Discrete math notes
description
Transcript of 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) &
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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
Cuna)e[ ) ueA 6l'" Cb,a)ql ,-l *{ (,el)ep : clc W
. fq,.) ert "y( (f,c) en I (c4,4 eq , p(affr
-fi1 ,,e",v6, Q =Z. +l A- q ' ,*, (a,b) et7
tely f/(, 0- lup ,i/t L,
-
fW; n "tah) n /ir-lv tt) tl1, pnN)( n br A ,ilrqt tmill. trJ = {"(,r, ---, n} , otr,or^il :\ N fi) in >}Vtto u, /(! ft i',n -pto l'tt ne N {,AV#r, u ,A t a) o)'K #^r/ i ;rQC,r
L^j 1,sf u,i.:.Qr, dr ,, ("{}t{ Nt, ') ;ruJil Nr u) B I A B1y ,lrtd r;nlD
f,X:,sf{y)q Jn
?,t yrtr t ',ttlr( ' ;tl/c p4{ r, F,rl4 AQE. h= 7 rkllt? 0'
.A f, 2t'1$g rr)n, 6 -r+t$' PG) "-l , ,Nh olnt A pr( l(rrqr)t (bp r/
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