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Relation-2nd Yr

The document discusses various mathematical relations, including Cartesian products, symmetric, transitive, and reflexive relations. It provides definitions and examples of these relations, as well as conditions under which they hold. Additionally, it explores the implications of these properties in the context of set theory and mathematical logic.

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mahapatrochandra143
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24 views33 pages

Relation-2nd Yr

The document discusses various mathematical relations, including Cartesian products, symmetric, transitive, and reflexive relations. It provides definitions and examples of these relations, as well as conditions under which they hold. Additionally, it explores the implications of these properties in the context of set theory and mathematical logic.

Uploaded by

mahapatrochandra143
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
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aia PSEIAIEON IND _IEUNCTION | Castesian pxesluch = =a | Let. Ten! “and \3'8" abe Qustnonsvempey seks. The seri ob al) odes pais. \Cd,b)°' such that} aen ond’ bee is Calle) the cadteian — psoduct ae Ser! with ser B and is donot by AX B- xg = §(@b)? aen nivel BF: Bxn = § bh) \bEB A aed? . | Relation = ra 5 y.¢Witd n |p Connection —behveen of 5 amor > angi | eq is farhies ..\ motes and ¢ hits) 1S, ys SALON § ‘Hosband\ andy (Wife iS welarion 2WESNY MEEd Wkuy} NOTE = fies Pe seatio has Some patter’ eat oy Cust y Seation ‘ envolves, fivionw ae 2 tal “fewvon in mrashemasical : ; hap i | 52-9 Numbes ‘pis grorest i i) Line ‘m ok nee i hai Ser A’ A as 1 Retation Wt xt ti | | C debinition ial a A wselotion ‘pe’ bxom ttn in te YnoPhes | ser Y’ is a Suaset oF x % a and fe wbrained | a. unigue. walarion a Statémeitt =hrivelving ae many jis gwen & ye x) and ab a The __) Rk ra fiays aby Ay a, Oy brs || ees OF Relation | slp Libary. Blocion <> Aoirnt | * 0B Belcion Wahid bound, | Leeann is called’ The vselation dekined sbyy(y iy) > : | | i % Ry err) Mey =< Wag 3 ONIN; Quit 10M) xX fs oF 2 Lye. (> ay F2 | | D Rebenive 4, aaa |S SummessiC V— pa | oD FTeansitive Ww) “Sh |. |i ge ye —> Were” Sea | Pp weloction R on : x ie? Laid’ to be | vehlenive Tk “and § Apel) pay oe } MY vex. Ha “fieal tek a aeidda “Wo, set | iby * gous “onswes f yes, the given, \esela tion. + 38) onan AEX => iQHad ER yy (yor | bexs wGbe) GR nyany sed “ka |o- Ce Cue) ER because. aN re ee Rom x ie aia ee oe ibbe: ay oa OWE R = 06”) eRe x ee ¥ ph ic j= fs! the: ee “Jobing, by x Ry aty is odd Yon 2h wed” angwes? bn ; vette Jus ty gor, oso eat Dconsides. an GORE cote Ing 24 iSuch that Ra Ra : ise ode feeds x Q3 | | But, ay ib even fx ao SIAM GESAENED gor taiaion — 9ebh 1 i VMS ig) Mt Odd ns 5 4 OR x, i ta} py , “Selation a a sehew ae selation a ‘Yehined “by, | on OR eyes ORY SD Spool on fe) Gerewve u sey Yous angwes ? 8 ‘ SP considessio gn Seiement) ue 3° dn NM suer- Pot sigs RM ey wool Sx tp, par Now, Mths - 272 y Y , Gag Pp 6K cand? ao, yd joo» tee “ OP 22h too, Oe ed “atk a “£8 WKB yO YH e / DBR ew - “Bede” pi # mah “sep lbidve ant nD se yh ge au mie (Bye meget — guyati af x m8 ; 4 | ee Sh 41 See” tl | mesic ik ‘ang > - om | | a my ey iF Steg Mt, RB cummerie on X_ , . ened yh ak ‘ WRY "es YRX Mos 2d EX ; MY) ER Ss C4; vw a = EI> ) he Relation why ee ¥ eg “Foahes FY ie noe | e _Symmersi > 0 the er, ot dures, dag 2) The SElotio9 — R debined, dy, RS € CabYS Chay SCID), Ch F a, KEL) bY Eso nop» Lyinrlersic”, Betouse. i | pa ER PY cow Ege 00 ‘ (> Q is the mininun syrectaes seidition on X. selarion> e defined * by x Ry beg Chey) Tustity 4 » }- A is) YG 7 : considers ‘ 8 2 Hees woand YN, = yh Now ee tore s ai = tty, 124100 an, ONIN yas LID Ri a i nee why gtx 2 foo Xx C * NM Commute | us) » Ry 2 es YK ot Heme, p 5 gymnepsic? MN by WA GB the setarion R deg nel” o 5 ‘ | * Ry S CHYD vis Ypveniiion Ky’, Lymmersic? i Justify gous answex?” Kune C Se consises ony 2 Alements x andy smn | $0 Ry, we ! ‘ none % Ry a> MY i. even on nN 4 Ry => tt iS en on N* Ths, ay PS Rx Hence, R i egymmersiC: Ge the Baation . R detined.BY, KRY > Cry) 2100. on zy symmetsic 2 Tusnly Goss — answes ? : * yp Now, By => KY 2 100 on a x Ry > —Ch-y) S700 HRY > yous £100 X Ry PgR t ls. por aggnmern 4 Hence, p iii) Teansifive A xeetin = R on , a ser “x > B, Said to be toonctive ik ox Ry and g 82 2 RZ » bow 4 4,2 Ee ech — . Ron x. is -Poansizive 4 eee ain O04) ch Y eoad GRez > ae ors Ree 6a 0m NC4,2) ER =the) ER € elation’ R ” Jpkined By} 3 % Ry SSM 1b bectho® oF gs “ig WO +banshive = selation.. 8) 32 2) The selations. Rs detined 2Y, WRy <=> is gv Bootes oF’ yo is a -+bansitive xelation- Vy $ Ego Dekined the Selotion Ro Byr REE Cay» CaybUyintbrA) 5 -chser? on exrh arb? Peansitive 2 Tustiky gous pnswes? n OY The seration 4s) nox faanisitive’’ because f Cara)’, che)’ eax Caje) er | se Re L Casa)» chyb) 3. On Kz EM rbeI$. Bans Stic) oo Yesy the given welation 18. #sangi-tive- because, (aa) cheb) -€R- ! a me an eyamples of, .selation Which — 15 methe velleuive now mnetsic not fsansitive ? 3) SP the selatlon ~ R_— dakined — bY Re tayo) Chr), Cayb)s Cher d 82 wive an example — oF xelation — which 6S serlewue — but “not. “4ymmersic and —-fisansitive? SP The selaction \"R ~ givén) by) Re § cba), CbybI), CHL) Cay), Chita? BZ pive an erample |b <“Belayibn eohich , *s grmmeric bition not: setlexive dnd ~‘¢sansitive 2 so The welation detined: yy . Re £ Cab) Ub); Card), Cb0), fobld S wive “ap Levanpie ‘BF welation, —eghlebs is «| | +sansitive bu not webtenive (ant « mmetste? (Sh me selorion’ defined hy, as ; Ref, Coa) 2 Cb) 5), 0tyh) I ne ove cans example. ok. av, blarbe wohicn' is’ , welerive and yan eric, «but si nots seaside 3 LZ ne weWeation, Ry. .jgiven bys 4 ee Ref Carady Cbyrb)s lose) ¥ Lab) ay bive ans example. of sedlation,, whic {is Symmetric. , tyansitive. but not .-weblemve.? SE me welogion R ebined i AY . Ref Cod), Chb)i% 9s bows 9 ives an example « oF \selevtlon whith iso Beltewioe, \-pyansitive bub’ net * syne tL? De the. seotion RR, delle By. say Re § aad cbrbly. Ct), caw} hivet an example % seltion ‘which iS * fie bh bymmetsic §=6oas wee at thanovteive 2 iin 8 ata li Such that x Ry. and | QR 2s lop the welations 4p deine’ buy, DORR e Re F Cayahy Chybhy Cesengz” WS x ds, they \weladiond Riv Yebined” by’, S j HRY SD Mey. So WOW On WO ReHERVE' cummesric A and t8an sitive 2 _Sustity 5, jour ansivet ? | ‘soe mea the elation R aye 225} f | : FY XE Yeo son HY. bic" \ Test ie OF eleuie = fx eS Ay oj I | con Consides © gn element we B2 “On! Ne Such | that x RL mvs ah AS F | NOW yw Ry eer ae -Erf00° *" ¥ | We = 52r52 slop 7 : Feel een) a A , >) Bat B2 & 100, 1S: x DP et x £ joo VWllse S — LLL ; 30 RB GS. not. speHenivers | Test So, sgemeteit : I Consides any 2 element. . : fate Ry, S uty Ss | 002. & P Qaot ~s Yin S 100 On N. CP iS Commarea — { | <=> R ‘ tive) | | Thus, Ry id Fe XS yes NH wer | | 59) R is Sot s GE ok pensive os | ensides any +hbee dements Me 60.5% yeases 250 | | Me WR YG => YY > 60 via0 YR = yr2 “Wt2 “> Got 60 But, "U9 -£ too Wt 2 “£5100 So; R's not = 26 450° 70 1.00 “2 oe et 7 ait an A ~ pransitive- AY aay 5 (SEAS 3 : N00 5 ® cOmarey is even wat | “> ua ty “even on 20 Ct i tomeute | \ ae ive oy Sy pn | “5 mRy <=> J RL 5, Ro Fs mer sr * Ad i the selation 7p dekined hy, a Ry MY ib ven OP Dr, Petterive, | gummetsic., and, transite ?' Justiby gous, ee, | so Debined the saad R by, 4 Fi | HW Ry Ko oMty | even ~ on 2/. Jest ok seblewve Ve 37 —Consides .an «Clement | HRY <> UTE SA Yen ot 200 Bur, 2h eve in Bk > WM syguns—HDr cap x Wt buts oes even “D> eK jy -eved 2 < ae 3 J $0) QR ig sehlenive - 54 ny symme tse x Consides 2 element + Such that ey - Let angitiog Consides canyi\ “3 ements | X Rays s doy Re [Nos SD. tty 9 is yeyel < FoR Api, PoMtye 72) ja dif, = im even -on 2 | Ady ut gt. 2 LF 2h) 2, +6 | Me 9 QO amd @ swe fer. | State a Bh + 2k, pt 24 2 2 2h + 2k, and: “Feansitive 2 \Tustiby | yous “answes? wt 2 | wees 2 +. Fae | 2 tt 2 1S leven’? bmi BAS ;. 2? «ke i navy; ake PR | Tus, KRY oA g2rMssn RS x | 5% pe! is -tansitive- ‘ \ - s ‘eds the SAation, az aktined jy wRy er wey is ddd on 2 Reflexive ; ‘pekined® the “wselation R ars : . | xy <> xty” is ele i me eae" Consides an @ement “y' in NOW, Soe achre, = Sax —@ [Bud 2H syiSev eve oh I et | =) win 3S even OF WOK 2 | eK is Mot.cedd y i | PX RE eS HSS | SR is pot weblenile. yoo . | Test ob gyrmpnil — —. encahhys consles any 2 eement = % tothabs ok RY bs Cy Sends MOM, Ry > Wty ie abt on 20 A ge 16 edd°on B* | ot Re 2 Fae a Ris monet xt - Es Ge teomsitive Ay SP | Consides an 3 elements oy Yoaw Zoi in gy Sieh sjcthaty, HRY and Y RB fp ok Hoey ICR <2 boty tis add Sine % | . a hUr 10 f pgke Ke yte Cig. ode on X bee ODM ass abe Bsa aR PIS ee | Add ing 2" OD and »Q .welget 7 \ WE TY Terie 2erl tee tl Sy Bre > Qe Ay t2 > Due > 2 6 KP T2F D U2 | IACKEKLT HhAYD \ > wiz" is even aditea as \ D wrza ds moti Mdeg y 4 ya aN So. ex Re au « 25 Se °s “pie fs nor, peansitive = nathan a op the action = R -debined.. by pave Whe % RY ‘<> % 1S, aN,..ynteqes,..0N 21 ReHenhy | Pansitve and symmersic 2, - Jest » Aous» Answest 5 ct SO Debined the ~ weledion. 2h = san Sothys an * metfers on. A 25h ASt ok. Rebtenive > :— oye Loony a 4 i (onskles an element woo Z : srieh stat sie | WD HRY dS yey is ay an ineges “on 2h A But, WM 20 nSu9 Bat Os en oh cintegés ee is AMG an Wenteyen Ax 2, 50, x RX . ie \y Tet a gee tM Considew ; thet Now, $0, ‘ CPBTC 2 sn fet & “en 8 “Consides a ical cence x Ry ord Gea. és ow. NO RY Fy. 5 can S Stages i Gers, gees yb can imeger ond bat » WE RN!» thay —gyin eke Phe Pee en iL ineges hy USO ON inneeE. ve | | 5, MEL Yn Z Fowl inreges’ 9 OM Zeon i D> “y-2 15.4 can integes antec i i > xX Re BS a eR O™ (Tus ye Ry ny pe =P RE (50k is paansitive + oh Js the vetotion eB dekined « AY tr WRY oS g is & poweo oF “2 1 On : petiedve, tymmersic “and -feansitie”? °Jusriby gmat angwest Sol’ deine the velatio” RB =: : ete -“ gq A pa oF: 2a Tet oe Rebleude + us considers © an! element’ Now; t gh MdRivih poe Buty hy SOE R20 | Bev paras Hoe ase at, HRS melo im B% Suh th 7 « | R j ; Test sii ae 4 ; Bveaae pain oe element %_ and, YM such, Frat oe RYS Be ioa powes of 2 ih ed end * be DP Bae oss Hence, RIS Summessics > * in Zr SUK : ie conides “omy z “dements thet, weg card iy RB wor! gy we a powes 2 > By ae ok D 4, ‘s cond Z pe Thus x2 Mey pe => “he Kener, py pounsitiV@—. 0 OX 6° ee eeuiion Revlebined. BY, Ry > AY IS divisibwe by “v2, Petteriv?, — symmersic and teunsieive? Tustihg a ous angwes? eo 1 DS detine the selfitn RY, _ % Ry WY WS divisibe by ‘ Tet © PRetlenve -— rng 5 | considers gn Element 2M De -Guch phe Ty ene 0 wes : Bibs Game KH REO ez 3a nk j . ce : Dh 1S jvisible ; who ROWE ® sortendde. S we BLE hammers / Consides = any 2 Clement “il ad, Y-he on GK 4 D dot, te diupibe by! Tous, ae" "he LOr pQ t Sy mMePsT Cc. [tee ok Pamir sa DN ns | consides- ony 3 eement> Aap yond BAN &. | uch that» “ye Rye andor BBS : NOP, Ry <> ey “igs givisible “ky = pe SS POR eV O ASMP) YR Hd “ye Ye divisibte ya n OD pee he Yk 27 —@ | padding 9" > ang OQ: HY TAS = MK ANky Dye ee = WlerK) ¥ ktky © 2/ 2. m2, iS divisible” by ‘n’ Soe Re tac ERE Thus, uRy on yee SO) RTs than sitive < ge the wAlariod > R° debined” by; mRy <9). X divides “ys Gn 37 peters Ue» Summer sic” anid -feMnsig ive 2. guctily ee answer > DP Detine the welaion R Ry oe Pa ag x Sep & Midin : ie divides 906 on 27 bor ac peri Cae “t | | WeRY oe! RX ‘ Hence, 4 is 4 50% : sacri est Ok prungigive cold ‘“Consides any 3. elements. X Gy and 2 Ine Such that RY and y RB: ‘ NOW, YX Ry <=> divides (Yo on 27 4 2 aR nl D> Yen »# KEM, —O GRE => Y dive 2 on x i 2 Sze Key Kee bom @ > BF “is 1 © 7 — | Again, 26B kY- 6, Zs Anke “Casing, 0. | Dem divider = ae Dh Re ey a | | Thus,” Ry. ae Y PB D> RES i | canbe 3 an ‘Penelene: I aaling 4 Se EELATION A vlaticn tchich’ ig: sel xNe > “dyomersit and i PRensitve iS cauted.. equivalence — Beatie “DA seorvo js parallel wis on arbi | equivalence —xélatito- ( ae | DA searing equas "0 = ig an= example | ene equivalence — - (1%) € el oy =p. x? ne hay Se pt ig vate Consides ‘anu ahs frat yw pty! Now ar my, , xe Bei is > cnsy) ER “D> cy eR! } 0 POY Pls) 5 MRM ee Heety eis gypinennic: ‘un ciive -

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