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1、PrefaceFormoststudents,thefirstandoftenonlycourseincollegemathematicsiscalculus.Itistruethatcalculusisthesinglemostimportantfieldofmathematics,whoseemergenceintheseventeenthcenturysignaledthebirthofmodernmathematicsandwasthekeytothesuccessfulapplicationsofmathematicsinthesciencesandengineer
2、ing.Butcalculus(oranalysis)isalsoverytechnical.Ittakesalotofworkeventointroduceitsfundamentalnotionslikecontinuityandthederivative(afterall,ittooktwocenturiesjusttodeveloptheproperdefinitionofthesenotions).Togetafeelingforthepowerofitsmethods,saybydescribingoneofitsimportantapplicationsind
3、etail,takesyearsofstudy.Ifyouwanttobecomeamathematician,computerscientist,orengineer,thisinvestmentisnecessary.Butifyourgoalistodevelopafeelingforwhatmathematicsisallabout,wheremathematicalmethodscanbehelpful,andwhatkindsofquestionsdomathematiciansworkon,youmaywanttolookfortheanswerinsome
4、otherfieldsofmathematics.Therearemanysuccessstoriesofappliedmathematicsoutsidecalculus.Arecenthottopicismathematicalcryptography,whichisbasedonnumbertheory(thestudyofthepositiveintegers1,2,3,...),andiswidelyapplied,forexample,incomputersecurityandelectronicbanking.Otherimportantareasinappl
5、iedmathematicsarelinearprogramming,codingtheory,andthetheoryofcomputing.Themathematicalcontentintheseapplicationsiscollectivelycalleddiscretemathematics.(Theword“discrete”isusedinthesenseof“separatedfromeachother,”theoppositeof“continuous;”itisalsooftenusedinthemorerestrictivesenseof“finit
6、e.”Themoreeverydayversionofthisword,meaning“circumspect,”isspelled“discreet.”)viPrefaceTheaimofthisbookisnottocover“discretemathematics”indepth(itshouldbeclearfromthedescriptionabovethatsuchataskwouldbeill-definedandimpossibleanyway).Rather,wediscussanumberofselectedresultsandmethods,mostl
7、yfromtheareasofcombinatoricsandgraphthe-ory,withalittleelementarynumbertheory,probability,andcombinatorialgeometry.Itisimportanttorealizethatthereisnomathematicswithoutproofs.Merelystatingthefacts,withoutsayingsomethingaboutwhythesefactsarevalid,wouldbeterriblyfarfr
