; TeX output 1996.10.04:1259m܍3$(܍3$6 3 cmmi10AB^B72_G\OR A\ORBX0AB_B `꫍:K_2cmmi8:U72_G\XOR A\XORBX0A꫍V:U_~B72!G\IMPLIES A\IMPLIESBX0A!B72!G\IMPL A\IMPLBX0A!B72 G\IF A\IFBX0A B72$G\IFF A\IFFBX0A$B!;t|{Ycmr8defk72 ,!G\IFFdef A\IFFdefBX0A defk ,!B72^:1:: c^G\ANDdots A_1\ANDdotsA_nX0Az1 !^:1:: ^BAzn72_:1:: c_G\ORdots A_1\ORdotsA_nX0Az1 !_:1:: _BAzn4,nG\is x\isyX0xny6I,NG\Nat n\in\NatX0n 2I,N68G\Forall \ForallxP(x)X08xfPV(x)69G\Exists \ExistsyP(x)X09y PV(x)ҍ6,"V 3 cmbx10The2\ANDMacro6ThistmacrocanbMeusedforthelogicalconjunction.wInadditiontothe\wedge6macroUitaddsmorespaceandtheform!ulastendtobMebetterreadable.6Compare#DCaꍍ|~x=1\ANDy=x#proMducesx =1B^yo:= x |~x=1\wedgey=xlproMducesx =1n^yo:= x|~x=1\landy=x,yproMducesx =1n^yo:= xC31*m܍3$(܍3$6The2\ORMacro6ThismacrocanbMeusedforthelogicaldisjunction.Inadditiontothe\vee 6macrofitaddsmorespace.Compare#DCaꍍc(x=1\ORy=x,yproMduces x =1B_yo:= x c(x=1\veey=xlproMduces x =1n_yo:= xc(x=1\lory=xlproMduces x =1n_yo:= x)ȍ6The2\XORMacro6ThismacrocanbMeusedfortheexclusiv!edisjunction.^Ithasnocommon6coun!terpart.Thefspacingislikeininalljunctormacros.5gmx=1\XORy=xlproMduces <1x =1꫍V:U_~yo:=x6The2\IMPLandthe\IMPLIESMacros6These macroscanbMeusedforthelogicalimplication.Inadditiontothe6\rightarrowfmacroitaddsmorespace.Compare#!aꍍkfx=1\IMPLy=x2proMducesx =1!yo:= x kfx=1\IMPLIESy=x!proMducesx =1!yo:= xkfx=1\rightarrowy=xlproMducesx =1!yo:=x)ȍ6The2\IFMacro6ThismacrocanbMeusedforthelogicalimplicationwritteninrev!erseorder.6Infadditiontothe\leftarrowmacroitaddsmorespace.Comparewv.kfx=1\IFy=x>jproMducesx =1 yo:= x kfx=1\lefttarrowy=xlproMducesx =1 yo:=x"6The2\IFFMacro6ThisnBmacrocanbMeusedforthelogicalequivdDalence. 5pInadditiontothe6\leftrightarrowfmacroitaddsmorespace.CompareT.]v=x=1\IFFy=xUhproMduces,x =1$yo:= x ]v=x=1\leftrighttarrowy=xlproMduces,x =1$yo:=xC32 :m܍3$(܍3$6The2\IFFdefMacro6Lik!efabMovebutwithasmall\def"abMovethearrow.Grĸx=1\IFFdefy=xlproMducesxzx =1 defk ,!yo:=x6The2\isMacro6This macroisfort!ypMesettinguni ers.Inthiscasetheprede ned\setminus 6proMducesftom!uchfspace.#DCaꍍJf\{y\setminusx,z\setminus4\}#proMducesMfynnx;1z5n4g Jf\{y\isx,z\is4\}`proMducesMfyNnx;1zn4gJf\{y\backslashx,z\backslash4P}lproMducesMfydnx;1z{In4g)ȍ6The2NumtbYerMacros6Thisaremacrosforthosewhoha!venoaccesstotheA,[wM S9-T,[wEB Xfon!ts.[It6mak!es~thesymbMolsforthenaturalnumbMers,Dintegers,rationals,reals~and6complexfn!umbMers.Theusualmagni cationcommandsapplytoitaswell.C33m܍3$(܍3$#6b ş} ffŤ} ff͟\tiny...6\normalsize...\Huge͡ ffX_X6zffz/Jq\bbBu}ffW%Aacmr6IBk+ITBL(o cmr9IuVBIUXBjI,B(0XQ cmr12IB\)9XQ ff cmr12IB0>DtGGcmr17IBCDtqGcmr17CIB;_GDtHGcmr17GIuBWޟ}ff`3I,BiITB\Complex\bbC͟}ffY˞݄]ߎXBɺCnU҄lΎlw@C̄Ovh CVƄt"C&(Mu|+sCʝ@{ h~C䐬aF~t}vC`A 0㎒NCXl ;Z󄎒VCCB=˄j$\bbOne ˟}[ffX1lm1Tl71uVl1UXlu71,l1li1l1l$C1l>ƦG1ulWݟ}[ffb1,liџȮ1TlD%GUnfortunatelythemacros\bbC,\bbG,\bbO,and\bbQUdonotscaleprop- 6erlyISwhenusedinsubscriptsorsupMerscriptsofform!ulae. ƥThefollowing6examplesfsho!wshowthesizingcanbMeachievedmanuallyf>\bbQ_{\mbox{\scriptsize\bbQ}}lproMduces`iBMu|+^Qh8HlΎfYKQC34m܍3$(܍3$6The2\Forallandthe\ExistsMacros6TheHgeneralproblemwithquan!ti esisthatafterthequanti edvdDariablethe 6follo!wingDyformulaisnotautomaticallysepMeratedwithasmallspace.This6canfbMeo!vercomefbythefollowingmacros.GThe-4\Forallandthe\Existsmacrostak!eoneargument.wTheytypMeset6therespMectiv!equanti erfollowedbytheargument(i.e.thevdDariable)and6 nallyasmallspace.Asusualtheargumen!thastobMeenclosedinbracesif6itOconsistsofmorethanonec!haracter.+OtherwisethebracescanbMeomitted.6Thisfallo!wsaelegantnotationofshortquanti edformulas.̩6\ForallxP(x)proMducesg+8xfPV(x) 6\Forall{x_1,\ldots,x_n}P(x_1,\ldots,x_n)lproMducesg+8xz1;1:::l;1xzn NPV(xz1;:::l;xznP)6\ExistsxP(x)proMducesg+9xfPV(x)6\Exists{x_1,\ldots,x_n}P(x_1,\ldots,x_n)lproMducesg+9xz1;1:::l;1xzn NPV(xz1;:::l;xznP)0fō61.2TThe6߆T cmtt12FormulaEnvironment6This en!vironmentallowstotypMesetlogicalformulas. M,Themainproblem6with1theeqnarrayen!vironment1wasthenumbMering.>Inmultilineformulas6m!yintentionwastohavethenumbMerinthemiddleoftheformula.Inside6thisfen!vironmentseveralmacrosarevdDalid. 6\beginfFormulag[*': 3 cmti10labpel]\endfFormulagQ`Startthelistofform!ulas.OptionallyalabMelcanbegiv!en.ThislabelQ`isfusedtoreferencethe rstform!ula.l6\=Q`Startfanewline.6\>levelQ`Start0anewlineandinden!ttothegivenlevel.ThisindentationisdoneQ`insWquan!titiesof\FormulaIndentwhichcanbMesetwiththe\setlengthQ`command.ThefdefaultvdDalueis3em.6\Form[labpel]Q`StartWanewform!ula.-OptionallyalabpelcanbMegiven.-ThislabpelcanQ`bMefusedtoreferencetotheform!ula(see\ref). GNo!wletshavealoMokatsomeexamples.,First,1weseeasingletwo-line6form!ula.NotethatthenumbMerattherightsideiscenteredbMetweenthetwo6lines.C350m܍3$(܍}Qn6$0.9jQn6\begin{Formula}[form:1] JP(X)?\IMPL6\= Q(X)?\IFFR_1(X)6\>1?\ORR_2(X)6\Form[form:2]JS(X)?\IMPL6\= \neg?Q(X)\IFFR_1(X)\ORR_2(X)6\end{Formula}C5##2aꍍPV(X)! ;Q(X)$Rz1(X)8> _1Rz2(X)!(2)%.S(X)! ;:Q(X)$Rz1(X)B_Rz2(X)!(3)IΚ61.3TTheNewTheoremEnvironment6MyexpMeriencewiththenewtheoremen!vironmentwasthatIhadacertain6sc!hemetouseit.!First,everytheoremgotalabMel.!Thus,everytheporem6w!as+followedbyalabelcommand.Optionallyatheporemma!yhaveaname.6ThisAnameist!ypMesetrightafterthenumbMer. Thebodyofthetheporem6allw!aysestartedinthenextline.Thislettothede nitionofanextended6NewTheoremoen!vironment. 9Theargumentsarethesameasthoseofthe6newtheoremen!vironment.Buttheenvironmentde nedbythisextended6command>tak!etwooptionalarguments.dThe rstoptionalargumentisa6labMelU tobeassignedtothetheporem.ThisU argumen!thastobeenclosedin6paren!theses.5TThen8secondtypMeofoptionalargumenthastobMeenclosedin6brak!ets.IIt5istypMesetin\smallafterthetitletext.IThethirdoptional6argumen!tisenclosedin<>.ItistypMesetin\small\bfandsurroundedby6paren!theses.6O&¤ 6\NewTheorem{guess}{Conjecture}6\begin{guess}[Fermat](thm:fermat);?There?donotexistintegers$n>2$,;?$x$,?$y$,and$z$suchthat;?$x^n+y^n=z^n$.6\end{guess}ffConjecture21>ϲF*ermatTherpe?donotexistintegersn >2,bx,yd,andzf3suchthatxn<+nyn = z{In#.6PGThercommandsusedtot!ypMesetsomeoftheoptionalargumentcanbMecus-6tomizedninthefollo!wingwaye.v Themacros\TheoremTitleand\TheoremNameC36;)m܍3$(܍3$6are_usedtot!ypMesettheirargumentin\smalland\small\bfandenclosedin 6paren!thesesnrespMectivelye.v ThismacroscanbMerede nedusing\renewcommand6asfsho!wninthefollowingexample:*ݍ46ߤN cmtt9\NewTheorem{theorem}{Theorem} 6\renewcommand{\TheoremTitle}[1]{{\sf[#1]}}6\renewcommand{\TheoremName}[1]{{\small(#1)}}6\begin{theorem}[Fermat](thm:f2):Theredonotexistintegers...6\end{theorem}ff6hTheorem217_F!ermat(conjecture)6hTherpe?donotexistintegersn >2,bx,6hyd,andzf3suchthatxn<+nyn = z{In#.C37F;m wHGcmss17oqGcmss17iGGcmss17eR6ff cmss12[ow cmss9Ls6cmss8Ks6cmss8H߆Tff cmtt12GDtHGcmr17CDtqGcmr17>DtGGcmr17 3 cmmi10'K`y 3 cmr10%m#R cmss10$