivative(afterall,ittooktwocenturiesjusttodeveloptheproperdefinitionofthesenotions).Togetafeelingforthepowerofitsmethods,saybydescribingoneofitsimportantapplicationsindetail,ta
3、kesyearsofstudy.Ifyouwanttobecomeamathematician,computerscientist,orengineer,thisinvestmentisnecessary.Butifyourgoalistodevelopafeelingforwhatmathematicsisallabout,wheremathematicalmethodscanbehelpful,andwhatkindsofquestionsdomathematiciansworkon,youmaywanttolookfortheanswerinsomeotherfieldsof
4、mathematics.Therearemanysuccessstoriesofappliedmathematicsoutsidecalculus.Arecenthottopicismathematicalcryptography,whichisbasedonnumbertheory(thestudyofthepositiveintegers1,2,3,...),andiswidelyapplied,forexample,incomputersecurityandelectronicbanking.Otherimportantareasinappliedmathematicsar
5、elinearprogramming,codingtheory,andthetheoryofcomputing.Themathematicalcontentintheseapplicationsiscollectivelycalleddiscretemathematics.(Theword“discrete”isusedinthesenseof“separatedfromeachother,”theoppositeof“continuous;”itisalsooftenusedinthemorerestrictivesenseof“finite.”Themoreeverydayve
6、rsionofthisword,meaning“circumspect,”isspelled“discreet.”)viPrefaceTheaimofthisbookisnottocover“discretemathematics”indepth(itshouldbeclearfromthedescriptionabovethatsuchataskwouldbeill-definedandimpossibleanyway).Rather,wediscussanumberofselectedresultsandmethods,mostlyfromtheareasofcombinato
7、ricsandgraphthe-ory,withalittleelementarynumbertheory,probability,andcombinatorialgeometry.Itisimportanttorealizethatthereisnomathematicswithoutproofs.Merelystatingthefacts,withoutsayingsomethingaboutwhythesefactsarevalid,wouldbeterriblyfarfr
