中南大学《离散数学》课件-第2章（英文版）.ppt

SetTheory

Copyright2007@byXuDezhi2AgendaConceptsRepresentationsPropertiesofsetsOperationsActually,youwillseethatlogicandsettheoryareverycloselyrelated.

Copyright2007@byXuDezhi3SetsSet=acollectionofdistinctunorderedobjectsMembersofasetarecalledelementsHowtodetermineasetListing:Example:A={1,3,5,7}={7,5,3,1,3}DescriptionExample:B={x|x=2k+1,0k30}

Copyright2007@byXuDezhi4SetTheoryx?A“xisanelementofA”

“xisamemberofA”x?A“xisnotanelementofA”A={x1,x2,…,xn}“Acontains…”Orderofelementsismeaningless.Itdoesnotmatterhowoftenthesameelementislisted.

Copyright2007@byXuDezhi5SetEqualitySetsAandBareequalifandonlyiftheycontainexactlythesameelements.Examples:A={9,2,7,-3},B={7,9,-3,2}:A=BA={dog,cat,horse},B={cat,horse,squirrel,dog}:A?BA={dog,cat,horse},B={cat,horse,dog,dog}:A=B

Copyright2007@byXuDezhi6ExamplesforSets“Standard”Sets:Naturalnumbers:N={0,1,2,3,…}Integers:Z={…,-2,-1,0,1,2,…}PositiveIntegers:Z+={1,2,3,4,…}RealNumbers:R={47.3,-12,?,…}RationalNumbers:Q={1.5,2.6,-3.8,15,…}

(correctdefinitionwillfollow)

Copyright2007@byXuDezhi7ExamplesforSetsA=?“emptyset/nullset”A={z} Note:z?A,butz?{z}A={{b,c},{c,x,d}}A={{x,y}}Note:{x,y}?A,but{x,y}?{{x,y}}A={x|P(x)}“setofallxsuchthatP(x)”A={x|x?N?x7}={8,9,10,…}“setbuildernotation”

Copyright2007@byXuDezhi8ExamplesforSetsWearenowabletodefinethesetofrationalnumbersQ:Q={a/b|a?Z?b?Z+}orQ={a/b|a?Z?b?Z?b?0}AndhowaboutthesetofrealnumbersR?R={r|risarealnumber}Thatisthebestwecando.

Copyright2007@byXuDezhi9FiniteandinfinitesetsFinitesetsExamples:A={1,2,3,4}B={x|xisaninteger,1x4}InfinitesetsExamples:Z={integers}={…,-3,-2,-1,0,1,2,3,