From 394589ce8b2e15c3c418c5e2af421a29b18b590f Mon Sep 17 00:00:00 2001 From: Rushil Umaretiya Date: Mon, 23 Oct 2023 16:38:36 -0400 Subject: [PATCH] fix formatting --- hw/hw6/hw6.fdb_latexmk | 8 ++++---- hw/hw6/hw6.log | 18 ++++-------------- hw/hw6/hw6.pdf | Bin 172285 -> 172288 bytes hw/hw6/hw6.synctex.gz | Bin 30584 -> 30346 bytes hw/hw6/hw6.tex | 6 ++++-- 5 files changed, 12 insertions(+), 20 deletions(-) diff --git a/hw/hw6/hw6.fdb_latexmk b/hw/hw6/hw6.fdb_latexmk index 950a2b5..e85284c 100644 --- a/hw/hw6/hw6.fdb_latexmk +++ b/hw/hw6/hw6.fdb_latexmk @@ -1,7 +1,7 @@ # Fdb version 4 -["pdflatex"] 1698093234 "/home/rushil/school/math/hw/hw6/hw6.tex" "hw6.pdf" "hw6" 1698093235 0 +["pdflatex"] 1698093499 "/home/rushil/school/math/hw/hw6/hw6.tex" "hw6.pdf" "hw6" 1698093500 0 "/etc/texmf/web2c/texmf.cnf" 1693368362 475 c0e671620eb5563b2130f56340a5fde8 "" - "/home/rushil/school/math/hw/hw6/hw6.tex" 1698093234 10597 99b46c2c3b3f88a118fc653eb5889b88 "" + "/home/rushil/school/math/hw/hw6/hw6.tex" 1698093499 10607 17031e54ea748adbb730bc9254f89a7c "" "/usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -150,8 +150,8 @@ "/usr/share/texmf/web2c/texmf.cnf" 1681034085 39561 34c98e380bf7c7201ee6a7909aff625a "" "/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1693368379 4623455 fa0568a71dd9a288d6c226ee477506c6 "" "/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1693368408 7880218 5644a8bb2704ddc1060e0c139ce5389a "" - "hw6.aux" 1698093235 113 e525f25688a88382d92e4852bbc94338 "pdflatex" - "hw6.tex" 1698093234 10597 99b46c2c3b3f88a118fc653eb5889b88 "" + "hw6.aux" 1698093500 113 e525f25688a88382d92e4852bbc94338 "pdflatex" + "hw6.tex" 1698093499 10607 17031e54ea748adbb730bc9254f89a7c "" (generated) "hw6.aux" "hw6.log" diff --git a/hw/hw6/hw6.log b/hw/hw6/hw6.log index 8f71e81..d65e151 100644 --- a/hw/hw6/hw6.log +++ b/hw/hw6/hw6.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022/Debian) (preloaded format=pdflatex 2023.8.30) 23 OCT 2023 16:33 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022/Debian) (preloaded format=pdflatex 2023.8.30) 23 OCT 2023 16:38 entering extended mode restricted \write18 enabled. file:line:error style messages enabled. @@ -479,17 +479,7 @@ LaTeX Font Info: Trying to load font information for U+msb on input line 134. File: umsb.fd 2013/01/14 v3.01 AMS symbols B ) [1 -{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2] [3] [4] [5] [6] -Underfull \hbox (badness 10000) in paragraph at lines 252--254 - - [] - -[7] [8] -Underfull \hbox (badness 10000) in paragraph at lines 272--274 - - [] - -[9] (./hw6.aux) ) +{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2] [3] [4] [5] [6] [7] [8] [9] (./hw6.aux) ) Here is how much of TeX's memory you used: 15641 strings out of 476091 318265 string characters out of 5794081 @@ -497,9 +487,9 @@ Here is how much of TeX's memory you used: 35749 multiletter control sequences out of 15000+600000 522110 words of font info for 69 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 - 102i,16n,104p,630b,420s stack positions out of 10000i,1000n,20000p,200000b,200000s + 102i,16n,104p,649b,420s stack positions out of 10000i,1000n,20000p,200000b,200000s -Output written on hw6.pdf (9 pages, 172285 bytes). +Output written on hw6.pdf (9 pages, 172288 bytes). PDF statistics: 112 PDF objects out of 1000 (max. 8388607) 70 compressed objects within 1 object stream diff --git a/hw/hw6/hw6.pdf b/hw/hw6/hw6.pdf index 1c9e67ebb31901b5212cd49a7cefca7b0585ef13..6b0e74b065b22eb5c48f1dc5094036d579ccba97 100644 GIT binary patch delta 2469 zcmV;W30n63!3u!F3b1z`0X35W1}T5NR^M;aFc5yvUtx)tL|Qzbf5bKtLLeAy0t8$4 zfVMq^Hr*oH29i|O{(CNQ)25qS+G+5TME2#o@9sOFkC9u#$cNzVAUHlzmRM>No{+(q zXil`0lnEe%5xEM_=ha<4iDF?ycsWhWtj?nt!uv?@FpX{oXU8X8_WlPD6pVjG^!E36 zc{Rj;G6WDJ{G8Tz&6fvUx3>X(3gUc_c182;sXv~B6gWz`nxX%cNDBx?yC5>t}>f zVGUWdR#PI;10xf15qt~S&fI?ir_yle`h`e58RC)(t~z?BtSN3Tw!2!xJ(zbEUt?fUq?>VQ4z(c zTh^3L#@Pf%RpMxlAZi8b5#nUmldNWuzc1+Q<0e{~J67P!6j|zDIbn zvg*hYa{e8iMLwCh)nPFk&4apd1T}h9#HoK5geOT41eyvk;VYlESY^sQ2P4_p{P2KCJTx1Ojyxza7@*2X z6c-yAi0x-kQL7H2^#3xwJKI>aOz>UZ<(VRbXBVsu4DCp)HC5K4S_uWYzn5)oQ0~zN z)geN{7B-q{X`gg#+Se9TucZGZNZ2z*bC9l4Y;0n}i@ccvJpF$G?=mj_y@>t0?D75K zZ5CEjD_t21yIoa-`mE@ATOgxqeJA*m7K6l6PGH%XmPJ3f%gStAl$o!WNhG(v5ub2n z(#P|X5*$GdNpR3HYUj|zNJ2GEI&N(0Ltp?xBLt2p26jPUC(`T5%Ljq8o{~?4K<}8w zg9GVq^^F-)p#>knz9VEiI&RJcOi+a#1V?X$lSQkC-WVcKg+GL(u4l8mA?N}DH` zlEsgS;Bl6GTEBh%n#;j?#sp!h1?kQ4pYv)1XMY?B5F-9PtM|SqsHGoTpU4PWB>MN-5*t%7Z&6+{ESaUtHISK&Jx)0uWFxIs_m7c(caMoCv>t z@ymMgdBI?vMhx#u370C`92Orx(P#_vZzD>CO`~(O>kvuMK%@I;wfJ*ECvC?#kts*j z&wq>zScppkXgYYS)5Alug+TLTMQ`yHnAd z6-LMM5Kxg)Vma?5{1oyoB`Iayj=Bs80Rym9bz5EqgrtNj-EH_05uNw$=mOG&B$lZ; zNZoo?Y(t1F5$O~l#$ZFQ1UTMtl*uBm+ka8z>ei&^99Y~)5p;MhNx1Fp|4#r2Cq|pA zyNpB_A`QQmFhr_|lKoX)1|)zr6rTlgZH@6ghVBW07)c;LG)=tPmkE&gJV{}t2xC(W zUP{eDd$jaPM6n^0MO+qcS(i&_P)WYSdGGXg!~NrUDX-hM7vZx zb|6wAX~Mx#s9r*}N$Uy{fh(}*zIa!=<6}E)y>H?gLpJPvnG(ayY(?sCdhkk4RgFsS zn!@UE0Oa?D*2HexVk@!Du-WF2?*&w$j*~-}{X`h1PyIsAl^Dy1b*d3p-GAVz6ePW* z$|tKOdajg6_I+tOW(pEbq)kRaqT$Z5TOJkeKBW#2(i_Clk{CY8?~xmQvDG?W`^VC*JVKNOi>%}`??iF&Yf~CM zzP>bq3KhR8@Zu1_?KmR{4%_?DW-hXY+Nmq^pkkj++%V4HzpU~|e4U1QaA7*h0 zx+-)D4i8z*=01nxpnrlVb^kZM{$eXg2SCzq{9!jg>k)deG*OUGBs_h=|IH=rwZ{Jd zsvP}u3T19&b98cLVS`L5hfFB}w@fJlvt}kZF*z+TG$$}HCn*XqO>bmGVRU66C`39k zFfubSFfubSHZwReIkzQ{0`U$}MnyS9H!?9bLohf)G&nLeGC?>&GdVazLPAD3Ge$)< zJ|H|sML9$_GBGtnFgQarI5IRcK{!D(IXFW?LPj_AUITWSTtwrlrb=SJr{jNTN)M5~XxFiyZkO+f8(v*ZRfG?mibPLn2}NG8ZI%=w+}PLT{^ zAb8MO4;dkoq@RqEA>tsJ5XYAu?jr*vn}1|MobS3ZMw}!E;{MdxILRZq5O0po@`(#l zy{OA+;wA-_@Lf{>Ouy{{fJf34L3 zNd&bTA*rNR6XZFj)eQMfX|({JU9DDNeO0Rsh^K!%1l$~Gg@Ko-Ry*(=(&~V5El6v1 z!h}Dxx?pD0?{>p1L+b`4aP__ zq=mqQQ#2T%tN46g-Q|-c<%Y+X)2wuLo}@IsPdJOS= z{`C9GGf<|-fCB@T~p>QO@mi(2=uj+q%eD8Pf8^^`G z^scj6wP+SxiQ7Wyl!gN9%bYTyFp(OiT!0QsvA>pyq;b*yj31ug;K?awbQQ_>1a6kA zjyxgn-_u#-lZjs)7PHZOxJ+2KOle-aZs}Lq6)a*DidGi%@h#m5Ac7MG5+sG?7wA?-;ZFk zuo_$8%aGgcsv6K|MK9O_36|?OA(XU~3MgeomyKyz^n*KBx^YpuP%jfeZv7%Y;mO30 z=OxJ*fEon14{IWJ4NU|otYOmeV^bdj0^k}cqCnPN2+)Z1dhmLT(2%gZNo5T9Dox|1GaJaK?du03qT(vU=}(GX9$?`oX!XUHCmK zHV#)-Zd2!Ff%B9K*vXzJODSdiwmi6V!cANrzu>w?3v@cLK!635i_U_Nf4*DeW=@3P zzIe4>d|fbDrxC;ZQo^N*HiyNh&otV?{QHO!Vbkc`>^ejeG|=ciS}p!s&`IAhPGrha z_0L9s20X+i0c$$*R;P!DWXaRGtXc=u`>bw{nZYqli8iJ?whIWhgqz;+d4v$P$rG8N?WD=#>D*JB%_}Zvm{E}NkDrN>OH;53cT7O zd%%v%Ha=SWcH{~D-W^=wCu^K6cOlJ8gJ3(Ej*O&M4;Z46mSJ-$@~hs3KhRA@Zylc?K@ywjvkE7=g{A=d6^}8~vXLK-*a#+Wq&a*n=-gveUwq%G z%~0qzuWHEm8YjmP_Es0c`X}M?bFWu^;fh;~0>2I}`I9q>P;->R^miM7af?#S9Hk7O z15((Te1R{<<$rJse`>k_Ue`Z@Z<N!|aAufNy|*Z?s4oj>g6XFakW98DC^iG-&wgul6jz1H|2 zP8t1h3T19&b98cLVQmVN9Eu2oS1E^ADFL@vDFUfxCNniPEig1EFfb=63NKA>WJF4V)Q<l{e%1IfdIip7}@sVm$L;NH_YDpcbCqdFcLL^KYNfRWpr0+FD zmM^qgASaDlt&r=SRvYALQL7#DbEOplygOPQzF#EclHR35 z>hXK;y!X%hYtET_=gzs`nQ!J2{W$XAadn>=l+XlX6@RFL;}L(hV#Xiwk7bt%c>Clw z*^xQwJ@+3uwNBh})*T1)ZV3&h-)jX$6dp#!G2H)#4Qu^dDQ*=d7|9L4y0c3uT$fVS z&8&7WWd5oijOB?;+qO&LK_%hJsvD*1I3}NdO;n9Plb0E>=_XJxh!-RNROGto97&M* z5_(l0nJoh3IHm}TP?5fNS6RDIe{`hWA0_if5T0;O**2a~c`&sX!4Q8*pGpl8z`fK2 ztmv=J5S2hHJIeCkUvaP0Nk0VP0;h|DXuT zH@@EM4RBrY+?8%cq$P~v38D4&@scV4>lVG?R36f(QA8X&U#2jgo&3VNov&w+;Q;X(bG=Qpulz z`j81%EsKpn6yhuQ^fnszh#?0{3{l6kU76c!ISd##+lq&|76mq@u&K@DAaVVndFg+x zYo%Pv?)rdm-dBVvbD7@VtoZQGV(V{IwF_D&;T51_g7kcYa-NuP;Z-O#RDqkQU7Cj<7cQlKh?%BKNouWo3VlA-BmQ2(9J$8@xjUJ+tABt4J6AczOWpGY;f4EvG zu?A~O{RD}=O}*X)EU}!#OKs;s75!QNIuHr;O6|I(>ykW6U4!RqvSdcoK-i6@x88DM zXr?CFURKi%aeztPRpTPdkP=c1Q!l7ea%0_^y|EZpB~pU}*`L(A$utEgF_%dmoC4R1SM zz#{u#mdPX*S(7n%FUNRA*BSTIK0!I~c?|2|K|>$@h}ldnmWDwPSJYn~)#C_|@~F^fSK{gXC8*neoz14CZG|#LMt_Sw6AkATZUPsS z3pY4{{H% zFacZs7&*AG?+Yo1sa5HgQ>VDmkav~#oVmkT1dA}>{Uu{{t84w!f_C>e8=XsLV-8_T zDI>#4i%m22Nq?aPPPpMKE|t#}_`YoRVSF+W9v{3)m@1z~m<|g~iN;OReJ8qOQh!?h zIi)Og*{Au#Tm=p6F#d|UZ}r)7YiFD^I`wg|1}&f{x|HkP8-gwkx2|uWI^tFnNtWF@ zmg@@y@hej}>J_ZP9J4hS~o=p6qCx@#;r}PQ3FLgKKw+Dr#;fmGlSwEbw@K;-t3{u z_YL1JgDyu5ZN^EyS3;fc;Bj{H>hkn?S&-}VZ`Gj%&U4sz)q&9EKZpGs+BI2iqFvpk z@7~@XvWas-i^fI`@6^mo3IP3-bO<5T_Z=P9&IRf*b7p(8XQBi|UDp+_o>-lyW=zU^ z3(mvX3jDWJzQk|6zH$%JSVp)qLl{EABdPpD$mJH_*QX|Y`IQsP5{57}?+X{^2LRTm zS@$+=RA_I{h{ccv-g2S9E5BoTyA@~d`L$L1p8_;CGYXBme^k?WMaG3jiR*cvwd|v% zTIS1_5%tHfeXg^X5cjPCyCl$*WrS{If_fqvUBQ%$vM+79$a76=X;fsBoS^i^3|f2C zZ?V&C?x9hRx@u??4Oh$O4RZH>8WlL$7_u2~bkW)jPMJA{^nW$|a*<6J+Tz_%+Tx83 zeMa&fjbcBXUj|d~H-pV9o7T`${3c!ncQj-HUw~EDYFtz4d3P>9Xr_J{!77dLN}9nG z245~l^d1blbuSwSm4P3Pn|fW0O!oW+UFgvdh&+0KT;z3cR`%eF{!Z}iLzj?OXjGsR z8bw^$1jepx28%p2$*Mr3m;XdUTSWe(|9{5+vWT1hqAS{`iS^daRZSk8kDdB@CA3_0 z``)Xkq8?sED0uQlZyE8-#QkFC-|l2>)s(x|LsoQUO}Y( zX@qes9z>G!8JP&-Oz2u@|VhUn_o8GaUCBucft(1gykoQ6=RCYNseUGK#wsVe>$_W>`U0P z4n9(=D!g!&0E?6U)2Zdjt2y-G#|;crH0(D9>;0MQ^)4A*NqP`bcHbRzxY-TEo8{bc`7O3 zQw&&*bH~5+f`_37wzdu^6@j5w{p~^%D1HMTN)ASfakFR9VmZFss2?t zl8Sg=RZp=bqJ2tvb>`1duCwUWeyqECR zoaj(25G-3@&$YvFrTV6$C4}-<2*JB!}{VWrPP5%aXZa3lF%72bJ-P zrYvyXzHTe%O@vkvUe0_v>Uc%OiC}k#p<&(GKa|bs&-|h|k@8NWisDr88v9@x`%Z8- zC(GGafn;d%dyjjpcxFsLzVBk9>&KMhIMU(k`Pp6L0m+ZPzjL!)ngY-23slGuI;!1- z!l*LPlV5#ugF7GFz5VAG@d7qg|F{W6@K`CUhdwh;QYxJnX8nm zfx*JGV5Jr(I39p{Roo(MBllxk4{$`-k_xo1ErrHX43h-$l}H*DJYdB8ItIZaTa81QbVoqJ3Xr zKe@Ze)V@g8NlDy2?{dIhw$EjYv(lt2!BsH2m2`xb2h6M#4wx6`T+ECW0~XUH{#}{n zeBm~djwQ?4u|pP=1&gvYX)1&a)${f#YQ_@$c zZ^t}i7^~l2W23WP6u0HzrL($1x7pD`1fGv{9P<5e-GsCM%Z^K!2D_h*f8*2F~-*GW(M|;uFdo^t; zo$f}UIBjO4b(Yq=u-+`ik!L-!;cIkR)x_os(|QPjC-Z@b|5PjDUMhPeUGMLWm+Lq} zNsU*i0@<@C^c#%)mo=wOdZ~_3y2G_M_k!|>9WkWMj<}MHB9wMnNixnD($2g+e0|oX zj{P7D7}AGyA&*O?!Cw?hUF7EG%oNzB!+29lz{&P?(@PY~mbFL9$%xrp)dTJ4l22^r z)0Ue)+wS@4_%jAeu8i=N&DHSpLyXsz+m+c$9CqLtFs>OBVZTDQh&H5owL%6RZ(}J6 z9fc_2v;%)f->W0<(|c%4x)x@%g>IGNvN?C?@J(y_mb~Tvwz1S60^?wq2%a3{N6lLj z%)Cdy8b-&d-wwy6`@gk3(GmhBzv*%w)VU!_HmefMgezo*moa^v-tyG{SNs1U=>K3f zn_o`EgUs7ITaqcZR|rWg-t+@PNbvGK@)-qcluD{-=T zPDW@UtqzU>a_0OGRZR$gnf@~r16)KWxCRv{i#&en2+%W$RLc1dB=VrhG_KXqB?Wa#ELk`$D#BIPco?`83AaF_)XPBfK8VFF$bk4Dt;hygN$-2ss zJogJ!{f9Z3VV+3Aty?y6NJDa@OajiMY9y$Wag2Dzh#+0pGCiVv0gLq@GRUouE+dqkpp?Qh7MRy*_%~;G+U9(U;3jWokKq=wpqO4 z>k249=8{2u4?i3B-7Nj!F)S{X5D%Mj8i}AgX%HB7Rl>ajv;n!{l&r>F%$L2V$BhiCU-0Ra*bFG&183CG> zL(=tW&@r=D%q$g(BQF$_{7$4SAuJ*^i*Ej}wo8*gU2J^ym0Q;sZJr0t& z(a8cM;R@1^^*4!hY}DWbr%jkBrCq9k*^;TH&w26*NtcNo$VpeGQ|8xP1M!Q#_D$K! zie-0vQx!S>rrdT9u>fY%5wBbvsm>{L0zBy6^XTaE!0HC87+*ja(cGzkiHzm9-X~sR za;TG9@_)9Edop;_3myiPmv@NtlX6j_b$Qk&RLpBW3sXUEyP@%+u3!<1iH=KUVJf6) zYKVEY;L*x^lKK)erMM$gg#+Nr%BRD`vi%6{5mU+^GAuNzCXiJh{ZVg|ZlQ`8v ztFe0b8nzU=|Mj*gn%!~T;%U!PXoKNx%kJcqG)101|N{zig zn%95BOqK9Lm?e2pOzA_iBS-g;%r7wXV!q!pQjxoq&0x9LQ`wq+TWlH*&kvFP5N)43 zF3WbJdG$`y!j{6RawKpZ_9_LAm0#rS8Ah}8u82J0Zx$!!V5i#rn_lS40=qP6(8^7~ zM$mn{w@SWqJGph!>`%bve?GRZBAZK?BYpU1A^A}cX&Q`+81bC*l|dqL1_AU9?8snY z>S5yMv~)ggAR16ngpIg!oyArVc}ni`i!~Qsxt|yPD7F)?#;IYdwPm8eol27N`VIYZPyb6CU{mm_>MPLV$81O18lEi3 zA^m?g%~0HWzsmMzY}7FV`mC}$t1l96QnP_S4#K`ozoJ*jUqo}BY7kZXxibrFbIlyQ z=caJ9b@4)mx}5Q886-J@GDV8{J%XoMn}GVkIkvCZ8LSRn$l@^rRk^v4;>}`aKRKlplU#59=97l*1 zYV%9_W$H5y(PNnFsAtSoisCMc6O1b507ra}{2+Q^ijrllvm^a_>kFKy1P@-+g&&9-Pr;k1K&Im#!1UY4 z4HFQJiVd-`>)Uvo96AT;IZ1s?Js_%b+|GbV9NnIpWdjX*edx#h3};M^3@eRTywJw& zpK)w*wLn_U+GDeNkG-o?@Q?n z8GcUd>9Y{NGqq0kRR!wFWME*5`VA-O)-}ydTMZ$s$J1dI-Kinz1YZw00+27L&0723 z#Wb}Scurc(RrA|jeKAh!M-o9X1(x~_R9S*H6pmToe{`e$qw6ZwT8gE;sjV#= z>kjpH>I}?`_5nLAtDjbHt%eK$@q5@>UMH6}EYkFX)mNgbDx$VD113x<%}T(~MFn6( zB&k6iPAL)8thLaoQ8zV&6RfJgi3CLEcTT4to?69xB&aiG{YE$-u}LXZ*`uQLJeITa zntwBVN~90_2XDXyG&1B;V&K6@9swS zbANIR4VHv;K7LaiFRRCa9HenwjBw_8`^7N)WWN$di$DDintiq!9x`xJxzFf^r zV`s^nA|}aQ23k}*FI_a(wpyJaFi0vNkO*q5@K!DxN-1YBk)|AS)YeqrN^fD$vmYLu z7QW#`=q^O&`zxa+b+Wbn&%`PZEr>3oMaNbF5Tbo-H52IGzpc-bR~BBf8b4tt#Tg)>VWOKEfh>E4^O`!h>q|HWZnSstMCpSMWBZl0szU~-wW%5~`+J>S&zc}1c zi&+^T#+uOjl+8V6xFCFG5xEj@ zd$30%J|>Cqy*NUSy1wOoCi39H)!~Ec-)4ZY$zf{+C(d@@?KS|5&On&BwUwSLK(*A3WIA4L51HEDrB*JIk^p9 zb*BMC_QLo9UsIg$1P`6(wcn0H-$Yp(*dHv2+Hb;{-YE9v-Wk#11l}hJ*jvWqx-$^r zYT3%1V6Rh#a0lwHcFRMlCQ6Z(O69&6Zj*yjA+i_aeErtqfkXUGP6;H3a`N!?&}OWa8(j>Jc>;N zL#XEPEh#b1bI^)~OL7ld2$q^y=b#^FboWQDQ6mdv$ceAw9*≫WCi*vP1;-l>@L? zbo{raSz3(Y=z7miuw4}!FFwBB1J@Dow+h*C@@qwr;a^tJC4;+obvJVZ+Q24p^qe*o zB<$_4C*@Y@R#LZxo<8z0lHE#LN8;l3Re{DtpBD^d=LbjC_dJ_^CCX@)6G(~jTm(h9 zK+H75vP&K+&=t9JUH7RF?AK;r1Wd$eOgq1WEm7}D7QqfCYot+3x)KcL2hxw{QALN`o}t4i(s1{p$Tu0j?5f&KtnoJQkja*)9+V>4RE*2&)^IT` z+t6MZ5sey~7Brc%6pVXAjtIuT_tPmKNg8SSA&8 zN@(gus&$kXu0Jrs3)Iu=zG^SRn%_L*;}ChgIw?jG!UhdpGZc%cD(%NT1cMXrzU)9Vgqn%=7WF4*P`>xXJs;>=0rNi&_buu(qMhZT z0r>9-_??3WX>&l+X*|?h|6J>1Tk6INj#`OGV_1gd zb0}{@o}O#+;BO9-d;q6u&;7=9^%t+$^#yFNk;_uTu1ue`iPF=4po%~QVNKn6kOju@ zO}@{NY8A(JZy6I#?yk(H70I8@tjx|bJ6wD-+Ur?Y-h`*<_;dr3ROpF_<03Ga9?|c1 zxijI|pN>`V?Z2knI68G0YPEqA)|QvRroY5S?!+3kj`e#!-2WXWm-e4mB#cvNRadu;#XHdqyG3WGCJ7@`@+*XhCsvE3b=kzY@FG`y8Z0jqmtbQBerr~;<=-5_Kn}6`$6K9Rg8AuwJdPF8G z=R3;EI=$C|0gzv%*r?d+(OIyYm{^J^>-jdxTgwOc&0VYJy2T$P1KcBy^}}uzw)DPe z54;FX-$b8HJu!@)&xx;`USJ?gf(d{8Jj z$^p>VhNI|Swm5B2x%XFTHN(MCJg2}&Cd%JlYb2H7Lriac&+{;p8@Vh>I0FRxXuzf%?#mw}Q9`j^1T3we%b zXo93^Y{sWpv__swIp1m^y^8W+590uKz~K^I*SymVUj?lf{8MRMYbI;lYF=82L`ke< zp!`Grsm$)dQKaxj_(OQp{{j9BeR71Oh)fnOME*|oZ2oh#a%@GL;O5^h??ZP3X1z?= z#D-m`yxO{GkY1HlGQ2nHj9gP*fN97n?_WhW6C|kRlm>YYu|iP8QIro@*KWIb)l@F~ zUz3!fBs%}q613|5Byx4Ksq@e}B=Z3;d58~$TFryxOMQlpt7=p)SHQMQr2WXF#nCvXOQjRR(K!a9GNykGH z>P&w9?gwvvjav#2K|cbuJ_n2Qh_mUs;@QNRvx-VQD@|aE4rVg*d;LZ2O{SJt1Wqps zvHC{4F595=(*y-AjAZR3o-vU&HkYe*Y-njXgz1>)YXwlul$G;FO33#+WUhwSGTNSn zjn%|?xC>-ksjF9lGLQlU1X&f^pVy=*wxsP_8WX>R&OXEqLIqGu_piPot4#3I@)A!b z*{4Zrzvn;b#6g(1Mm{p$+(`61P04YWDIN!hO4Y8|2 zFcMrw;eVTS%^x802d%%`&{U@1eBw-X*SQ+P)C5hWIiW<283_V0S5&l0_YFsmR;5RQ z%8l^owD`B9>*b&$rP0K<%A|wA*}pTPSEZ&zu-~QW+UaWJ!dHAb{*PAgUZ#e;B8m9T zvCCaA#bVys!ho|r@Q2W!jQK2LQKU`JZ|SxAF<;OmU&5*LdBRoTZ~$ylGyU^Rwrc&z z;)KKa7V+qc4_I_};QLJY5hXLsz)IIOW;A3-)hb#hwO`DSwsDbVX>Yj(w}fDMwjL z_)7R6ET3Duvc;u}1|j10A0KBl!Pq;$5E%3CsEKzSuANO&k5eg)vh9@m1YIsZ;L1$p zp;~0ZGVK$u^KSmgDb{1HRIF~s`cqcu-N}=Lw2jIg!N%eBw^6y55wTw1OGB9W!VNCy z6$pPlgNi)^Vr!y2vbocIkOFL)94HVOaGqfB)78am&PTP4q&Ua3Yx1aJZ#&DxE|X=8 z^A4spAad85zkcnf^y~ao-X|(>%ek#0$Mv8lIgpR3=6Op}p6lJc`9j_PjQi8EJD3v5 z(A?Thp0mO1CLIH!I=Y4jkr0O!bT7Sq);4?y4sQi~s8z^p1Ulm?Sl?H`@!wln2Wl3; ze%hLKuG9s5#Or&0P!38C{|s#!*e+1stj)@N0TjWOz;E(t8->X?P77qVjVfR=1~AT? zFm*$WQ)%;{?L3pT5OH*pPcwdNoEKPTqqa+SqxNUVfbD!>>PY<^UpVPO%*nynT3Cm7 z!U%x$euQy?E^O8^uJYv+1vuv3g_3iL@GIAUMbE5{$HZ2*-y*OTv1>3eDPA9e28jzw_vPJByv#*^yf z^9)%j%R7aAAJHqca_lYnX#1g&2bHu?RaOqeQY=$xqUh!;f2--`ECeERC%#IhE-k;oSBJld6?51 z!+=G;CiGX9^d2SHzV^%j4C*~&R`Lg_j(irt(MbOlljOIr`o1S7NohYKWkf{Ik6KdK z?2%;tBbsA=t@f9gs)A-_NukmX;C;@2!T;s|pT-Vj(x$y)7|mV7HfOV4i|mbASig}C zYGdw+(R{LSbSdRs$n!yk^XIj*TKTTT@JAKS&HRdxd1@UBXtO0R zmw{e+@WvR0vc-j{$=!(LE%t>9G-~M(L^;M@$kz|h5rI)9r<;y+m8UVOV908gk93hk z-bzDQN{rqd#+9VaP^SP732-ih$*|L*2HcR|B409=*{)8Tt|kt9t@EImV-6N# zduPnfXJjVS=qucYAo4lGF3V9vR#)$i=D8exX>9LgnCr8kfL>Q_(4?ruULTBSbqIds zj%VaXnWCOjL;x47PUGo4ZS~Kc7wS=ZT$hAiyxIv6`?Er-^+4A*Tm|UFPIfe@Y^-G3 zxyp&vRt}e^-cBYE?3+YIGW!K&BR}Jkd{!NC=3t>e>ap+N@o;Pc*}ZEGscbHM8z#Dz zaK!kIJY}g$kp+kQvMTwf=6j~=grCL!x$6&sMZz1wWL`aHa>$C~%6wP8MxHgJ5{_wr z#8P4Fr>|YYTy=w6YW+Z^)afR*E1l7R%1w6~NuFt_lUJ*+Xv8Q+cNGi4Wfc5HZg$}D zJ8%DVhzcFI>(x}RF2w+nb6muRQ*V&fG-rG2p{e#P-6Qmm;A;AF9t4B~(``S=gFw@?_RK=e!O2ZRHMt=| zNf#T21ZMWk#N-nYsd)RttQdTf@sIiv8=WfYvF`2ke`aSt_0MVSBNunsNjXU--%fFX zDjc5T;-yl%T@L^z3r(@)J10eh8mPgQb+CX#`8;L?m$2dL#B1Q#IVeeU-sKnWj`2{1 zJ4u;HerpVN}@D*d|LfU^@w|WSHIK0nWo%nUDcu{)DX=F#`)%7^999wl*n#wLf(K{zz5% zZQNn}zP%P`+Wb^&!F2t(DQ)`|shesh{}1XQ$)2IT`kk-Ls_CCNiaAU|idw^NhIAX+YdK`(Nc#Yh0nDlf#nTj( zm=N=1?X_#wxmn8yJWEk7lzcQe(j-GAi?C(8pneK?$arSupf)BygOD;NWTiK$+1kU2 ze&EnFNkhLLQ4SaL!Oo>jD9TETy=aZmQQA|`6?XZP(~J&W9<>Z4MyDN_bW)kmSM>`G zoVH%e8Db!wsc-c#R>{vPxz%!PkHs0P=)u+VC+U?l&xm1x-zz22f>&+lW|{%f9^h!T z9y|aF@H1VDDfUh8n+hTPL2nql6E?Ky9!^G8!Mu-gl5`)-1!#~!B|q^VQ_6%I7cL+O zN;sS(X4h~%;8A8$BAtFtZ-^CC6x`LsPjiIfssl~W+SD$e@inqnHZQ{^_4BrE=s%Fq z{%1(pY5fW)6Bx*$+k}dqy)tnuAKN4f0eGLgKbsb=+BbsCTgI4-m@h9OtaPZDOj*aDwT$w zAFw2~K5}csy#AVkq}XlO5O+Fzernl#B3&1GWew zs(ERCiKd6K2iddzraH#1(_l(t*^zRPaSDFN?2iBosIr8bPF+g}@& zX!l74!M-J9!N+{$z8Z?1|GKZ!P5N3XsfzXUfyhx8%RHg6Y^TC!`T?PB)r{?ZLyK4b zk$h6$T(ryIkE>(sMkL*p?C^=Pfa&c@dKr5HJH45o@?%{j{6kC5gFffBErOfEbu)gF zu33BbVm0v7nWH&>>WSB3%>uneCl<{bf<2^(?dV?QTLYsX0yTZAX>V4IB8{&KA=J5K zO{UZ~%xwD{v2ggF^#f${81n0V&H{7<(BwYjCxig?pU#a|nciPcg zh{z@2(?&4g$AFU>3R+Xk$Z~igZ#+7SSB;phaV)pK;Ctpo?yEHyT%7EbH~ff|_`|1Y zOB~By1R4z&kd zDV3}k=gvHxbX`Xxk(063@~|S$U-SHHyA7m0ofI9jcB($E^?}M)1C`kRnZj9LEHpEx z4oMSkkqZxDv4&Km>Z_kd!JTZW{ghe;LfT$NQ>FR$GB5vNREoL-hWf(&`J@ybiT~!O zgG?Q}R)+h!ojd$o$4tk~XT?=AiZEx+OfL4VtG`yYbp?D~=>k8AG`v@&<>_({!c;#o z{eUORwE%2z2<&KLG{?H!HfxsN;K!A@Kov0KuzjR~(dJ<&p2oMu6*sW-YY)be{=2t< z&gN1h9tjXV`pi|JZP#X>OimN=tsmO}g9p@De0<#L2T8DkBsW+n=tJ5XeoA{vE8%-$ z8yHeSMtKOAOL9>|oprchKx5~hb1DjsGBXXvBk5tz2{BFBLiv*H#J=91daDyy*aXZR zVeNRK`-GO=jF57umzVgYsydRBJp|0C!L6>*A+0gn;teE=W2dHm9NMJn&84sT?uPgo zOm5%!`mu=?%j7!4|8EJk^2( z%oF}No^FFYvASKidojHmS9>qlmvQ3ciQBe0E^%O|6S$u&!a| z(ED4@i1Xh+geVSY%@jl0+Ltzx`|-EMH^>K5N$&R10k?;^g*1n$j=bmLOGR0SwG|@F z?Inj3a3Uv~?AVH(Nrr}V&W}zDmgc7glR2#)M5i7}B`m|;$)ZUj7S4%LmRpEVX;|HO zrM9G=#J^3FYqX}Hl(io1eqc#8AmKnaj+8MlST4H&H@x`bzN&uvWQaMe_xkmUz67v? zH-c+#Rujc-u@O_!w*g?l!m55EF+%*=ic}+9!fl_CpZ*+T*pHnNS&CUX7QG0MIq|yp zb#)PqbH=oXNDTn3WkzCHqa?-BK3|^77Y2m?ERb}VH>XlMsZEJYKom=na+1epFBK_V z9WVkpgfEw$I~Ew=j6jl1ZwTV6u0T4+tpZwGq|)hMG~X8St`=p=y%Xbh)_s6GyL(Mg zHpCYFxY8obEnJ1W^DM}ou_(We2>f9je+Ub&osnjwJTk9@isx3Bi{cBk*kv65dH3*x zsL0ZYMGmb_UYuBUNq#Bfj^w@A3eXC@(Mh=C~IxzvmYr=81~fvD@_4}&uegyEpi8kcZ*4mtG23!KxJ=0AAo z3DWq^$~Y+&gelzRIl8!Z5);@x91D9V+??sBC@j*u`r1|FhCVSEilG;IGZr##9fp|! zGsyM76JvcJZN7TDZfEWzY3Cj0X)M{%dAdSmn{LuJ!Tl}q@8QB~Q(HG?Y6rqNTfqIT zmR3}hk=9U*i)K)g^OXCW=iJ;s%D`j9$~)}zFhi=abpqbNZFxH~~@gGgwhHxVfz=aaH%dz6p=bX@}m&<^aAoXGroK0CPs`z$O*&Xu#k!HFo*_VxlTX SrUSS+W1jQO;gvSR#`+&l*0kUN diff --git a/hw/hw6/hw6.tex b/hw/hw6/hw6.tex index 2fc7646..6bd61cd 100644 --- a/hw/hw6/hw6.tex +++ b/hw/hw6/hw6.tex @@ -249,7 +249,7 @@ from \(\AllReals\) to \(\AllReals\). \end{enumerate} \pagebreak -\prob{36} If \(f\) and \(f \circ g\) are one-to-one, does it follow that \(g\) is one-to-one? Justify your answer.\\ +\prob{36} If \(f\) and \(f \circ g\) are one-to-one, does it follow that \(g\) is one-to-one? Justify your answer. \begin{proof} Yes, we will conduct a direct proof. Assuming \(g(a) = g(b)\), \begin{align*} @@ -269,7 +269,9 @@ are functions from \(\AllReals\) to \(\AllReals\). \end{align*} \pagebreak -\prob{74} Suppose that \(f\) is a function from \(A\) to \(B\), where \(A\) and \(B\) are finite sets with \(|A| = |B|\). Show that \(f\) is one-to-one if and only if it is onto.\\\\ +\prob{74} Suppose that \(f\) is a function from \(A\) to \(B\), where \(A\) and \(B\) are finite sets with \(|A| = |B|\). Show that \(f\) is one-to-one if and only if it is onto. +\parskip=0.5em + A function \(f: A \rightarrow B\) can map each element of \(A\) to a unique element of \(B\). Since \(|A| = |B|\), if \(f\) is one-to-one, then every element of \(A\) must be mapped to a different element of \(B\) and since \(|A| = |B|\), every element of \(B\) must be mapped to by an element of \(A\). Therefore, \(f\) is onto. In the other direction, in order for \(f\) to be onto it has to map every element of \(A\) to a distinct element in \(B\) since \(|A| = |B|\). Therefore, \(f\) is one-to-one. Since we have shown both directions, \(f\) is one-to-one if and only if it is onto. \pagebreak