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1、DiscreteMathematics:ElementaryandBeyondL.LovászJ.PelikánK.VesztergombiSpringerPrefaceFormoststudents,thefirstandoftenonlycourseincollegemathematicsiscalculus.Itistruethatcalculusisthesinglemostimportantfieldofmathematics,whoseemergenceintheseventeenthcenturysignaledthebirthofmodernmathematicsandwast
2、hekeytothesuccessfulapplicationsofmathematicsinthesciencesandengineering.Butcalculus(oranalysis)isalsoverytechnical.Ittakesalotofworkeventointroduceitsfundamentalnotionslikecontinuityandthederivative(afterall,ittooktwocenturiesjusttodeveloptheproperdefinitionofthesenotions).Togetafeelingforthepower
3、ofitsmethods,saybydescribingoneofitsimportantapplicationsindetail,takesyearsofstudy.Ifyouwanttobecomeamathematician,computerscientist,orengineer,thisinvestmentisnecessary.Butifyourgoalistodevelopafeelingforwhatmathematicsisallabout,wheremathematicalmethodscanbehelpful,andwhatkindsofquestionsdomath
4、ematiciansworkon,youmaywanttolookfortheanswerinsomeotherfieldsofmathematics.Therearemanysuccessstoriesofappliedmathematicsoutsidecalculus.Arecenthottopicismathematicalcryptography,whichisbasedonnumbertheory(thestudyofthepositiveintegers1,2,3,...),andiswidelyapplied,forexample,incomputersecurityande
5、lectronicbanking.Otherimportantareasinappliedmathematicsarelinearprogramming,codingtheory,andthetheoryofcomputing.Themathematicalcontentintheseapplicationsiscollectivelycalleddiscretemathematics.(Theword“discrete”isusedinthesenseof“separatedfromeachother,”theoppositeof“continuous;”itisalsooftenuse
6、dinthemorerestrictivesenseof“finite.”Themoreeverydayversionofthisword,meaning“circumspect,”isspelled“discreet.”)viPrefaceTheaimofthisbookisnottocover“discretemathematics”indepth(itshouldbeclearfromthedescriptionabovethatsuchataskwouldbeill-definedandimpossibleanyway).Rather,wediscussanumberofselecte
7、dresultsandmethods,mostlyfromtheareasofcombinatoricsandgraphthe-ory,withalittleelementarynumbertheory,probability,andcombinatorialgeometry.Itisimportanttorealizethatthereisnomathematicswithoutproofs.Merelystating
