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Note for Discrete Structures - DS By Ktu Topper

  • Discrete Structures - DS
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  • APJ Abdul Kalam Technological University - KTU
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Text from page-2

B={x: 1≤x<10and( x%2)≠0} I fanel ementxi samemberofanysetS,i ti sdenot edbyx∈Sandi fanel ementyi s notamemberofsetS, i ti sdenot edbyy∉S. Exampl e  −  I fS={ 1, 1. 2, 1. 7, 2} , 1∈Sbut1. 5∉S SomeI mpor t antSet s N  −t hesetofal l nat ur al number s={ 1, 2, 3, 4, . . . . . } Z  −t hesetofal l i nt eger s={ . . . . . , −3, −2, −1, 0, 1, 2, 3, . . . . . } Z+  −t hesetofal l posi t i v ei nt eger s Q  −t hesetofal l r at i onal number s R  −t hesetofal l r eal number s W  −t hesetofal l whol enumber s Car di nal i t yofaSe t Car di nal i t yofasetS, denot edby| S| , i st henumberofel ement soft heset .I fasethas ani nf i ni t enumberofel ement s, i t scar di nal i t yi s∞. Exampl e  −  | { 1, 4, 3, 5} | =4, | { 1, 2, 3, 4, 5, …} | =∞ I ft her ear et woset sXandY,  | X|=| Y|r epr esent st woset sXandYt hathav et hesamecar di nal i t y , i ft her eexi st sabi j ect i v e f unct i on‘ f ’ f r om Xt oY.  | X|≤| Y|r epr esent ssetXhascar di nal i t yl esst hanorequalt ot hecar di nal i t yofY,i ft her e exi st sani nj ect i v ef unct i on‘ f ’ f r om Xt oY.  | X|<| Y|r epr esent ssetXhascar di nal i t yl esst hant hecar di nal i t yofY,i ft her ei sani nj ect i v e f unct i onf , butnobi j ect i v ef unct i on‘ f ’ f r om Xt oY.  I f| X| ≤| Y| and| X| ≤| Y| t hen| X| =| Y|

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Ty pe sofSe t s Set scanbecl assi f i edi nt omanyt y pes.Someofwhi char ef i ni t e,i nf i ni t e,subset , uni v er sal , pr oper , si ngl et onset , et c. Fi ni t eSet Asetwhi chcont ai nsadef i ni t enumberofel ement si scal l edaf i ni t eset . Exampl e  −S={ x| x∈Nand70>x>50} I nf i ni t eSet Asetwhi chcont ai nsi nf i ni t enumberofel ement si scal l edani nf i ni t eset . Exampl e  −S={ x| x∈Nandx>10} Subset AsetYi sasubsetofsetX( Wr i t t enasX⊆Y)i fev er yel ementofXi sanel ementofset Y. Exampl e1  −Let , X={1, 2, 3, 4, 5, 6}andY={1, 2} .Her esetXi sasubsetofsetYas al l t heel ement sofsetXi si nsetY.Hence, wecanwr i t eX⊆Y. Exampl e2  −Let ,X={ 1,2,3}andY={ 1,2,3} .Her esetXi sasubset( Notapr oper subset )ofsetYasal l t heel ement sofsetXi si nsetY.Hence, wecanwr i t eX⊆Y. Pr operSubset Thet er m“ pr opersubset ”canbedef i nedas“ subsetofbutnotequalt o” .ASetXi sa pr opersubsetofsetY( Wr i t t enasX⊂Y)i fev er yel ementofXi sanel ementofsetY and| X| <| Y| . Exampl e  −Let , X={ 1, 2, 3, 4, 5, 6}andY={ 1, 2} .Her esetXi sapr opersubsetofsetY asatl eastoneel ementi smor ei nsetY.Hence, wecanwr i t eX⊂Y. Uni v er sal Set I ti sacol l ect i onofal lel ement si napar t i cul arcont extorappl i cat i on.Al lt heset si nt hat cont extorappl i cat i onar eessent i al l ysubset soft hi suni v er salset .Uni v er salset sar e r epr esent edasU. Exampl e  −Wemaydef i neUast hesetofal lani mal sonear t h.I nt hi scase,setofal l

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mammal si sasubsetofU,setofal lf i shesi sasubsetofU,setofal li nsect si sa subsetofU, andsoon. Empt ySetorNul l Set Anempt ysetcont ai nsnoel ement s.I ti sdenot edby∅.Ast henumberofel ement si n anempt yseti sf i ni t e, empt yseti saf i ni t eset .Thecar di nal i t yofempt ysetornul l seti s zer o. Exampl e  −∅ ={ x| x∈Nand7<x<8} Si ngl et onSetorUni tSet Si ngl et onsetoruni tsetcont ai nsonl yoneel ement .Asi ngl et onseti sdenot edby{ s} . Exampl e  −S={ x| x∈N, 7<x<9} Equal Set I ft woset scont ai nt hesameel ement st heyar esai dt obeequal . Exampl e  −I fA={ 1, 2, 6}andB={ 6, 1, 2} , t heyar eequalasev er yel ementofsetAi san el ementofsetBandev er yel ementofsetBi sanel ementofsetA. Equi v al entSet I ft hecar di nal i t i esoft woset sar esame, t heyar ecal l edequi v al entset s. Exampl e  −I fA={ 1,2,6}andB={ 16,17,22} ,t heyar eequi v al entascar di nal i t yofAi s equal t ot hecar di nal i t yofB.i . e.| A| =| B| =3 Ov er l appi ngSet Twoset st hathav eatl eastonecommonel ementar ecal l edov er l appi ngset s. I ncaseofov er l appi ngset s−  n( A∪B)=n( A)+n( B)−n( A∩B)  n( A∪B)=n( A−B)+n( B−A)+n( A∩B)  n( A)=n( A−B)+n( A∩B)  n( B)=n( B−A)+n( A∩B)

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Exampl e  −Let ,A={ 1,2,6}andB={ 6,12,42} .Ther ei sacommonel ement‘ 6’ ,hence t heseset sar eov er l appi ngset s. Di sj oi ntSet I ft woset sCandDar edi sj oi ntset sast heydonothav eev enoneel ementi ncommon. Ther ef or e, n( A∪B)=n( A)+n( B) Exampl e  −Let , A={ 1, 2, 6}andB={ 7, 9, 14} , t her ei snocommonel ement , hencet hese set sar eov er l appi ngset s. Ve nnDi agr ams Venndi agr am,i nv ent edi n1880byJohnVenn,i saschemat i cdi agr am t hatshowsal l possi bl el ogi cal r el at i onsbet weendi f f er entmat hemat i cal set s. Exampl es Se tOpe r at i ons SetOper at i onsi ncl udeSetUni on,SetI nt er sect i on,SetDi f f er ence,Compl ementofSet , andCar t esi anPr oduct . SetUni on Theuni onofset sAandB( denot edbyA∪B)i st hesetofel ement swhi char ei nA, i nB, ori nbot hAandB.Hence, A∪B={ x| x∈AORx∈B} . Exampl e  −I fA={ 10,11,12,13}andB={ 13,14,15} ,t henA∪B={ 10,11,12,13,14, 15} .( Thecommonel ementoccur sonl yonce)

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