| Number Aystim — ao Decimal number db Binary number GHD Octal nurnbasy GY) Hexadedmad number >» Convevign t- * Decimes +o _binow Convent ce fe (22), = Gs al 32 fo 16 |o fo aa ° B, = (loooee), 0 ——— I at 4 = | BI | I } @& (43.t25),.= (1) aie ait¢ tof 25x22 = 7 250%2 = +250 + Soo —>o — jo bo bo Jape po Riis To {CON 1,000 —7 1 = (lelout. ool) G& (%),= (1), 5o 23 Jo typ is Sid ea a a bobo = (ro),— #* Decimal -to octal $- 2 (039),= Os SPER] -emL & C®,.= Us [533] 15331 _ ql és [5 %{ 3 12 = (1025 | a te A ——— * Decimal to Hexadsdmall Decimal “to Hexadedmes. & CDe= Cr ie 349 {6f 23 [0 Ewe) (178), eee ae oe mal _Conveusign. <- 2 (CoVCDnG))ic GQ @lol) = 0) a 2 10, (2 Onn 1x2 Ox2'+1x 2°) 7 NT DRO 1.97 > (b+or4to+1 alee =@, zh a= = B42, le ea veag HAH = We (Fad? xb! + SH) Qad*+ 4x0! + 6x99) = 4W9+244+5 Cee = GED a = (166), a (Ba8%,= CLI. Azto eet po G {ha62-+ fox lg? + We 1g! + lox les) = (44902)0 Se 4 Binary _-tp__Octoll ce & (Monod, = (1s ge oe LIONS (ss) ae + octal to Binary oS. G L= ©}, 2B =% & (sua = O), iN ier ee Oe => (leiiovole), oe Binosy +p Hexadutmal 2 Q (onod,= 0, Ke G(Uuowo + (GD), eaeSRO moe On asy_ o = ee) @ (Sea), = Oa i 5 62) T Roo ona > (o1o110t1 001%), a z tr Octal +o Hexadedmal {- Q Gry = We 6 pee Fixst we Convert fm bPmosy Ram fm Hexadedimall, 642), Gyn J 12 jo O10 (liotee 10), Haxadeeimall p20(I}0l00 O19), 4 @ A XD, a 4 Hexadsdmal tp octal t & (co), =(), @c),, re eT Alojiyoo), x 190) ey, = a. race @), oOlv ilo = (lou), —_—— ————de Numbow wit di ffewnt Bare Cradix Co) ocean eel seeaura Bare.e | Bonet) oO a 1 i" | cain 3 3 | qy emacs 5 5 rl 6 6 | = an lo % u 93 12 lo =A oe el 14 [A =e is Eee 14 IL 10 I64e fb 1b She 15 Wty IF 15 =F + Complement of radi Chasdy 3- © Dedmal C1) @ Binary (2) Ls 94 Complement LN Complumant © 10! Compliment Ls 2% Complument © octal (v) G) Hexadecimal (14) Ly ay CompLumant Lo 15's compl & 94 Complement > 16% ComplomeFr Using 15 Complement G (toro!) — (ololo)=? m " 15 Comblement of A Clolo = lojol tan, nt 1K Complemad af N lolol +tolol Bey PAS — & loro) = ( joo!) = 2. 7 i VA Complement of N = Ol0lo0 M+'4 Complemest of N = @lale 401010 | o100 again 15 Complement of Sum =-(lon) dam tamt- note er ae ae = a important mote = > The prow of Finding 1/6, dy Gnd 15'5 Com blemends Will semair Aare. | * “The prvans af finds | Pay aes F ee "a OA, B® 16% Cam p lerrerd 4 Using’s 2's Comblemenb 2ig e1o!0)— Ciotoi)= 7 Vs Complement af N = ololo os Mat 22% Comblemed = I's +1 | = Olon M+ 2 Comblmedt of w = Olola Oot RO Corey ie Ove numbex Ao, agaim, 2% Complerext of Lum 184= Soloro ah = Atl = —(oten) a (lotor) — Coteio) =9 ™ TN 1% Complement of N = folol 2 Comblyment = I%+l= lol M +25(omplenent of N = lolol + totio Notows | ae (CRY ble discard Cosy 3. AF) \ 3 will ba final angus. 97 = (lon lugs Using 2'3 Complement , Aolve it — © (Mloooe) — ( Joio) =? M N = Ololo 2'5 — Comblement of N= Us Comblemed of 19 Wal= Oloti +Rar, M424 conblomedt of W = Iloooo oo1O il te ‘i| "A Complement ef Aum= 0001 00 2's — Complemend = Ors rates Oocolo) *A = = (ovolot) 2 Using 10 Complement , Solve it 7 + (9742) - (Gu) =2 ™ N 9% Comp Lumet of =e 9915. — 064) (ow complannack Beraraieians = 344 Bae 9359 M+ [9's Comblemert af ND = 9442 +9359 (D910! Comy im FRA We discusd Comey & o cy * (64) —(s5u2)=2 ” © 34 complement ofN= 9993 —9F42 Pay Ran, ieee M+ 10% ComHemercd ofN = oé6ul 025% aes IDA Comblamad= 92+) 025% nO Coy ic. Cue nwmben again Q'5 Com plement af parame? 3} 20 10% Complament = giod+! Nrausn = -(9101) ——ae CODE %— > A Aytem of rebresentation Of Numeric, Alphabsts or ssat Charadou Pr ; Apec eo Chanadin pf bin asy fon for Prowning and trammissiion wimg Digital techni quus : > Tiybe_of Code © Binoy Code 2 The numbex Aypstom LWitR bose 2 1% Known On bimaryy Code © Thae aw O amd. © &ha uel g fetid Code, £s (%o= Clove), j Coded decimal * also Known os “BCD Cod’ > A Code or we repos emting g dadmal mumbow fn Which each decimal digit Ps sefmasenttad by 4 bib bin ony Code - , also Known as motina) BCD > Decimal digit (0-3) an rebrrented by Rix mectinad bimary mumbsr equivoleits using UY bit. > Each decimal digit of duefanad mumbir f4 rebnesented by HRs Ubit code fmdividually. This cods i also Known as -“BH-21" Code or Ale bly BCD. Ths fs also a dacightid code &. Decimal number BD_Cose_ oO 0000 ( Ooo 2 Colo 3 oOoll 4 Ol 09 & ol ol 6 o110 q oll 8 1000 e) lool aaOO S¥ Bp cod of (12) YS (2) a (cel o0!2) LS cot 001 ode) @ Exuw-3 code (Be0+3- excess 3.60 | 2 to TR bf © A Bed code fowmed by adding 3 a 7 Foo equivalent of fs decimal muimber — i eaR decimal digit th obtained by adding 2 +, 4R matured «BCD code of Digit a. 10s cect Ont) oes 23 ghg a FIR Vole = (oN oon Wo 10s) 1 olll Oo @ Gray Codes A code tm which only ome bit chang, betuawm — Aucunive awmber Ee (iMHIS), L 1100 (111000100) _ @ Octal codes A code fm wohicR each group of Re bit Atooding fom Lsg pec by ih equivalent octal digit. ipsa Se Cll) iG ee ay | Hexadoctmal Code t- g =a, OS *~ A metRod of ‘i fm hic eaR group of 4 bit shuns pd ia excl 2% (ello otto), = (1), ° [g= 2® = ©,Binary Code 2 Binary ] RGB, 8, Q000 Ooo) Colo ool) Oltoo ° ° !ol ° tio Omit) looo | lool lolo toi t100 flo llto tay |} ©9000 Ooo} Ooto Excum-Fcod | roy BEE | OCR ool C0v0 O100 eco! Olol eo fe) i} j) (0) O10 © iii) Oho looo o1n lool Olotl \oto 100 You lloo 'loo lo} tii io lolo lou (V5 lool re Lovo 17 g >eburert — decimal mumboy tn (0 cliy Binosey Code BCD code Th) Gxuss~3-code iv) Gray Code Octal code Hexadecimal code © bincwy tore wing — 396 © 4o3¢| om 7 O Ye = Glow), o. be e- 2 Goiwom) | @ | +33 4‘ Now > Colo tole) ) y=) @ woAhon) =¢ @ gin Agi. a 7 fe (1oMo) 2G. ne © 3% 36), = Contove), O ax Rasch Es ool ‘ pal bacelio (Cot Joo! ono, @M 356 pean f : x (oor x Keclize es!) {10 WN)008) _() one tiee @) 0 ® Cotinoos) (one els dul tol =(6FO (#8 (I1ooe100) 7 € = © 409 ( (Hoge), = ( [000000000 000), GD Hos ont Lk Sono = (lec covefvoj 019), @ 4096 +330: as ; RgS a, > OM con tiocioe) onl oon Noo ® (io 000000 00000) J Wid, tooo ; eee ‘All p00 090 069008), =@) —a2—< lo © zd{looooee gocooy), = ws ( oD), (io ooo) aae Hamming _Code Hamming Code tf used for evar diction And Corredion ,9¢ ts olso used Conebt of Addition Parity, bits. (2's aa Ret Where = pani ty c eee ee m= menage bite hengl® of Hamming Code = m+p Position of Hamming Code = 2" (4 1,.2,4,8, 16...) @ G Bac caters BCD Even pasty odd paxity DcBA e BDCGA P Sp C8 FA (CL ORCTOR Om OMONG IG! a> eOma oo OOOl=> 1 Oo =p Og0 © © | © | OF Om O) a OU© © «eG Ooli- ogoot1 3 eC} TOmmian © | oo-= io (too ea © © | oH © | O lisp Ome Out =e t @ il ot (One tO aes © ©| | © >pr (© Ao) © i liq? Ori -—- oom 10 00> | lI ©8o = 0 \omoro: loOl-> © 1oot uct git aimowontfor Oll0 2 Wrimg Gy Quer Detinnne $e hamming Code ponity ad Ollo oud > may > o> parmel Q°2 p+5 P=! a2 6 (PH Pe2 (Ce) 3 G2 Cvabdd) [P= }> @e,% : Longi® of Hamm Cods= mp = U49= 4 Com = 2r(1,24,8-) P,P > position Of Hamm > Hamming Cod, ees 4 8 Ct Pie aha te} (Py et (ee) O-» g) iB p, o> o Oo oO oa, ° ° ' <> ° I ° oF ° \ ! a ee ( O + 0 ones, { Oma oe ! ( OoFor Ps (4, 5,63) Ser _ chick. 4564 fm 11. fox eum ®,=8) Qu Find hamming Code using Even Ponity sc e (Ho foo) m= + ormin ge gcocl yaa Oona > 2P= p+d if 24> 12 Ve>12_ Cvoltid) —> Pi Pa, Pa, PR hong tf of Hamming Code= 449 = 11 7 position af Homming Code : 2°(1,2.4,0, W--) > PLA Py C0 PL eee, ° ° ° ° ° I fo} ° ° | 2 ° oO ( ° 3 Oo ° I i 4 ° | ° ° 5 ° | ° 1 6 O l { 0 3 ° I \ | 3 \ Oem ome: g { oO Oe aa lo \ ! © o I { ( © fj ae for Pee (ays asa) for check » | 3 5) eet Sa , ia jagomel for aun> foe BC 8,910"), To chk: ==> a 9 Piet fe fF oo Cl uO Hamming Cods :- O10 lollool Ay + Logic operation :— 2 > Baric logic operation <—> ts > ES Univeral Logic Opevat? gq, ———> NAM a oe Apedah bogie Open —> ev_og FeMag lo o |S ee peak SoP form (Aum of produc) POS fern Gn SOP form Input conti]. gn j famed of Sum) oe POS fosm Input Combination -mation i+ token Covretpand| 0 5 -rq to Lohich outpub.iga] ' taken Comtpending to 4 Which output fs 0 AGliaa ol ® AlE Ila ete \ a @ Aymbe iS } F 8 @ Auitching Civuitt ® Butf table aa Vv ® CG) F © Algebytc Function r a Jn Pos @4A). (848): 48)—————— SS i rs * OR OPERATION ea Coe 3B © Symbol. a 8 oy pe vottehie Chreus: Brey al « © Bw table @ ae ik : A} BLE. —sTofo 8 a aie | F ronal Ns Te ntatar © Algsbytc_finction [Fase Teese Not OPERATION 2— (£C-F404) also Known os * INVERT, @ Symbo a——_f>—_ F © he tel @ Auoitching _Cireult Ale ay t A ilo “@y * G © Algebrte function art * Buffer OPERATION (34 Inoue 32, am atid of ® Aymb Whilr provide outpud), s—[>— F D) Tule tab © Mane fation Ahe + gtx NAND OPERATION 2 (c¢- 4400) (Wand = aNp-rNoT @ Aymbol mo . ——f>»—_-F = i © uf table SEEM = © Auwitehing cyst o|o 1 o;t i) I ° ! Ue ee) Vs GIF © Algebyte_fundion mn © Aymbol => NoR OPERATION 3— (Lc. 4402) {il @ Austtehing chrait vO @)F| SF Speco Logic open ott *% €X-OR OPERATION, S— @ Symbol = Aeb * €X-NoOR OPERATION a. @_ SymboBooleam _Ederd * Dual +Reorem — @ eal 4-0 — OR logic wb GI) OR Lage —> AND logic » 2 Demorgans AR rem — AND Logic Demorgan's See Ce oe the Line. UW) AyBFT = ABC Gd) BE = T4B4T ae OR. AND. <> A+0=A asi=) aiy WEA 1-+AraglRing = | Aili> AFREI # A@O=A vy Aoo=A [Aine a | <> AOA=0 <4V> A@A=0 “vi ne . a Rei a> Aol= A peta Aon =| (A+80) +8) (aro) fom RUS (+8) ( A40) = AA+Ac+ BAL BC = A+AC+ BA+BC = A(I+ C+8) +86 Be = A.1+BC= A+Qc age Oe ce > AB+Ke = (ic) (HB > ACA4B)= A > @A)A40 =AC+Eg > DERG = FA (AiG) = AH > Ab Re 4Qe « kere ES > oe 7 69 R40) (x)= (MEO AB+AB= A > (A+84B) = A _ o°°° TTT. _—>(J Boolean algebra @ K-Map {-( — Tabulor metfod ( Qume Mack huky mtd) * Boolean algebra - we +Rought dom't con Condition {4 e Aolved boolean algebra | Edamtity 4 to rremomtar, Expumfon. cam be oniatenttoston, tA only fin SoP fern © heart mintmixatton 4 no amy Guoanty Qurs- Atmplify following O Fluxyz)= ge + Dre + Wye + Wyz >= =(g4 WH +N nyt wy) > 2 [5+ + (R41) + wy) 4 eas 1B (G+8). Gey) +9) > 2 (9+ W(xry+wy) Sz [o+ WX + y(wsis) Sz [gy +Wi+g) Sz [7+ 07x44) > Z- (+3) + Wx] i, () Yason= ( aBcbene > ABCD + ABTD >RBD G c4+T) a oferta! } > ABD ae= eae (ii) AE + aac + AT +057. ElAra) + A(B4BO T+ A.(a+B) (4B) e4 A.C ce) 3) T+ ACtAB = =F + ac(edd > T+ At ae i = (€+a)@e) + AB T+H +AB +A 4B) > T+A ae ad (Rj +2) +z +7 + W2=7 / >[(E4D ay) 424ayt wz “ll zea ZG + 28 eG 44 ny twee / ie ies a ary = / ZAG +R + thy) + wy +a YG / Sz tarry EL > RY +242 + WE boty =tWe +azy RUM W) 4G 04 va Zany lia) * 7 2+ 704 ( AY = YT RYSeen a Alt oy Af Se9, ae Ahould be present fo. exons tim Faec)= AB+8C AR=I So mm — Canamical —ferm F(a.8.0 = ABC+O +(A4A) BC Le a ABC + ABE + ABC+ ABC =) ABC + ABC + ABC iy to Oo‘ @ S | F(@@0= 7MCo.19.0.5) |—[ PS] SS F(aa0) = (A48) . (840) in__Canaai cal fern Q+83.C9) - B+c4 aR) = GrOE) (0480) Green) (8+) = +840). (884). (74+e40) (000 ) (0 od) a 0b) FAB.0 = Se: \ 4) | —3/Pos aa Bo= &m (2.35.43) yalmn SoP & FOS ¢ 4 Que Express edo Camanteal form & Fogg = G22) eur) (yszs (mG +2) orszes). (oye) (aay (0497) Reages) (marge (yt OF 9) sd AYAZ Ory, Stes) (eee) (sury+2) ty 42) (74942) Ow on > AtyeZ) Gay) a sor) ‘0s o10 100 Of — (Pos) | Foud=an[o2, 41] Goes 3,5, 64) —(SoP) (ay2) mC & ail a A+ AB+8( diy YO20 = A (248) (40) + AB(OHD + GHA) BC = @B+ ABYC) + awe + AGT + ABC +ARC FABC + ABC+NOC + Re ABC+ ABT+ ABC +4 ABT ABC + ABT ABC ABT 4 AGc gi hie lot loo Del J dCA8,0 > Em(4, 6. 5, el Ey) Y (08,0 = 7MT ota a@ J= ACAB) ( a4O+d) = (A¥8B+ C2) CA¥B4 67) (A840) =(A+BVOHB) (A4842) =) (A404) CAF B4C) (A4B+O CAT) (04040) = (A+Q+) (A+B 40) @B+10 (44847) DRCOme Om la) (ey) Se) 4) “Yo. 3/h) F(A BO= =n MBs, 23] ae shen ve Em 45563] Kermouigh Map (k-Map) + Systematic muthod to mintmise boolean expsusicor, for ann Vantable mp.af Cts one @, *Adjacnt 1/o +p be Grouped. | SoP , 0 -> fos €2,34,5 vai abk Expswsion Cam be imimfmfted am k- mop gray Coda sebrusentediOn % Wed, f(m@)= 2= 4M $ (WO, c)= 29= B cath S(ABoD= 216 els L(AGGDE) =25 22 ely 2_ Vani able_k- Map SB. Leag= 2%=4em eee“il 6 eit | aii eaalene ee ONG nan oo Pe le aa ty fue fu alee 100 | to! ofa {ts dnb: >~4- Vastiable Fla nom= 24-16 as D As Mceeo ean i) AC eee | oe se Ted Go We] Re I] slig alae els L stafjoooo Oorere DE 6c 22. 1 ji to 0) Om \ teat imme b os {4 5\4/4 "To ] vais |is 10) Se See e LS © De c BS\ 00 ot H oe f 16) 13/19] 1S Of |Ro] 2 | 23] 22 | ah} 25 [31 [3° 4 jo [24] 25124 | 26 F oe ag gee 00) ol ola TNO OW Jo) loo eeece] Colo Tojios | of One| 11000 | wi0e Joyoo foooe orf Co) {cfc | g [au |av De 16 at Oovol] colo | ool | o loo fOoy \ilet 1olol © lo0o0} a ae Sy 25a) 25 eae 9 Vile eo ee Oe Ww Tor lool 3 15 Wil \ 2s \) Si aes 19 || Seatefoetto folio [OTofo | Iolo ite vou 72 y | Jo | 2¢/|) Sous "6dr Rulu for solving k-Mop © Fivt prrfereme for octal ( group of 3 ab) Accord prtefertomer flor Gpaxd ( gnoub cof 4 ¢ ~~ y y 4 (9) @ xd preference for poly ( group af 2 wits) @ Last refeoron ce fer Aing Qe Ee @ F(A8)= Sm(0,1,3) —> SoP wg F Zo ay A i | 0) H(g@)= 248 @ F(ABO= Em(0,1.24,5) —» sop OS BE ge Hm ae Gow Lome Ro ie etd F( 0,8, 0804 4 TF naeIGE F(aecd)= Bt pe 13,15) ot = 16 als Pes cm cP fd & Io ae ABoo Re of AB 1 68 10 [ iy faecn)= Fe+D A 2, — (A,8,C,d) = AB ¥ . CD) = AB z y) C+ABTD +ATH +ABCD4+TD +B @ebep =< cD ¢ oan or Ir $2 ABooGd. FURBO = AMCo;s1,5,54) \; a NG? 2G tie rn o| @\@)| 04 RB il Xo | o- F(08..) = @+8).E) 2B _kaMop wil dent com 3 * Some — imput Conditions might mot t 4p Aystom output omd Acme fmpud Condition might — mass OCU, t Tere fm put Conditionm asu dan dent case Condittey Can be Use high/ fos output Conditions OS per Owl Comvimemce. Dus FAB)= Em(0,3) +¥d (2) Neale al = 1s Tn) Aso cts x A! Sea (Apy= AB Flagi= FG +as J a = ACB o ACR Qustiow ¢ | (YD F(a8c) = Em(0,1,6,3) + Sd (3.5) © F(aacp= Em(01,64) + Ed (34.5) OW E(wryz)= Em(.34,nis)~ AG+ RG a OD. FawcD= ém(or.3) TEA (94/ ) A fLCasipe AB+AE 2 Flw.yz) = Sol1.90 19) +d (22,5) ee teaeest 1 | a HEPES at i of at Wr oO} for minim’ Aakfon, Wx if A = Wx fo a Fav,gn)= Emla>__ following by K=Map ?- (OF 80D = EmlO42 5,64 19,n,13,13, 18 = F(naGBE) = Em( 041,3,5.64 3,116 18 19,20, 212 st _mdfod 20,26) Soh ose ne pe Be oe oa | L Bcol \> Bc 1yo) @ Y= Em(.0,4,484, 10,1624 25.2% 277 » 25, 26527 25,3) J @ Ye Em(0,5.6/8, 910 WG 20,9 ‘Om Aten mithv cb 8 ee fete Sore 5 asc Maceo DI + ABCD + ABeDe\ BBE RBC ol Q0° oo} oLL roo, (PT fei | : | pe! | @ ve 12 | a6 oo t & coat (On AB NY AB \o Ye RODE+ RoCde+ AB THE + RECT +OD + DEJ a= F(aecDe = Em( 0,5-6,%, g,lo,}. 16,20, 2425 26,2,2, a) Fla i Bebe (OBpe= BC+ NeEt BEDE 4 ABDE +ABE +. abate. A BoM Lo. B.GDe)= . >) Zn(%,3,10,0, (3,15, V6, j%, 21,24, 25,26 29 30,31) Are DE a O09 Co! ON) __ oJ F(A, = +8C = (A8¢DE= ABE +Re + DAB + ABE CE+ ABCDE\S oN GO Ow ira Ce Wee: oe es fee po| se Ase) oe ez | F(ABGDE= ~M (6, S\M, 13,14, 1420, 25, 2b, 29,30) =) (att) (oO) Bae) Loi S Woo, seam £(a,9,0.0,.¢) = (BE) <¢ 8%) % (Be) xaacE ) PEBErAC) + (AGED) _— -We, Marks THE Quine — MccJusty mubod { tobulan aed) wy > HL Shouts 42 Capabitily OF Hamdading Lengt mba, a Vooti ables. ~> GE Shoudd emswrs minimised — Exporarston 5 SM should y Suitable foy Compiiler Aobdi@n Ques: F(a,8.60)= im (OsL2,8, Jo,n, 14.98) Solve by tabula mel fod O~-> 0000 i= S'o on il 2 | Olt IFS CIE I> 1110 aa al Orolo) 1s > Wt 7 On, Von & Jo? & 10) b8 210 10, 11,14, 5] 12,14 11,15]13% 36, Flo, la, 13)+ 24 (2,3 15 fig Ryza 2,10 j 5);3,0,6|— \- i Gyles |p oma Olea -1ol ak = mL aoe a? 7) 14 ee Minti | 3 \ wT Az3021 CK a Dg wyz< 3.3 “i 47,3 ayn 2431] oO I>, Boe 5.4199 f(wy~y,2) = xX aa D4 + wytaye +79 a