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DISCRETE MATHEMATICS Dr. Patitapabana Rath LECTURER IN COMPUTER SCIENCE CENTRAL UNIVERISTY OF ODISHA n ES yy a in fiw The \ameateate | 4 he Ciao Vx iy Pete — Bt) as odd , the s Aare odd. By ew ber Kas « partie lan nu be Bhatn add - Rs as add Yu,4 = > odd , He x? a odd fe p(n) 2% odd, Q(x) > a? a odd. dk Ke poche Aunken Bak oO ddd. Form ig a You, af POU. then Q(x) Pw) fev o pov uelw iS ACK) By UP) Pee anyone de HM arrgrent Prove the Sty alidily oh, te i ae at George doer nob rare eight ys Shan he’ Georg. Pr soy sesect Guay temeg’ “ae dat ap 4ndeet Sei". Yak iF Geory ‘ io Wicee ae m4? oF _ i= ee i ee fey “rng tegp The yore aagunnt wb vealed. Q' ley RG) i x 04 rsh be 6 Exp ser an N@: % hops term 4 Van Whis domad SuReuP oy Muh Onte Mie Every Aabbet Rops- Van (RO) Hm) ) (bp Every anim, fae a re gs Anbbil yx (Rin) A HOD) (ty There enced rabbit Wey Phat Aspe qx ( ko (4) Thin oncsts % animal ah Ae pe > Hl) ) whieh <8 SE iG Rin) A Hx) ) Q. dee Poy) = * ad mives 4 Kr’ (yea Vg) PO) = Every om (vn) (Ag) Poa) rege loves low eo (4+) Cy) Pliny) = Someone Lives (4 (Gy) Peps Semen Aa ver Qype We) Povare F (vy) (3) Ge Eveghedy a Cag y) FOr: C4y C50. POO One “dF On Yhene ay ARE Me loves Ti 5 ee Dennen le Anime baheck 4f ib a @ nabbot and over tae i ~Aome Orne erreayp one Armet-oty ? , bedg + lover by every Jeved ” Bomions twhema @veg- hogs “oe Pom ane Jy We PCy) > V4 J9 PZ) EUzay SETS qy Wx PO, ¥) Fuzzy pevarion > wane Coy 63, FD ; js ee ag “ ugeeeg Parti) Pela) o> Fame? Se yr Chy Poby FG) Pals 4) paca cre, | vray PCy) (by mt of U4) Belay ee F (pre) Paltn a) Hatete) 2° PRO ae a Dawe. T Se eae ass bales sie Ch rat Ye Ly, ¥2, sa eee) i INVeRse OF Fuz2Y¥ RELATION pelt) Para) ¥ (Capp ane Q Let ROY) = fuzzy aelation OH Me (iep oaks, teple tinged ba 2 te™ 1 M,= | on R com os 5 ™ Lot 0.3 le a My a Pg Ait Inverse of R(x ¥) Maar Met = 41/1 os oY 42 )02°0 0-5 OPERATIONS ON Fuzzy. RE LATIONS am EL i ee Oe wf * ES L 4)= x f Pe le, &, (4,4 ) Rus Meet) ys ie Cae Ms ») Qn: facs d= Man (Freep, sa) ) detut Mtevel Ry se deary Set My Ne | rad7, 4 Stary dint ow dingy hs cats ae ate (>| paer> a af i a } a ; sks ‘ : ‘ees os a aa ‘ \ : . 4 ‘03 faa Ee, A ae { 2, 3, Ty s} a wa 4 pee et Cs ag 4 45} . a (i ry oon ea ty Aiea 7 at ic Sum OF Two Fuzzy SETS A ANDB, Say Keel > 49 sseay ' Sopp Cde pine) “preyed. YAcwAND 8.4» end Yanvendwe. Pry to hy ZZ 7 RELATION Ret ‘Ese A 7 ates fe 4 aad BA te 58d A atath \ : fom ke ua eel +f ae ca GEG 5 4 - S| product A XBE EIN UA, oie 1 Af; y Mathemabseay \ R Eee fenpixen, yb, Ot Az{t2} , be {223 hye Axae (1,2), 0,3),0, », rae 337 logins rations hip = ey vRetatin R= fc», Chany, C2, So} ol A wtbala St % at. ; mAdsigh ®. Domews: yz vr aeh and (yg) ER fe Se ge i Dance ASS i fa, rer Renge: — Range = B= HaaiSnesscen A + oh f Y (A.2y f ¢ Range oB. ioe i ‘36 “ad hate oA ra | ‘Bisa f) Be f 1,2 mA 2 (ay 4 e a Ray of R = {23% ate m sj 4 P , 95 5 Trvense Relation? (aca ae Ley: ae roi A « aA * Gy.) a oR 4 (2, et G, > er ohn) ian: A ated bt, Refatom > 1? =f Gyap ren Cy), aay Vead Relation ev E Rote. pep. Properties 4 eae ‘ Re Relation ty v tA ielabten’ OD Refteniver Cay eh fer a! ach. Rete tin : Oo reper Ss ] (oe) GR Peal tdi} 4 ‘ G) Noo Puptintv Relate + (a, ER fre ere ® Symmretse Relatim s ( (eb) ER (Go) Asyamabee * . (wimg ® ~ n> iter ges CinG Ax Rettim * © A. 1 G2 Gm ma ce ye : j, Cease RA (hae Ao * = ab E fv ott o bEA, > (bh GR gor att a, beA ( fi vali 0, BEA. ” @) Transitive Ridin (ase) p Bens c lek mee) ee all 4) LEA A pliabet eeglad ef | Equivel wu Relate: Fence Fulatin ip dk lb eftnive, dyroebnce a brain tives « Pavtccd Ordiv Retatens A Puletem js glentted pee’ A crdev Sulatem if ib tb Aefleneve, antigens And Biansctive. 5 Che Rebtin = Refhentve + Sam mele Oke AE £12, 8t \ Bot ec , cad, 4,24 Ref 01, p (17,2, 20} hase S =f Cp), 04,22, oz) DN! pe ates, 4 C19, G10 20k ae 63, Rosey Read & © Compe stim Of; puedo & pe fU,v, 2,2, &, 2} praRea=t % 44,0, (2,1) ath Rk Geph 4 Relation’ dq Vextecen, 2, Y , eh pale atee pte Nodes ar DeAached Gre fren oe) Fle ex oe 8 Re ic 7) Cae % 2,0} ry CLOSURE ! PS habia ive eect A Hodirs) Fag - AeHlenive i, Somme bac - Dranitive ae Subset aa Called has Defend | herr, { 0.2), 2,3) GY BYMMETRIC | CLOSURE Clodur Cosure ; ‘ 1) 2) (3,29) =f O)2% 3,2, C2) 2D, 30) a 4 we called feu —— dae Ae } : > \ 1.14 “ttm 2), “te 3 (3 ‘ye fe, » G2, C13)} sped Chey ( 1,2), , Dy Ge 2), G2 @2)} fs 4° ee Mepemstooty, pl, Ru R” { G9, (22), GR} a 23, (22, 10) 4) 01,32, (2,19, UP Se (0,9, 2@ 394 4 4 3 a cRoRe 4 ¢u2) (2a), C309 Pa 14 C122, (12, G, 1).@,22 (2,22) (3,13, Gay 227 4 Rue ve At ye xeR|x po od xz2y} us XER|* 3 FH { ee end? ax7-7 ae) a . piso {a bis forrs} r of reR] Xy3 os ax-[225}V4, z= 5 y usner| xg 2 ed arrizes} men |adeca pat a: * 23 -{ xer|x723 Lysbyg vaste! i | pe oat 2} (xer]xe-2 on gx% 42) {xce| n>3 ad n= 7B YU LER | po-2dx i ad ue ti) UG adeR| xer2. OA mete Be NA) {ayy eo. 1 (mp Ye 11 CBA Bs | Re cunnen te RECURSION Relation % (R a) ee 4 3yeees oy a, ai 2 Febbraccee / Tn? 8Tad 4 97% 4 Sy hin TY ES Ot (hy 2785, ‘> dhogie 2 paged for $l @: ky 4&8 4 men, 5, Dll 12, so} pert n Sor: Q: Te Tet? €°22) When Ty 251 fang RR Ted (2/8 So deeener Tye Togttoen (73) Pad Ted Te Tad font Coal fat 4 age il An 224 forst foe fem af aga Cenayt ty Cy = ae \ i 2 32/2 jue tans op epee Kye ape kal ae ; 
