\(\newcommand{\footnotename}{footnote}\) \(\def \LWRfootnote {1}\) \(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\) \(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\) \(\let \LWRorighspace \hspace \) \(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\) \(\newcommand {\TextOrMath }[2]{#2}\) \(\newcommand {\mathnormal }[1]{{#1}}\) \(\newcommand \ensuremath [1]{#1}\) \(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \) \(\newcommand {\setlength }[2]{}\) \(\newcommand {\addtolength }[2]{}\) \(\newcommand {\setcounter }[2]{}\) \(\newcommand {\addtocounter }[2]{}\) \(\newcommand {\arabic }[1]{}\) \(\newcommand {\number }[1]{}\) \(\newcommand {\noalign }[1]{\text {#1}\notag \\}\) \(\newcommand {\cline }[1]{}\) \(\newcommand {\directlua }[1]{\text {(directlua)}}\) \(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\) \(\newcommand {\protect }{}\) \(\def \LWRabsorbnumber #1 {}\) \(\def \LWRabsorbquotenumber "#1 {}\) \(\newcommand {\LWRabsorboption }[1][]{}\) \(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\) \(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\) \(\def \mathcode #1={\mathchar }\) \(\let \delcode \mathcode \) \(\let \delimiter \mathchar \) \(\def \oe {\unicode {x0153}}\) \(\def \OE {\unicode {x0152}}\) \(\def \ae {\unicode {x00E6}}\) \(\def \AE {\unicode {x00C6}}\) \(\def \aa {\unicode {x00E5}}\) \(\def \AA {\unicode {x00C5}}\) \(\def \o {\unicode {x00F8}}\) \(\def \O {\unicode {x00D8}}\) \(\def \l {\unicode {x0142}}\) \(\def \L {\unicode {x0141}}\) \(\def \ss {\unicode {x00DF}}\) \(\def \SS {\unicode {x1E9E}}\) \(\def \dag {\unicode {x2020}}\) \(\def \ddag {\unicode {x2021}}\) \(\def \P {\unicode {x00B6}}\) \(\def \copyright {\unicode {x00A9}}\) \(\def \pounds {\unicode {x00A3}}\) \(\let \LWRref \ref \) \(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\) \( \newcommand {\multicolumn }[3]{#3}\) \(\require {textcomp}\) \(\newcommand {\toprule }[1][]{\hline }\) \(\let \midrule \toprule \) \(\let \bottomrule \toprule \) \(\def \LWRbooktabscmidruleparen (#1)#2{}\) \(\newcommand {\LWRbooktabscmidrulenoparen }[1]{}\) \(\newcommand {\cmidrule }[1][]{\ifnextchar (\LWRbooktabscmidruleparen \LWRbooktabscmidrulenoparen }\) \(\newcommand {\morecmidrules }{}\) \(\newcommand {\specialrule }[3]{\hline }\) \(\newcommand {\addlinespace }[1][]{}\) \(\def \LWRpagenote {1}\) \(\newcommand {\pagenote }[2][\LWRpagenote ]{{}^{\mathrm {#1}}}\) \(\require {colortbl}\) \(\let \LWRorigcolumncolor \columncolor \) \(\renewcommand {\columncolor }[2][named]{\LWRorigcolumncolor [#1]{#2}\LWRabsorbtwooptions }\) \(\let \LWRorigrowcolor \rowcolor \) \(\renewcommand {\rowcolor }[2][named]{\LWRorigrowcolor [#1]{#2}\LWRabsorbtwooptions }\) \(\let \LWRorigcellcolor \cellcolor \) \(\renewcommand {\cellcolor }[2][named]{\LWRorigcellcolor [#1]{#2}\LWRabsorbtwooptions }\) \(\require {cancel}\) \(\newcommand {\intertext }[1]{\text {#1}\notag \\}\) \(\let \Hat \hat \) \(\let \Check \check \) \(\let \Tilde \tilde \) \(\let \Acute \acute \) \(\let \Grave \grave \) \(\let \Dot \dot \) \(\let \Ddot \ddot \) \(\let \Breve \breve \) \(\let \Bar \bar \) \(\let \Vec \vec \) \(\require {mathtools}\) \(\newcommand {\vcentcolon }{\mathrel {\unicode {x2236}}}\) \(\newcommand {\approxcolon }{\approx \vcentcolon }\) \(\newcommand {\Approxcolon }{\approx \dblcolon }\) \(\newcommand {\simcolon }{\sim \vcentcolon }\) \(\newcommand {\Simcolon }{\sim \dblcolon }\) \(\newcommand {\dashcolon }{\mathrel {-}\vcentcolon }\) \(\newcommand {\Dashcolon }{\mathrel {-}\dblcolon }\) \(\newcommand {\colondash }{\vcentcolon \mathrel {-}}\) \(\newcommand {\Colondash }{\dblcolon \mathrel {-}}\) \(\newenvironment {crampedsubarray}[1]{}{}\) \(\newcommand {\smashoperator }[2][]{#2\limits }\) \(\newcommand {\SwapAboveDisplaySkip }{}\) \(\newcommand {\LaTeXunderbrace }[1]{\underbrace {#1}}\) \(\newcommand {\LaTeXoverbrace }[1]{\overbrace {#1}}\) \(\Newextarrow \xLongleftarrow {10,10}{0x21D0}\) \(\Newextarrow \xLongrightarrow {10,10}{0x21D2}\) \(\let \xlongleftarrow \xleftarrow \) \(\let \xlongrightarrow \xrightarrow \) \(\newcommand {\LWRmultlined }[1][]{\begin {multline*}}\) \(\newenvironment {multlined}[1][]{\LWRmultlined }{\end {multline*}}\) \(\let \LWRorigshoveleft \shoveleft \) \(\renewcommand {\shoveleft }[1][]{\LWRorigshoveleft }\) \(\let \LWRorigshoveright \shoveright \) \(\renewcommand {\shoveright }[1][]{\LWRorigshoveright }\) \(\newcommand {\shortintertext }[1]{\text {#1}\notag \\}\) \(\newcommand {\LWRnicearrayarray }[1]{\begin {array}{#1}}\) \(\def \LWRnicearrayarrayopt #1[#2] {\begin {array}{#1}}\) \(\newenvironment {NiceArray}[2][]{\ifnextchar [{\LWRnicearrayarrayopt {#2}}{\LWRnicearrayarray {#2}}}{\end {array}}\) \(\newcommand {\LWRnicearraywithdelimtwo }[2][]{\ifnextchar [{\LWRnicearrayarrayopt {#2}}{\LWRnicearrayarray {#2}}}\) \(\newenvironment {NiceArrayWithDelims}[2]{\def \LWRnicearrayrightdelim {\right #2}\left #1\LWRnicearraywithdelimtwo }{\end {array}\LWRnicearrayrightdelim }\) \(\newenvironment {pNiceArray} {\begin {NiceArrayWithDelims}{(}{)}} {\end {NiceArrayWithDelims}} \) \(\newenvironment {bNiceArray} {\begin {NiceArrayWithDelims}{[}{]}} {\end {NiceArrayWithDelims}} \) \(\newenvironment {BNiceArray} {\begin {NiceArrayWithDelims}{\{}{\}}} {\end {NiceArrayWithDelims}} \) \(\newenvironment {vNiceArray} {\begin {NiceArrayWithDelims}{\vert }{\vert }} {\end {NiceArrayWithDelims}} \) \(\newenvironment {VNiceArray} {\begin {NiceArrayWithDelims}{\Vert }{\Vert }} {\end {NiceArrayWithDelims}} \) \(\newenvironment {NiceMatrix}[1][]{\begin {matrix}}{\end {matrix}}\) \(\newenvironment {pNiceMatrix}[1][]{\begin {pmatrix}}{\end {pmatrix}}\) \(\newenvironment {bNiceMatrix}[1][]{\begin {bmatrix}}{\end {bmatrix}}\) \(\newenvironment {BNiceMatrix}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\) \(\newenvironment {vNiceMatrix}[1][]{\begin {vmatrix}}{\end {vmatrix}}\) \(\newenvironment {VNiceMatrix}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\) \(\newcommand {\LWRnicematrixBlock }[1]{#1}\) \(\def \LWRnicematrixBlockopt <#1>#2{#2}\) \(\newcommand {\Block }[2][]{\ifnextchar <\LWRnicematrixBlockopt \LWRnicematrixBlock }\) \(\newcommand {\diagbox }[2]{\begin {array}{l}\hfill \quad #2\\\hline #1\quad \hfill \end {array}}\) \(\let \hdottedline \hdashline \) \(\newcommand {\Hline }[1][]{\hline }\) \(\newcommand {\CodeBefore }{}\) \(\newcommand {\Body }{}\) \(\newcommand {\CodeAfter }{}\) \(\newcommand {\line }[3][]{}\) \(\newcommand {\RowStyle }[2][]{}\) \(\newcommand {\LWRSubMatrix }[1][]{}\) \(\newcommand {\SubMatrix }[4]{\LWRSubMatrix }\) \(\newcommand {\OverBrace }[4][]{}\) \(\newcommand {\UnderBrace }[4][]{}\) \(\newcommand {\HBrace }[3][]{}\) \(\newcommand {\VBrace }[3][]{}\) \(\newcommand {\ShowCellNames }{}\) \(\newcommand {\tabularnote }[2][]{}\) \(\newcommand {\cellcolor }[3][]{}\) \(\newcommand {\rowcolor }[3][]{}\) \(\newcommand {\LWRrowcolors }[1][]{}\) \(\newcommand {\rowcolors }[4][]{\LWRrowcolors }\) \(\newcommand {\rowlistcolors }[3][]{\LWRrowcolors }\) \(\newcommand {\columncolor }[3][]{}\) \(\newcommand {\rectanglecolor }[4][]{}\) \(\newcommand {\arraycolor }[2][]{}\) \(\newcommand {\chessboardcolors }[3][]{}\) \(\newcommand {\ldots }[1][]{\dots }\) \(\newcommand {\Cdots }[1][]{\cdots }\) \(\newcommand {\Vdots }[1][]{\vdots }\) \(\newcommand {\Ddots }[1][]{\ddots }\) \(\newcommand {\Iddots }[1][]{\mathinner {\unicode {x22F0}}}\) \(\newcommand {\Hdotsfor }[1]{\ldots }\) \(\newcommand {\Vdotsfor }[1]{\vdots }\) \(\newcommand {\AutoNiceMatrix }[2]{\text {(AutoNiceMatrix #1)}}\) \(\let \pAutoNiceMatrix \AutoNiceMatrix \) \(\let \bAutoNiceMatrix \AutoNiceMatrix \) \(\let \BAutoNiceMatrix \AutoNiceMatrix \) \(\let \vAutoNiceMatrix \AutoNiceMatrix \) \(\let \VAutoNiceMatrix \AutoNiceMatrix \) \(\newcommand {\tcbset }[1]{}\) \(\newcommand {\tcbsetforeverylayer }[1]{}\) \(\newcommand {\tcbox }[2][]{\boxed {\text {#2}}}\) \(\newcommand {\tcboxfit }[2][]{\boxed {#2}}\) \(\newcommand {\tcblower }{}\) \(\newcommand {\tcbline }{}\) \(\newcommand {\tcbtitle }{}\) \(\newcommand {\tcbsubtitle [2][]{\mathrm {#2}}}\) \(\newcommand {\tcboxmath }[2][]{\boxed {#2}}\) \(\newcommand {\tcbhighmath }[2][]{\boxed {#2}}\) \(\let \symnormal \mathit \) \(\let \symliteral \mathrm \) \(\let \symbb \mathbb \) \(\let \symbbit \mathbb \) \(\let \symcal \mathcal \) \(\let \symscr \mathscr \) \(\let \symfrak \mathfrak \) \(\let \symsfup \mathsf \) \(\let \symsfit \mathit \) \(\let \symbfsf \mathbf \) \(\let \symbfup \mathbf \) \(\newcommand {\symbfit }[1]{\boldsymbol {#1}}\) \(\let \symbfcal \mathcal \) \(\let \symbfscr \mathscr \) \(\let \symbffrak \mathfrak \) \(\let \symbfsfup \mathbf \) \(\newcommand {\symbfsfit }[1]{\boldsymbol {#1}}\) \(\let \symup \mathrm \) \(\let \symbf \mathbf \) \(\let \symit \mathit \) \(\let \symtt \mathtt \) \(\let \symbffrac \mathbffrac \) \(\newcommand {\mathfence }[1]{\mathord {#1}}\) \(\newcommand {\mathover }[1]{#1}\) \(\newcommand {\mathunder }[1]{#1}\) \(\newcommand {\mathaccent }[1]{#1}\) \(\newcommand {\mathbotaccent }[1]{#1}\) \(\newcommand {\mathalpha }[1]{\mathord {#1}}\) \(\def\upAlpha{\unicode{x0391}}\) \(\def\upBeta{\unicode{x0392}}\) \(\def\upGamma{\unicode{x0393}}\) \(\def\upDigamma{\unicode{x03DC}}\) \(\def\upDelta{\unicode{x0394}}\) \(\def\upEpsilon{\unicode{x0395}}\) \(\def\upZeta{\unicode{x0396}}\) \(\def\upEta{\unicode{x0397}}\) \(\def\upTheta{\unicode{x0398}}\) \(\def\upVartheta{\unicode{x03F4}}\) \(\def\upIota{\unicode{x0399}}\) \(\def\upKappa{\unicode{x039A}}\) \(\def\upLambda{\unicode{x039B}}\) \(\def\upMu{\unicode{x039C}}\) \(\def\upNu{\unicode{x039D}}\) \(\def\upXi{\unicode{x039E}}\) \(\def\upOmicron{\unicode{x039F}}\) \(\def\upPi{\unicode{x03A0}}\) \(\def\upVarpi{\unicode{x03D6}}\) \(\def\upRho{\unicode{x03A1}}\) \(\def\upSigma{\unicode{x03A3}}\) \(\def\upTau{\unicode{x03A4}}\) \(\def\upUpsilon{\unicode{x03A5}}\) \(\def\upPhi{\unicode{x03A6}}\) \(\def\upChi{\unicode{x03A7}}\) \(\def\upPsi{\unicode{x03A8}}\) \(\def\upOmega{\unicode{x03A9}}\) \(\def\itAlpha{\unicode{x1D6E2}}\) \(\def\itBeta{\unicode{x1D6E3}}\) \(\def\itGamma{\unicode{x1D6E4}}\) \(\def\itDigamma{\mathit{\unicode{x03DC}}}\) \(\def\itDelta{\unicode{x1D6E5}}\) \(\def\itEpsilon{\unicode{x1D6E6}}\) \(\def\itZeta{\unicode{x1D6E7}}\) \(\def\itEta{\unicode{x1D6E8}}\) \(\def\itTheta{\unicode{x1D6E9}}\) \(\def\itVartheta{\unicode{x1D6F3}}\) \(\def\itIota{\unicode{x1D6EA}}\) \(\def\itKappa{\unicode{x1D6EB}}\) \(\def\itLambda{\unicode{x1D6EC}}\) \(\def\itMu{\unicode{x1D6ED}}\) \(\def\itNu{\unicode{x1D6EE}}\) \(\def\itXi{\unicode{x1D6EF}}\) \(\def\itOmicron{\unicode{x1D6F0}}\) \(\def\itPi{\unicode{x1D6F1}}\) \(\def\itRho{\unicode{x1D6F2}}\) \(\def\itSigma{\unicode{x1D6F4}}\) \(\def\itTau{\unicode{x1D6F5}}\) \(\def\itUpsilon{\unicode{x1D6F6}}\) \(\def\itPhi{\unicode{x1D6F7}}\) \(\def\itChi{\unicode{x1D6F8}}\) \(\def\itPsi{\unicode{x1D6F9}}\) \(\def\itOmega{\unicode{x1D6FA}}\) \(\def\upalpha{\unicode{x03B1}}\) \(\def\upbeta{\unicode{x03B2}}\) \(\def\upvarbeta{\unicode{x03D0}}\) \(\def\upgamma{\unicode{x03B3}}\) \(\def\updigamma{\unicode{x03DD}}\) \(\def\updelta{\unicode{x03B4}}\) \(\def\upepsilon{\unicode{x03F5}}\) \(\def\upvarepsilon{\unicode{x03B5}}\) \(\def\upzeta{\unicode{x03B6}}\) \(\def\upeta{\unicode{x03B7}}\) \(\def\uptheta{\unicode{x03B8}}\) \(\def\upvartheta{\unicode{x03D1}}\) \(\def\upiota{\unicode{x03B9}}\) \(\def\upkappa{\unicode{x03BA}}\) \(\def\upvarkappa{\unicode{x03F0}}\) \(\def\uplambda{\unicode{x03BB}}\) \(\def\upmu{\unicode{x03BC}}\) \(\def\upnu{\unicode{x03BD}}\) \(\def\upxi{\unicode{x03BE}}\) \(\def\upomicron{\unicode{x03BF}}\) \(\def\uppi{\unicode{x03C0}}\) \(\def\upvarpi{\unicode{x03D6}}\) \(\def\uprho{\unicode{x03C1}}\) \(\def\upvarrho{\unicode{x03F1}}\) \(\def\upsigma{\unicode{x03C3}}\) \(\def\upvarsigma{\unicode{x03C2}}\) \(\def\uptau{\unicode{x03C4}}\) \(\def\upupsilon{\unicode{x03C5}}\) \(\def\upphi{\unicode{x03D5}}\) \(\def\upvarphi{\unicode{x03C6}}\) \(\def\upchi{\unicode{x03C7}}\) \(\def\uppsi{\unicode{x03C8}}\) \(\def\upomega{\unicode{x03C9}}\) \(\def\italpha{\unicode{x1D6FC}}\) \(\def\itbeta{\unicode{x1D6FD}}\) \(\def\itvarbeta{\unicode{x03D0}}\) \(\def\itgamma{\unicode{x1D6FE}}\) \(\def\itdigamma{\mathit{\unicode{x03DD}}}\) \(\def\itdelta{\unicode{x1D6FF}}\) \(\def\itepsilon{\unicode{x1D716}}\) \(\def\itvarepsilon{\unicode{x1D700}}\) \(\def\itzeta{\unicode{x1D701}}\) \(\def\iteta{\unicode{x1D702}}\) \(\def\ittheta{\unicode{x1D703}}\) \(\def\itvartheta{\unicode{x1D717}}\) \(\def\itiota{\unicode{x1D704}}\) \(\def\itkappa{\unicode{x1D705}}\) \(\def\itvarkappa{\unicode{x1D718}}\) \(\def\itlambda{\unicode{x1D706}}\) \(\def\itmu{\unicode{x1D707}}\) \(\def\itnu{\unicode{x1D708}}\) \(\def\itxi{\unicode{x1D709}}\) \(\def\itomicron{\unicode{x1D70A}}\) \(\def\itpi{\unicode{x1D70B}}\) \(\def\itvarpi{\unicode{x1D71B}}\) \(\def\itrho{\unicode{x1D70C}}\) \(\def\itvarrho{\unicode{x1D71A}}\) \(\def\itsigma{\unicode{x1D70E}}\) \(\def\itvarsigma{\unicode{x1D70D}}\) \(\def\ittau{\unicode{x1D70F}}\) \(\def\itupsilon{\unicode{x1D710}}\) \(\def\itphi{\unicode{x1D719}}\) \(\def\itvarphi{\unicode{x1D711}}\) \(\def\itchi{\unicode{x1D712}}\) \(\def\itpsi{\unicode{x1D713}}\) \(\def\itomega{\unicode{x1D714}}\) \(\let \lparen (\) \(\let \rparen )\) \(\newcommand {\cuberoot }[1]{\,{}^3\!\!\sqrt {#1}}\,\) \(\newcommand {\fourthroot }[1]{\,{}^4\!\!\sqrt {#1}}\,\) \(\newcommand {\longdivision }[1]{\mathord {\unicode {x027CC}#1}}\) \(\newcommand {\mathcomma }{,}\) \(\newcommand {\mathcolon }{:}\) \(\newcommand {\mathsemicolon }{;}\) \(\newcommand {\overbracket }[1]{\mathinner {\overline {\ulcorner {#1}\urcorner }}}\) \(\newcommand {\underbracket }[1]{\mathinner {\underline {\llcorner {#1}\lrcorner }}}\) \(\newcommand {\overbar }[1]{\mathord {#1\unicode {x00305}}}\) \(\newcommand {\ovhook }[1]{\mathord {#1\unicode {x00309}}}\) \(\newcommand {\ocirc }[1]{\mathord {#1\unicode {x0030A}}}\) \(\newcommand {\candra }[1]{\mathord {#1\unicode {x00310}}}\) \(\newcommand {\oturnedcomma }[1]{\mathord {#1\unicode {x00312}}}\) \(\newcommand {\ocommatopright }[1]{\mathord {#1\unicode {x00315}}}\) \(\newcommand {\droang }[1]{\mathord {#1\unicode {x0031A}}}\) \(\newcommand {\leftharpoonaccent }[1]{\mathord {#1\unicode {x020D0}}}\) \(\newcommand {\rightharpoonaccent }[1]{\mathord {#1\unicode {x020D1}}}\) \(\newcommand {\vertoverlay }[1]{\mathord {#1\unicode {x020D2}}}\) \(\newcommand {\leftarrowaccent }[1]{\mathord {#1\unicode {x020D0}}}\) \(\newcommand {\annuity }[1]{\mathord {#1\unicode {x020E7}}}\) \(\newcommand {\widebridgeabove }[1]{\mathord {#1\unicode {x020E9}}}\) \(\newcommand {\asteraccent }[1]{\mathord {#1\unicode {x020F0}}}\) \(\newcommand {\threeunderdot }[1]{\mathord {#1\unicode {x020E8}}}\) \(\newcommand {\Bbbsum }{\mathop {\unicode {x2140}}\limits }\) \(\newcommand {\oiint }{\mathop {\unicode {x222F}}\limits }\) \(\newcommand {\oiiint }{\mathop {\unicode {x2230}}\limits }\) \(\newcommand {\intclockwise }{\mathop {\unicode {x2231}}\limits }\) \(\newcommand {\ointclockwise }{\mathop {\unicode {x2232}}\limits }\) \(\newcommand {\ointctrclockwise }{\mathop {\unicode {x2233}}\limits }\) \(\newcommand {\varointclockwise }{\mathop {\unicode {x2232}}\limits }\) \(\newcommand {\leftouterjoin }{\mathop {\unicode {x27D5}}\limits }\) \(\newcommand {\rightouterjoin }{\mathop {\unicode {x27D6}}\limits }\) \(\newcommand {\fullouterjoin }{\mathop {\unicode {x27D7}}\limits }\) \(\newcommand {\bigbot }{\mathop {\unicode {x27D8}}\limits }\) \(\newcommand {\bigtop }{\mathop {\unicode {x27D9}}\limits }\) \(\newcommand {\xsol }{\mathop {\unicode {x29F8}}\limits }\) \(\newcommand {\xbsol }{\mathop {\unicode {x29F9}}\limits }\) \(\newcommand {\bigcupdot }{\mathop {\unicode {x2A03}}\limits }\) \(\newcommand {\bigsqcap }{\mathop {\unicode {x2A05}}\limits }\) \(\newcommand {\conjquant }{\mathop {\unicode {x2A07}}\limits }\) \(\newcommand {\disjquant }{\mathop {\unicode {x2A08}}\limits }\) \(\newcommand {\bigtimes }{\mathop {\unicode {x2A09}}\limits }\) \(\newcommand {\modtwosum }{\mathop {\unicode {x2A0A}}\limits }\) \(\newcommand {\sumint }{\mathop {\unicode {x2A0B}}\limits }\) \(\newcommand {\intbar }{\mathop {\unicode {x2A0D}}\limits }\) \(\newcommand {\intBar }{\mathop {\unicode {x2A0E}}\limits }\) \(\newcommand {\fint }{\mathop {\unicode {x2A0F}}\limits }\) \(\newcommand {\cirfnint }{\mathop {\unicode {x2A10}}\limits }\) \(\newcommand {\awint }{\mathop {\unicode {x2A11}}\limits }\) \(\newcommand {\rppolint }{\mathop {\unicode {x2A12}}\limits }\) \(\newcommand {\scpolint }{\mathop {\unicode {x2A13}}\limits }\) \(\newcommand {\npolint }{\mathop {\unicode {x2A14}}\limits }\) \(\newcommand {\pointint }{\mathop {\unicode {x2A15}}\limits }\) \(\newcommand {\sqint }{\mathop {\unicode {x2A16}}\limits }\) \(\newcommand {\intlarhk }{\mathop {\unicode {x2A17}}\limits }\) \(\newcommand {\intx }{\mathop {\unicode {x2A18}}\limits }\) \(\newcommand {\intcap }{\mathop {\unicode {x2A19}}\limits }\) \(\newcommand {\intcup }{\mathop {\unicode {x2A1A}}\limits }\) \(\newcommand {\upint }{\mathop {\unicode {x2A1B}}\limits }\) \(\newcommand {\lowint }{\mathop {\unicode {x2A1C}}\limits }\) \(\newcommand {\bigtriangleleft }{\mathop {\unicode {x2A1E}}\limits }\) \(\newcommand {\zcmp }{\mathop {\unicode {x2A1F}}\limits }\) \(\newcommand {\zpipe }{\mathop {\unicode {x2A20}}\limits }\) \(\newcommand {\zproject }{\mathop {\unicode {x2A21}}\limits }\) \(\newcommand {\biginterleave }{\mathop {\unicode {x2AFC}}\limits }\) \(\newcommand {\bigtalloblong }{\mathop {\unicode {x2AFF}}\limits }\) \(\newcommand {\arabicmaj }{\mathop {\unicode {x1EEF0}}\limits }\) \(\newcommand {\arabichad }{\mathop {\unicode {x1EEF1}}\limits }\) \(\let \symsf \symsfup \) \(\def\Alpha{\unicode{x0391}}\) \(\def\Beta{\unicode{x0392}}\) \(\def\Gamma{\unicode{x0393}}\) \(\def\Digamma{\unicode{x03DC}}\) \(\def\Delta{\unicode{x0394}}\) \(\def\Epsilon{\unicode{x0395}}\) \(\def\Zeta{\unicode{x0396}}\) \(\def\Eta{\unicode{x0397}}\) \(\def\Theta{\unicode{x0398}}\) \(\def\Vartheta{\unicode{x03F4}}\) \(\def\Iota{\unicode{x0399}}\) \(\def\Kappa{\unicode{x039A}}\) \(\def\Lambda{\unicode{x039B}}\) \(\def\Mu{\unicode{x039C}}\) \(\def\Nu{\unicode{x039D}}\) \(\def\Xi{\unicode{x039E}}\) \(\def\Omicron{\unicode{x039F}}\) \(\def\Pi{\unicode{x03A0}}\) \(\def\Varpi{\unicode{x03D6}}\) \(\def\Rho{\unicode{x03A1}}\) \(\def\Sigma{\unicode{x03A3}}\) \(\def\Tau{\unicode{x03A4}}\) \(\def\Upsilon{\unicode{x03A5}}\) \(\def\Phi{\unicode{x03A6}}\) \(\def\Chi{\unicode{x03A7}}\) \(\def\Psi{\unicode{x03A8}}\) \(\def\Omega{\unicode{x03A9}}\) \(\def\alpha{\unicode{x1D6FC}}\) \(\def\beta{\unicode{x1D6FD}}\) \(\def\varbeta{\unicode{x03D0}}\) \(\def\gamma{\unicode{x1D6FE}}\) \(\def\digamma{\mathit{\unicode{x03DD}}}\) \(\def\delta{\unicode{x1D6FF}}\) \(\def\epsilon{\unicode{x1D716}}\) \(\def\varepsilon{\unicode{x1D700}}\) \(\def\zeta{\unicode{x1D701}}\) \(\def\eta{\unicode{x1D702}}\) \(\def\theta{\unicode{x1D703}}\) \(\def\vartheta{\unicode{x1D717}}\) \(\def\iota{\unicode{x1D704}}\) \(\def\kappa{\unicode{x1D705}}\) \(\def\varkappa{\unicode{x1D718}}\) \(\def\lambda{\unicode{x1D706}}\) \(\def\mu{\unicode{x1D707}}\) \(\def\nu{\unicode{x1D708}}\) \(\def\xi{\unicode{x1D709}}\) \(\def\omicron{\unicode{x1D70A}}\) \(\def\pi{\unicode{x1D70B}}\) \(\def\varpi{\unicode{x1D71B}}\) \(\def\rho{\unicode{x1D70C}}\) \(\def\varrho{\unicode{x1D71A}}\) \(\def\sigma{\unicode{x1D70E}}\) \(\def\varsigma{\unicode{x1D70D}}\) \(\def\tau{\unicode{x1D70F}}\) \(\def\upsilon{\unicode{x1D710}}\) \(\def\phi{\unicode{x1D719}}\) \(\def\varphi{\unicode{x1D711}}\) \(\def\chi{\unicode{x1D712}}\) \(\def\psi{\unicode{x1D713}}\) \(\def\omega{\unicode{x1D714}}\)

Exam version: su26a

David W. Agler

June 23, 2026

The following exam consists of 7 questions, for a total of 100 points. Read each question carefully (note: answers may break onto the next page). This exam tests your knowledge over the material from Chapter 5 of the course text, lectures, videos, handouts, and discussion. You may write on the test itself, but place final answers on the ``answer sheet'' (last page) provided.

0.1 Proofs

Provide proofs for the following syntactic entailments. Be sure to setup the proof correctly, number all lines, and clearly indicate how each line is justified using the rules from the deductive apparatus.

Q1. \(Q\vee Z, M\wedge (Q\wedge R), \neg B\wedge C \vdash (R\wedge \neg B)\wedge M\)

--- Answer: --- Answer: \(Q\vee Z, M\wedge (Q\wedge R), \neg B\wedge C \vdash (R\wedge \neg B)\wedge M\)

(A natural deduction proof with nine lines. Line 1: Q or Z, premise. Line 2: M and open parenthesis Q and R close parenthesis, premise. Line 3: not B and C, premise. Line 4: not B, from line 3 by conjunction elimination. Line 5:

Q and R, from line 2 by conjunction elimination. Line 6: R, from line 5 by conjunction elimination. Line 7: M, from line 2 by conjunction elimination. Line 8: R and not B, from lines 6 and 4 by conjunction introduction. Line 9: open
parenthesis R and not B close parenthesis and M, from lines 8 and 7 by conjunction introduction.)

Q2. \((A\wedge \neg Z)\rightarrow Q, S\wedge \neg Z, A \vdash Q\vee \neg \neg W\) --- Answer: --- Answer: \((A\wedge \neg Z)\rightarrow Q, S\wedge \neg Z, A \vdash Q\vee \neg \neg W\)

(A natural deduction proof with seven lines. Line 1: open parenthesis A and not Z close parenthesis implies Q, premise. Line 2: S and not Z, premise. Line 3: A, premise. Line 4: not Z, from line 2 by conjunction elimination. Line

5: A and not Z, from lines 3 and 4 by conjunction introduction. Line 6: Q, from lines 1 and 5 by conditional elimination. Line 7: Q or not not W, from line 6 by disjunction introduction.)

Q3. \(C\vee S, C\rightarrow (G\wedge F), S\rightarrow (G\wedge \neg L)\vdash G\) --- Answer: --- Answer: \(C\vee S, C\rightarrow (G\wedge F), S\rightarrow (G\wedge \neg L)\vdash G\)

(A natural deduction proof with ten lines. Line 1: C or S, premise. Line 2: C implies open parenthesis G and F close parenthesis, premise. Line 3: S implies open parenthesis G and not L close parenthesis, premise. Line 4: C,

assumption for disjunction elimination. Line 5: G and F, from lines 2 and 4 by conditional elimination. Line 6: G, from line 5 by conjunction elimination. Line 7: S, assumption for disjunction elimination. Line 8: G and not L, from lines
3 and 7 by conditional elimination. Line 9: G, from line 8 by conjunction elimination. Line 10: G, from line 1 and lines 4 through 6 and lines 7 through 9 by disjunction elimination.)

Q4. \(\neg (Z\rightarrow W), S\rightarrow W \vdash \neg S\) --- Answer: --- Answer: \(\neg (Z\rightarrow W), S\rightarrow W \vdash \neg S\)

(A natural deduction proof with six lines. Line 1: not open parenthesis Z implies W close parenthesis, premise. Line 2: S implies W, premise. Line 3: not open parenthesis not Z or W close parenthesis, from line 1 by implication

equivalence. Line 4: not not Z and not W, from line 3 by De Morgan's law. Line 5: not W, from line 4 by conjunction elimination. Line 6: not S, from lines 2 and 5 by modus tollens.)

Q5. \(\vdash (R\rightarrow M)\vee \neg (P\wedge \neg P)\) --- Answer: --- Answer: \(\vdash (R\rightarrow M)\vee \neg (P\wedge \neg P)\)

(A natural deduction proof with five lines. Line 1: P and not P, assumption for negation introduction. Line 2: P, from line 1 by conjunction elimination. Line 3: not P, from line 1 by conjunction elimination. Line 4: not open

parenthesis P and not P close parenthesis, from lines 1 through 3 by negation introduction. Line 5: open parenthesis R implies M close parenthesis or not open parenthesis P and not P close parenthesis, from line 4 by disjunction
introduction.)

Q6. \(\vdash (P\wedge \neg S)\rightarrow ((\neg Q\vee S)\rightarrow \neg Q)\) --- Answer: --- Answer: \(\vdash (P\wedge \neg S)\rightarrow ((\neg Q\vee S)\rightarrow \neg Q)\)

(A natural deduction proof with six lines using nested subproofs. Line 1: P and not S, assumption for conditional introduction. Line 2: not Q or S, assumption for conditional introduction. Line 3: not S, from line 1 by

conjunction elimination. Line 4: not Q, from lines 2 and 3 by disjunctive syllogism. Line 5: open parenthesis not Q or S close parenthesis implies not Q, from lines 2 through 4 by conditional introduction. Line 6: open parenthesis P and
not S close parenthesis implies open parenthesis open parenthesis not Q or S close parenthesis implies not Q close parenthesis, from lines 1 through 5 by conditional introduction.)

Translate the following arguments and then create a proof for it.

Q7. If God is real and God is good, then there is no evil in the world. There is evil in the world. Therefore, God is not real or God is not good. --- Answer: Translation: \((R\land G)\to \lnot E, E\vdash \lnot R\lor \lnot G\).

(A natural deduction proof with four lines. Line 1: open parenthesis R and G close parenthesis implies not E, premise. Line 2: E, premise. Line 3: not open parenthesis R and G close parenthesis, from lines 1 and 2 by modus

tollens. Line 4: not R or not G, from line 3 by De Morgan's law.)

This is the end of the exam.

0.2 Extra-Credit

Bonus Question: Absorpotion law for disjunction: \(P\vee (P\wedge Q)\dashv \vdash P\). Solve only using intelim rules. --- Answer: Tips. Proof 1: \(P\vee (P\wedge Q)\vdash P\). Use \(\vee E\). Proof 2: \(P\vee (P\wedge Q)\dashv P\). Just use \(\vee I\)

Bonus Question: Commutation for disjunction: \(P\vee Q\dashv \vdash Q\vee P\). Solve only using intelim rules. --- Answer: Tips. Just use \(\vee E\) for both proofs

Bonus Question: Let's define two rules. First, \(\oplus E1\) is defined as follows: \(\phi \oplus \psi , \phi \vdash \neg \psi \) or \(\phi \oplus \psi , \psi \vdash \neg \phi \). Next, \(\oplus E2\) is defined as follows: \(\phi \oplus \psi , \neg (\psi ) \vdash \phi \) or \(\phi \oplus \psi , \neg (\phi ) \vdash \psi \). Now prove the following: \(A\oplus B\vdash (A\vee B)\wedge \neg (A\wedge B)\) --- Answer: (Proof)

Derivation rules

Definition 1: Conjunction Introduction \(\wedge I\).

\(\phi , \psi \vdash \phi \wedge \psi \) or \(\phi , \psi \vdash \psi \wedge \phi \)

Definition 2: Conjunction Elimination (\(\wedge E\)).

\(\phi \wedge \psi \vdash \phi \) or \(\phi \wedge \psi \vdash \psi \)

Definition 3: Conditional Introduction (\(\rightarrow I\)).

(Proof)

Definition 4: Conditional Elimination (\(\rightarrow E\)).

\(\phi \rightarrow \psi , \phi \vdash \psi \)

Definition 5: Reiteration (R).

\(\phi \vdash \phi \)

Definition 6: Negation Introduction (\(\neg I\)).

(Proof)

Definition 7: Negation Elimination (\(\neg E\)).

(Proof)

Definition 8: Disjunction Introduction (\(\vee I\)).

\(\phi \vdash \phi \vee \psi \) or \(\phi \vdash \psi \vee \phi \)

Definition 9: Disjunction Elimination (\(\vee E\)).

(Proof)

Definition 10: Biconditional Introduction (\(\leftrightarrow I\)).

(Proof)

Definition 11: Biconditional Elimination (\(\leftrightarrow E\)).

\(\phi \leftrightarrow \psi , \phi \vdash \psi \) or \(\phi \leftrightarrow \psi , \psi \vdash \phi \)

Definition 12: Disjunctive Syllogism (DS).

\(\phi \vee \psi , \neg (\psi ) \vdash \phi \) or \(\phi \vee \psi , \neg (\phi ) \vdash \psi \)

Definition 13: Modus Tollens (MT).

\(\phi \rightarrow \psi , \neg (\psi ) \vdash \neg (\phi )\)

Definition 14: Hypothetical Syllogism (HS).

\(\phi \rightarrow \psi , \psi \rightarrow \chi \vdash \phi \rightarrow \chi \)

Definition 15: Double Negation (DN).

\(\phi \dashv \vdash \neg \neg (\phi )\)

Definition 16: De Morgan's Laws (DeM).

\(\neg (\phi \vee \psi ) \dashv \vdash \neg (\phi )\wedge \neg (\psi )\) or \(\neg (\phi \wedge \psi ) \dashv \vdash \neg (\phi )\vee \neg (\psi ) \)

Definition 17: Implication (IMP).

\(\phi \rightarrow \psi \dashv \vdash \neg (\phi ) \vee \psi \)