Relations Introduc3on to Rela3ons. 8.2 Proper3es of Rela3ons 8.3 Equivalence Rela3ons. mjarrar Watch this lecture and download the slides

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1 Mustf Jrrr: Lecture s in Dcrete Mm6cs Birzeit University Plestine 2015 eltions 81 Introducon elons 82 Properes elons 8 Equivlence elons mjrrr Wtch th lecture downlod slides Course Pge: More Online Courses t: hfp://wwwjrrrinfo cknowledgement: Th lecture bed on (but not limited ) chpter 8 in Dcrete Mm:cs with pplic:ons by Susn S Epp ( rd Edi:on) 2 1

2 Mustf Jrrr: Lecture s in Dcrete Mmtics Birzeit University Plestine 2015 eltions 81 Introduc-on el-ons In th lecture: qprt 1: Wht el-on qprt 2: Inverse elon; qprt : Grphs; qprt 4: n-ry elons qprt 5: elonl Dtbes Wht eltion? LessTh 10 Equls 100/10 Hge 20 MoreTh FrOf li FriendOf dm Mustf Smi FriendOf EnrolledIn Knows Tes EnrolledIn Workst DMth2 Birzeit University LoctedIn LivesIn mi Birzeit I 4 2

3 Wht eltion? Definition Let B be sets (binry) B subset B Given ordered pir (xy) in B x relted y by written x y if only if (x y) in x y (xy) 5 Person EnrolledIn Course li S Lin I DMth lgo DB Domin Co-domin 6

4 Person FriendOf Person li S Lin li S Lin FriendOf I = {(li S) (Sli)} 7 Z 1 2 LessTh Z

5 Z 1 2 E Z Define E Z Z follows: ll (m n) Z Z m E n m n even 9 : rel-on on Power Set Let X = { b c} Then P(X) = { {} {b} {c} { b} { c} {b c} { b c}} Define S P(X) Z follows: ll sets B in P(X) (ie for ll subsets B X) S B h t let my elements B Is {b} S {bc}? Is {} S? both sets hve elements {} h one element h zero elements 1 0 Is {bc} S {bc}? {b c} h elements { b c} h three elements 2 < Is {c} S {}? both sets hve one element 10 5

6 el&ons Func&ons Definition function F set set B B stfies following properties: 1 every element x in re element y in B such (xy) F 2 ll elements x in y z in B If F function B we write If (xy) F (xz) F n y=z Y = F(x) (xy) F 11 Let = {2 4 6} B = {1 5} Is function B? = {(2 5) (4 1) (4 ) (6 5)} P 12 6

7 Let = {2 4 6} B = {1 5} Is function B? ll (xy) B (xy) S y=x+1-1 Mustf Jrrr: Lecture s in Dcrete Mmtics Birzeit University Plestine 2015 eltions 81 Introduction eltions In th lecture: qprt 1: Wht el0on qprt 2: Inverse el8on qprt : Grphs qprt 4: n-ry el0ons qprt 5: el0onl Dtbes 14 7

8 Definition Inverse el-on Let be B Define inverse -1 B follows: -1 = {(yx) B (xy) } ll x y B (yx) -1 (xy) 15 Let = {24} B = {268} let be divides B: ll (x y) B x y x y x divides y Stte explicitly which ordered pirs re in 1 drw digrms for 1 } = {(22)(26)(28)(6)(48)} 1 = {(22)(62)(82)(6)(84)} 2 B 2 2 B Describe 1 in words: ll (y x) B y 1 x y multiple x 16 8

9 Person EnrolledIn -1 Course li S Lin I DMth DS DB 17 Person FriendOf -1 Person li S Lin li S Lin 18 9

10 Inverse el/ons in Lguge Wht would be inverse following s in Englh SonOf -1 =? WifeOf -1 =? Workst -1 =? EnrolledOf -1 =? PresidentOf -1 =? BrorOf -1 =? SterOf -1 =? 19 Mustf Jrrr: Lecture s in Dcrete Mm0cs Birzeit University Plestine 2015 el:ons 81 Introduction eltions In th lecture: qprt 1: Wht el0on qprt 2: Inverse el0on qprt : Grphs qprt 4: n-ry el0ons qprt 5: el0onl Dtbes 20 10

11 When defined on set Insted digrm c be mo so becomes Insted seprte s so it becomes Insted representing representing seprte sets sowhen itit becomes representing seprte defined on set digrm c be mo When becomes defined on setdrw c be defined on set digrm bemodified modifi so When itbecomes Insted digrm representing seprte s represent only once s represent only once drw c reltrss s represent only once drw so it Insted representing seprte its becomes seprte sets soso itwith becomes ordinry Insted representing representing seprte setsor s represent only once drw s ordinry ordinry digrm with only digrm s with digrm s represent once drw s represent only once once s represent only drw relted reltr s with ordinry digrm 11/25/18 ss with ordinry swith with ordinry digrm digrm ordinry digrm ll ll s in ll s x y in s xx yy in ll ll s s xx yy in in ll s x y in ll s x y in re yyy re xx yyy y)y) y) re xx x (x(x (x re xx y x y (x y) Grph re re x yyelon y)y) y) re xx xyy y (x (x(x relted relted itself itself loop loop drwn drwn extends out If relted itself loop drwn extends out go If If defined extends out When on set digrm If relted itself loop drwn extends out If If relted itself loop it drwn drwn extends bck it bck it beitself modified so becomes bck If it reltedc itself loop extends out relted loop extends ouut goe go bck it it bck bckitit ll s x y in bck re x y x y (xy) Grph eltion mple 816 Grph eltion 616 Grph eltion Definition 16 Grph eltion Grph eltion ple 816 Grph eltion eltion e Grph Let = ={ { define follows: Let 4 =4{ } define on llll x yx y y Let }8} define on on follows: llx set follows: Let = { } define on follows: ll x y LetLet = { }8} on on follows: y Let ={ { 455 8} define define follows: ll = define on follows: llxll xy yx xx y) x yy y 222 (x (x y) (x y) xxxxit y 2 (x y) importt dtinguh clerly y 2 (x y) y 2 (x (x y) y) between set on Drw grph Drw grph Drw grph Drw grph which it defined Drw grph grph Drw Solution becuse ==000 0Thus re Solution becuse becuse = re lo Solution 0000== = Thus Thus re Solution becuse = 0 = 0 Thus re Solution becuse loop == Thus re Solution becuse = 0 Thus re loop lo itself Similrly re itself itself so itself Similrly re loop loop itself so for itself Similrly re loop itself itself itself itself itself Similrly re itself sos itself Similrly re loop 4 itself 5 itself so forth itself Similrly re 4 itself 5 itself so for difference integer itself 0 difference integerwith withitself itself difference difference integer with itself difference integer integerwith difference integer with itself 00 == 222 = ( 1) nd becuse 5becuse 5 becuse = 2 ( 1) nd becuse 5 55 = ( 1) nd becuse 55555= nd becuse = 22= 5becuse becuse 1) nd 55 becuse 5 5 5= 555becuse becuse ==222 ( ( ( 1)1) nd becuse =1Hence re 555 The 2Hence 1 Hence re 5 5 The oth 2 22== re The Th = re The o The = Hence 1Hence Hence re = re 55 oth ss in in grph shown below re obtined by reoning grph shown below obtined by similr reoning s inin below re obtined by similr reoning Let =grph { } definere re on similr similr reoning s grph shown below re obtined s in grph shown shown obtined by similr reoning s in grph shown below re obtined byby similr reoning follows: ll x y x y 2 (x y) N-ry eltions eltionl Dtbes N-ry eltions eltionl Dtbes N-ry eltions eltionl Dtbes N-ry eltions eltionldtbes Dtbes N-ry eltions eltionl ry s form mmticl mmticl foundtion foundtion l dtbe ory binry NN-ry s form mmticl for l dtbe ory N -ry s form foundtion for l dtbe ory -ry s form mmticl foundtion for l dtbe ory N-ry s form mmticlfoundtion foundtionfor l dtbe ory bin bin NN -ry s form mmticl for l dtbe ory ubset subsetcrtesi Crtesi product sets similrly y y ubset Crtesi product sets similrly n-r n-r sets similrly n-r y reltio ubset subset Crtesi Crtesi product sets similrly n-r yyreltio subset Crtesi product sets similrly n-r y product sets similrly n-r relt subset Crtesi product n subset Crtesi product n sets subset subset Crtesi Crtesi product product n n sets

12 Mustf Jrrr: Lecture s in Dcrete MmEcs Birzeit University Plestine 2015 eltions 81 Introduction eltions In th lecture: qprt 1: Wht eltion qprt 2: Inverse eltion qprt : Grphs qprt 4: n-ry eltions qprt 5: eltionl Dtbes 2 N-ry eltions EnrolledIn(li Dmth) EnrolledIn(Smi DB) Enrollment(Smi DB 99) Enrollment(Smi DB ) Enrollment(Smi DB F) ( 1 2 n ) Binry (2-ry) Ternry (-ry) Quternry (4-ry) 5-ry n-ry 24 12

13 N-ry el+ons Definition Given sets 1 2 n n-ry on 1 2 n subset 1 2 n Thespecilces2-ry-ry4-rysre clled binry ternry quternry s respectively 25 Mustf Jrrr: Lecture s in Dcrete Mmtics Birzeit University Plestine 2015 el7ons 81 Introduction eltions In th lecture: qprt 1: Wht el0on qprt 2: Inverse el0on qprt : Grphs qprt 4: n-ry el0ons qprt 5: el7onl Dtbes 26 1

14 el'onl Dtbes on follows: (1 2 4) ptient with ptient ID number 1 nmed 2 w dmitted on dte with primry dignos 4 eltion Ech row clled tuple Ptient ID Nme Dte Dignos ( John Schmidt thm) (57429 Tk Kurosw pneumoni) ( Mry Lzrs ppendicigs) (00852 Jo Kpl gtrigs) ( John Schmidt pneumoni) (24488 Srh Wu broken leg) ( Jml Bkers ppendicigs) 27 eltionl Dtbes on follows: (1 2 4) ptient with ptient ID number 1 nmed 2 w dmitted on dte with primry dignos 4 eltion Ech row clled tuple Ptient ID Nme Dte Dignos ( John Schmidt thm) (57429 Ø Notice Tk Kurosw Tbles in th pneumoni) wy re ( clled Mry eltions Lzrs ppendicit) (00852 Ø Informtion Jo Kpl sred in th gtrit) wy ( clled John Schmidt eltionl Dtbe pneumoni) (24488 Srh Wu broken leg) ( Jml Bkers ppendicit) mjrrr 2015 mjrrr