Discrete math notes

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  • dtx.luLoltkg@*,1 l5i ac il-*: ?

    l:, .[' . A,= {s, +,.c]

    J oo trv( e

    "?o?

    --r ,s.nf p')-l rfr''otl xJ-

    J'6F,9" ojro {AO

    ryGrJj,lfuf

    tz. "to. 13

    y@jN',Jt - '' ;lE,q,r)prnv/( b fut{o, z,rJ

    ,'r0A/

    A;tx llsr:

  • fP(A)=

    ,0, Fl ,

    , xfa

    la c*J *-,l'

    .'., 'i.,r i

    {"3, #J , {,,^J ,0n,Et i A. p[*) n

    ?'pvl rxe,4 ,6f

    pt*) = fs

    I o r,t,4'l o

    h ,6 .,x1e^ 8,' L,nJ 8tn2 A I rn+}i)

    'V( A: {t,a,jJ , nW{r,$, [r,*, dFt zBrnl

    -/)p CI i)< Y0{ ,)oatcpcxk aH bna (r) Jalo Idrtttg-ntn^t'B -J A nllaV:m+D

    .XfA f,Inul XeB d0

    " AtA={tf. Ano -tv I

    (xertJ - g q,tr< A: fa)(xef,tj A 7rn A

    AnB=4 ""c

    I tilo)./(u rf

    A \e fx I (xex) /( x{o]

    , ltlta2

    ttts 6-l A tlhpr9/0 AA , nQrol /p orsr;r

    til", rn9o tr,l I tPl;t)

    t{ ,"a_)t;tl fi

    (xe,t) V. itL

    (xeD AAi

    .t,lAtg + 0\A !&r , Aoo= bnA , 4119= gUA\,

    i gt , Aqg - O '^.1 r -,r'/'aio

    -nl e-1 A. lA ugl = l*l+ lr,l

    lrnN/c w^t -/t 1,,3il lc) , (- 8n? ?/ir Y.,trr)

    ,lt Jt^DttGA ug= / u(nil)

    (n nst U[nt,q) = 6:vq / -n rriiti ,r, 6 D +.s/r f o)Jv "A "s rr osr /21, [ir, ??,c

    '0'"1'o ^ml[n'oAAS

  • . 6os)i ' lol =

    "iam = l{l "ltl - lAnsl.:

    ^ (Brd =4

    iAn6J + /6i41

    !,!( , ),A)o -rB)A?

    .9t --attl,?0 y1 X

    /c M Pa),rlloa

    t u.ttTo,t,)

    ,d( ll#lttllC.

    r,t/tlt

    B-!trJD[

    A, Av6= ft-,ut Jr^eA,6*s-J r vtCJ , ;l?taV I -,l'h6 = l*l IgJ

    {r,4 x {tpJ : {t^,rt, ('t,d , (s,-r) , (aaJ

    $u6n, xlnc t}{ (t9 u(*ax61 n(cxp).

    (x5) e Gnc) x (enp)

    Gxe)uCcn) y (*udx(sLp)rnt' C?1'l

    tAx0 n (cxn) - (;nc) x (gno) tnrG1 \fi4r5) {r\Lxu): \^!tt/ ^\Dttu/. I e8no - I XeAn J( , 1.,,1, do (X4) tA dt ; ) y,> /*/ l:rnv;rJ I v\/ t - U

    Prt ul(^ (hle ,4x6 $ Bnpe B - ,' Anc sA, CYrflu (AxS) o (c *o) + (x5) e C xo p,A

    'J tg /r,) 6N

    . A* b ^ {[o,b\ ] ae ,{, be$ J -nltw /,6 "lysCo?- l A*Bl B*,q

    :ny$/(

    , l/ vgt = lB *,ql$y,l r-r'r&g) &

  • 0 ,s-or'--ryru P -1$yo\"(fr ftrf .lr.tB I'l( -;nt'>9D

    . A*S A tinl )D )J B ^ (f z{ -N o)'tr ltitl )o I Fa,t1't P., on' PDiu' :,?tnV B t Ppoiln -r'?lg h /rJUl)

    ( ir'll nqlc )/o" rsoiJ'/rrt,liow, 'prto -- A :/ruU I k[lw' 'NQt' = 6C c A *B n),J

    .d

    , rla er-b# l'+tw,s S Jr( C -n ryJ 6tel r.?t 6{ =L" C: {to,u) l,ro'ic b stu'l TD 4

    y3= {(u,o') l,r,;r. .'-l AJ , D"AxA A*' tal)= A ri /CIa (xv.f3B lD [\ yd li-n qs4)

    ''il' J@/ A ^,r

    "Y/t 8)' 'r )sr''''

    F= { [o,e) I

    ,ir>Qt' 'xttU\ = I l-[ /dllt I./l,ru'D 1t t16qNi lc = [

    -D Cntco,n? , F.AxF. 1^Qsmllk A -^r

    ",tt{'b Clttco J1";,1 $,'

    I p?,h f ' alt llf yJ

    \- ,r4Jtt1"i,lchf

    J,r4 ittr lrh lc lra- D9') 6J I p) ?ttr1t'Jo C

    J,l{,h h2uf,

    ,f ,A -)B

    ulfl) f an?,1\ le8--^ Ytul '

    A $nNf6f r,yrliwDlJ cev,tn

    >tlya r:lll?r0ns -f(u,t)er

    ur8U"

    b= d(a)IILA

    lto)

  • 'W to,fl

    lll -^t t,{i R\ {4

    , IrJ pio;l 6,N0 p,nul t

    /r) d&l = {

    ftmD lr'tl) A-cftG't'f 6 noa)

    E -{i, (ilwv'rtl

    fa ;'Jlol-n,) = fv'

    il^llD ){}kt , fR' t,lD 4

    itasik

    t ptoJ 'J^ , t?7t6 'f ' A + B(il= fl.-BIlart s.t. {h)=b!

    e'f({ ti'ttc,: { p{Ay>,tu fotn)0., an , f(x);x I {,

    .r,{,&t fr/,\ ?uNr;t

    :lVtil o turl,,

    () "4{(

    -*,,^, ,,1 gat r- t+,2 S 4!t- 7, nt,t'&h a+_y,

    ,;;"p/t rro3l;)

    0t, v> t.tl

    :

    Yta,

    /-)

    t,,)$'^t)

    lLs R.

    4DftrqJUl 0

    6

    [111.,1i'e /ors to o^eJ,h ,

    ^ &,1'Cl.z

    fir) .l(-r) U^^ tn '*"Mn

    4

    r"ntr ?n^?h kl'), {' 14 -t8 7l,a t nr t){(a,)' ffar) - I Ct,, /nA wJr> ,ot(

    l" fxl=x r

    {, [l)=Vx

    a,

    '[tl-t 'it,a { 'A t 6 nlt (t}

    A ,ft fue f'AtB Fc o)(r,n,r] u {r,oJ ir,tttt

    ^ t," tsfi fr V,a )r

    /,r , Jl g lR. u(^t'll ft4:'R-'R

    fth^)'Jt

    6-

    e

    tsrlecti,r,r lctnt, ) {i/rry'') f'/tB rrrsiD+io).L ^a p +rA p'v beS Sf '4

    f P dr- j , (oW 4 P'fA 'c) f /O {, ;r/r^r1rio

    J*l = ls) 'Jk , -rfDD S1 AM ,u'cFi E-l,t;J ) , ,tt/UJ 1( i

    , l*l > /ol

    1{i;)"{i

  • __rl3ptoae4 6fip NB D F Sn? tf))s f,A+A

    ,nt6ry)b!,6Anan t\\it A .,.(ct,DeF n) p U%.6,.,6)eF io p (*eA p,? le8 tr a 4 -ieN f

    .6t) pnJ6., tr6A Bv?rr dd 61or; -f[ar) x))e ,sr t, 0-,1 " C(a 'Vt,

    )0,10 ",t?

    aEpn :ryab p,V'sEptan u

    ^J;r!>\n-t),r) { Q'

    v

    ' * l,'+l -.JBi ,J/t \^ ,?tta f,A *B dr .,t:.

    i.t1, "(e* qk wff f i y,, /^nryt 6 o)

    .Ff,ul=lA:A-N n0ln Nlutl0,Ntar,nh,[ f puJr..'/f*) sg ') l[6,)

    .(fu-!ut-tt ) l*l=l8) Jr ft tn, f ,cr.s','

    ,lhi , h0 R t{' f(,n)= xQ {'ll*[t tiJrr,ft] [t. riti t{ gc R nn f ' ,y, &r) :x} f

    'l|",+ ftb 6

    ,,K s, tn,"1 f fid,=*. f, Rto-, lR)o C

    ' ,D-

    , (W="l b ffi,

    d,lil )o-rl1b

    f ^ tt -r/'(

    f' A-B A Uut> riD?t)#'A -B : il/o)D .,tlin) rllaDeI

    l'$

    fl+lR I +r 7,a ^f ^D

    ({yh,lVp,0h 'hD fr ro,Jtr,t ll

    l,+)={6e8i JaeA,l{o)=b} ft,)"( A 4,fi}ry DcB ^I

    - +-! ( o)'. { ae*l f[a)en]lu,,ru,Jt$"n. I cr

  • " l{'' fp) l> lpt. lcl > l4r.l)Uc DsB 'l

    ?,AD -t/( CeA BJt

    fr frta d, A -E r'r ,p)JC,{p }j tff> $-t ,Hc

    -Irf G ^01 co U(, DcB , f 'A-B ,o( t"l,tr') ;tta-nrrai t{ 0r)

    *e.A ^l.f i(. yen

    .x+{'(n) { ."{n) - ,*, $',n a .\

    . it"rr 5l "ru)e J[f '(all ,i'i

    va J 'ifk)r-) : ivl 5- j.t):) ruf E[

    ,irrl ,4qrl *J - r

  • ,PI

    d , lat& , 4r, A^ -

    l"r

    , 4 ),t) fa.li-,,o pt {.1

    A/ / nr ,Cn iNJ l,;*rrilu A )-d Wn 7c' a1t C

    vn /a _l

    t6 4fi, /t t h;tft C/rD 'f,u{a /( t T onC

    fiau,,: ('{r.1,)k) =z * p> \eA,t Fl 264 D l' ,\ ttlC'-wtt

    , x.r. h ,J"l ^ ,{, &o) ={n (N) nilJ ya,ys e A1 J,.,\,f^ ,) 6 t+cxrl) - -[, rd,(x,l) ,J/( {, &Yt,$) *(

    z({.of) (v) \ (r{to/.r) =tr,Ir.Ye $ ,r;, fao|" -l

    t/( f'Jrt Yhn

    W {n' u ntd Pn\t Virr la {su,f "

    (ilo n,a\'

    \:

    , (nr J

    R pll ln 'er"tD f -) fj

    .lrg + iJ & g"C=iJ ,t p),, inlD: ,D,o f ', I -+ B t1t)tsl) i

    ( =a' + d(*) = {{,a') Plr }D^ f frp cLeA P,? le8 6d prt. R { hrJ,lG)=a ^d -r"{t, iln'l-n G

    P .1 tA-/

    -tt)"? lr ';)YA^ f ft

    iJuof= f= {";lA :D)t)D (s

    W f r;d( ,ta; f,A -B t(aonl" trdo n0 N't\) (c) Jtv) lnrn/D

    &

    f"; = ;Jc1"f = iJ,n

    .4Irn

    ".'

    R, f et fu =.U, g"f=lda ir iln,.')n4-l o-J Wr

    . fi/ 6iln JngtWW

    mfit

    . ,rG)

    -!

    Y.,t

  • ,;))OD t'.d: .;;

    ,fto)=A -v p G ('? b.6(ao. r"({D) . 4(46) =.5 -u P

    .

    3tbl = fta

    ( .M h(il=litxl-u ti\{ / B ,ntt, lr,h' , X+Y

    "\e .X gC

    I

    I

    ^\[? 0 ]^D + (->)I

    6d' ,k I -,t j')a.g w 5 eg 6( u.,',,'' ', ,

    4 5 ' B-'A ot, . -n'- "*,P,, J6)7 J ^U ry(U

    .fll?1lw ltlrr't 'q; rarP)6eB 't, : foJ = iJs k

    G;)(A)= {f6a) - $can) :5 - 'ds

    (r)6 ' r/t - CI.er{ 'D' t 3# = iJ* (i

    ^ tZ ,.,n J tvfrtD J/( , aroD f, ht B ,qt+

    .J=$' ^Qt .tr) 6 0 tt'>>n, \)'

    ^,^ J 'b t'ti)

    /1'----...' i t)'D)D @ -)Yilflyqol\^) oil Iw )A .---s;B''rH?- ($"f),J '= Jofo;'= J",,f'!:],",,

    idlo.j goiA6A JiJ, ? J,

    \J J

    "

    P ))l _,J

    w). 'w

    ,fl'?pn l"t,CgD , 3 ,B:-)c ,h"fS"f) b :-(LrT).f

    --roQD 'aff ,j JI,

    APd

    @

    .hcoo 3nt'((

    (h..g"f))rc')=( (J^g),' f)a)

    . f ) (u) * (ho t5.+)/r* I(a)) =

    ^(J(f(*))),1(f{o)) :h(J(fioD) t'

    ,afif:M))D'

    f,A'8I

    '-ft

    aeA ql t)DL2:i JYVfi

    , UU ?3/

    rit)'6

    h(d3

    "j),t),1)

    t)Ih,

    $.'fr0

    T=[

  • i"o

    h (x) =\i4 ,g h)=rr , f(x) -- u(3il : (tU)ofG3)tx)= hig[x)) - h (x') = x2,4E"D

    (x) = g [f{x)) = 5 [Jx) = (rr)'- ,lx IMJ)" f/(x) . \) (fat) = {^1)(rn) = Uil'*4 -,

    ,,(h"(qof))(x)= t",,(gr;Xx))= it(u;*; * lQr')*',t=v"\*4 '' ,

    "D,o) f'A rB JA,,) t+ lAl:lAl nlnl - A,B rDD' fr 61n {,Ar13

    -r,t,p g IAI=lg) 113[irr ,d ,^o,>

    ,stAD {t}r-g t{"? F>

    t'J, h , fR- fl( il(^l{Jt-)ttett lid

    v

    v

    ,

    ctt 4r - lq o),) -.,"'f,Ufr , 'rt',

    * ,9t gJl t

    , P s AxA .9;02 )) /tt;t A tpnt fr-fi-on' r rnal)CqA)eR eryn ofzb Qtttn wla{ tW

    B={t r,g.rkxllIxiyj ,,4=ll{ vr,,t,,qd,'so2A) 'X

  • A'4 fi, ,9tnp Jf) h,;t(rt-,q)xGx*l ti

    r (.e) Fhc

    _

    J].4o,"4q,fioDt

    A qn' .l#lg 4 'j)i) lflrrJnlct, Atq fr oh) irtyns

    A{ifn2 JY)

    (a,t) e R

    (L= [tn,r),1.t,il,(t,a),Q,il, Cr,,r) , fu,e), &,u$/ P----------t

    'd) J(r\t\ \i: l: uq

    "A f oN f( , ,J,rE,A. (q,q) e fi ( jvlr.t aeA 6,1 ,4( 'n'ollA. A a^'i)

    ( Cr r^) Ffr, rol crcA p'l )tA) p,;uw tg{)), (bp)eg e (c,,}ep. ) a,be,\ 6{' gc oQ,t,o, (( oh,)( Crr,n)+( fic (o,I)e4 -u p ar6e,{ Qr'| & t'W M)

    (-u.L) ,Us-)erz- b p C1,A, c e/ 5l ,nc irvfiUre f( otti)a,\CeA u' dc {1w tcP) -fti,Oq P'i:u)^b P)

    ' ( &,1:) #P Rc (o'b), (b,c) eft

    (a,t),(b,a) efr -u p e,M 0l ,sc ,6,o ,O/\ ft- o/y;)-a,) rD plta q,kA \ti /< e,pnt t1) .Q=L f'W

    .(,6u,o ,ilt, s uti - r,atrJ .( (op,$,il ef ? nbt'oL\f\n

    4=x, b =X g fu,|') = (X,x) ^u p X4 gt n?..rl;r orl ,lv(,^l.(6,q),(c1,D R e fr:tr e

    , Ca,c) e[ /i (o,0,$,0 ql lst t ivrttt ., ], ,'t ,*C,a1tC..:.. .

    . G,i)= [rr,c) e( /l 6=C ' U .+t , 6. L a frf,) ,fi/jN r,l), 6+ , p U,u), (q,l) eK -u rvil t,wi)

    ., ), ?'r8o ,C)k

    ,,frr,e,cJlc-' r!,o,f,,o*r:'#^L

    . aeF

    " (h,otre l^h-f.

    .,+tIU &,k4

    1u,ri) eA\A-R

    J{ (a!)e( fJ(

    A=A*A\ tw) | ), ? 'a'a?dalt rwi) I ), .) ryj1ts, Ntl lyt )'n'?{nC

    Nra dl ) r1(4e ,8/((a,L),6 a)e,\rh= IZ,ilt

    i'd ivilD !*{ LrRev ,,)Cfit'a Oj:, {{-

    R= { tug) eft.xlR I **tl . h=f?- .j"(Fr)^^ q * -^ ,t ,r) '(:t"te( /) ! ryvlD lM ? 'd.o,lat

    -a'*-,t ., (e,',i+k^42r-3 ,r (ls)eR :^n/) t^f ,'r*oi ,) G , t ) , (A, il* tnnn ,tJct /,a6sltQ

    (*), # rt (D (t, *) pn &r1Gu'0 'C) lt tit ^ {UrO

  • ft- ffry) e,+*,t I " "y$-- , lr-tr, -'nJ .q

    = f (4il , (t,-a) , Gr, ,t) , (*'t;t\J = ,(x/{/u.ro'Y)

    . ,{A'v '6J1 d; , 'dCu-LJ:.1C , ,}ars, oat$r ll

    t(tq,il Civ 0., on,) R. = t (x,$ e L, Z I ,nX::J.(n,nl

    (i h-^ = ,,P (W rLeZ 6il t )t\/1) ) P

    .

    rir! fv-m ffi^ (n,^)eR n{J ;)rr,/'D .'P/

    .,e,ll5 r,,rk ,h-h 6 (w,,r,\r(rn,k)Q ihl)il) lJ)

    , A=v 'sI

    ',a"ol$)? nru

    , f *,1) ef. t+q O( 0n,^1, b,q e(. I u .t 1nl 2 'fi'ct '6JJt

    (J1PW 'eD' ,

    A fr R oDt.;9n7 A ,\r) rnQt;l. ,'l'C,J)10

    ')frJ,e, wofa), r,Qtt 1N i4^lei? il' , &: Tt , Q= axA rlltnlf t? (t

    :rO W,s{) x ft 2t{tpn ' ;1,{,w X6A tDr 4 I L'N oD) A 'D' !N>}?,

    t,{ro'fu : ,i 1;3,, Ci"qo fa)a = ft,rJ[:Jn: {:,vJ

    -:AR

    R=

    /Lu- { fqJn = t g,rj

    Nc t'ilVv aN lcJ

    {n,* , {s'u}J G

  • .fUJo-

    /Rxe !k]''r

    fxln Jt(

    ttfiv o)>/ {lY\eA 6t

    J'DJa &t

    ,;>' I Nl QCI.

    (a

    trvilD

    ^/:

    -.1 .

    ,x[UnlSJrt= [xJrNrB , &e bJo

    ,sCy"t0 {l , (v,3) e(

    (x,x,b( ltftft Jft]n,4 rlSJas&)a

    e Jt tIJ(. 3e LxJg

    xor/n A^ u t,lJ

    t -nryJ

    ,fotJ^w0

    ',)D ;u+|rre p01)

    aru(g

    " #. |g,r,tj tNti' ft[: { {t], tt,sl1

    o?2)e

    .(j,a)eR- ft,*( y(l

    , (v3) en il/6 , 5 e LxJa p) : [uJn . kJo rw:tlMtenu rw ,f xe t"ll^ : (5,ileR ptl ,frlo il* U rq4/ Mtrrt : [X]a E IyJr. ^ u W) (nytt/pn ,tAan )wc; fxlo.}l* tpt [yJa s &J,4 e J.fx&

    [X]ntg=.[ttr f,tJ-bl tuh. &lB, IJJ*" Jlo )A 0r_-"{,7u oD)-L lnpt,ir)

    " lyJ= Bl -u 4cI [xJ o TF *0 - a nJJ (ut/),'f .eJ *LrJ = zeT ,r#u& ura 'f,. z:e txJnTxJ

    ,/Mlc

    v

    A ft --oEnp w 6 oga? /rD A 6 Dlh .nQn;, A

    (4U c,,ttf) ry) hf r,,i bezl 6. f/\- "i ,tugtqn C le ?tvta *'^ A tplal.J A- W

    , 6n c ^f e,poi.,, totJtu B,C e A 6f

    ^ cVlnzir) {{tJ, {',J} nvh.nyfit /d {{t,r;} ,pJ ^t/0t t{ I {t,rj, {v"r!}

  • "[rJntFV * atq, [x],Lx.k4/,)'.na'vs :t1w^tN wJ Jtxi# 6on/ X e IrJa ryv [xJe ap 6f " p) :4

    _laO, 2t -n&,tta tJ ^f FO nel 6 t) fU rJf, \N{h A

    .ry{h Juo

    tA R ohlI

    B11 = {c*9eAxA f

    zLrqr,) :t,8^ f,

    --rtLy'a lrl/)al

    Y( ,A f jt\lw olv A rent^liptt

    "D3),1D P/U1D -ilt V11J l2ya,/tJJt

    '4i-I)

    ,,10I

    lser\ -t*l J

    , nlm2 A ,an tdtrI= U (sxs)

    Ee*A

    a.r'tl Rg i'J/('I tttTu

    cr?CJ

    X,J"6

    A

    r{

    ' ,,ttba oD lll 'w wtJ lrra/;). (a,o\eA\^ 'f? . q-A 'r' , 'sor&) Qy -u nt4 -1

    . dLeB *er p 6e-A ?') " Vth L -i rI).

    (q,q)e R* i{ 'ae$ -/ ,Le6lB . zL J,t,) ,J[( , (i,,A e Llt ,)/ t , g,fo QA ^u {r>J - a

    , &,o)e 2y\ yA b,oe1 e . uftr -L f).

    (a,c)e (zt- ,t} Ca\8,(b,c) eLt ]';t 'ry04J* Rt " u rto) -34,5C -u P Ce A JN"r) - f1t).efttt

    .L,ce6 vp BeA 4N') e ft,c)e(a

    a, ce9lSnc+c/ e be6nc t-/\y)\\uk ,1S

    ^/h A -l )/').(a,c)e Kt e

    l'Al A fr' ttba ,oni,, h't fr) 'f^, ^uo) e// ,,1( , t^l ^A ,if)

    . A b -_-//hft -'rr\ A({ -

    {t?u ,n, }| * k tO,*^- rt | - t ;.,1)}i.r rpJr^Al

    ..

    1, .tilbo utflA, CaVa /''9)4 ) l,.i ''?''i* -t

    ' Dis\"tkl't\ ' ri:t 0' ''t (p'z tao'tt]]

    "r'f rro ( lnJrP

    {-..-.,Ath)= A

    7

    kut=n

  • , h,bez ,( rW^iD ftt

    ', p,,ND ,R cAxA oD/ d A dr,p f

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