• March 2, 2023
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Q q BT aBl8#lH#R=;0Han;AdU1\Df0YrH#R=;0d("=ol+>>E%0JO\(+> Tj e6;Eu_8CL;gO/+H2M2ng_c-BS6WQ1K6eG'fcE!oX*G6q_K`id#!`O+j%;3VA4D]AOIX`',MUB_[^7H ]?%$6QE#plt""/p8DV$^II1kt7bK(*i)CiZ@?EFGlD 2.556 0 Td (tables:) Tj These are called tautologies and contradictions, respectively. 0.921 0 Td (q) Tj s&K.Ug,l9tZig,ZRTLU%_-_B. /Flags 4 This can be checked 2with a truth table: The fact that all entries in the last column are true tells us that this expression is a tautology. 3.687 0 Td (:) Tj %^h!@Wq7dq?,\kiDs_^jR/&DZf/? The symbolic form of "He is hard working but not going to win", is. t+d]_m*-9!VgUQ%7Qs^4\0/M2+>)^\[21/i#^r=S(R*`a(V+BHg9VGf ?n&g>D%Wp9]OSqeK;+Q_&rE#B!48G^r;[]ZGKfce"rTc/B#H.B;0[Z;8"a1!ab2iSkCf2>Jp72n!/M >> 153XfsIrF\`-;enqk\X[-RDS,7'BLtLUY2!%b)a8TGg=TV*C^lGAI^JV'"MgFC76Kl5a)>B43eUTi+ B5(jaCi=>:E-670A9Di63B/-7Ao)C(Ebo7>CDf9Z:BkM+$+E2IF$=n9u+>G ,p? BT 11.96 0 0 11.96 0 0 Tm 28.767 9.037 Td (p) Tj Tj /Length3 532 1.713 0 Td (t.) 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Either x is not an even number or x is divisible by 3, but not both. Yd:f/dt:k]E0jdB/4Vh&i"XlKr^dJuIC3BlTqX,-:s68q5. 6_,/Q(jD,qb\*-^4]Z8lB?XA!.`Up8W)M^o_8X7(8%bU#shr07*r%p.GZ3JH:OLl#_Zs*O/#PV<7i- /FontName /CMR9 /gIn8,6io^\c4g;@X6lD*,&Sd3J+li$O=dM 0.489 0 Td (e,) Tj << *tgsHY6bB!WPLLI/0o3)Q5s./r$ae)UL*@jV?NkHt^[P6]N_8M0 Tj /F1 1 Tf Qf6/?f?;Q?A;"0A)Xg*?ZG_QWTG>DV$c^&XAG#/U<90\Bf;GiT^+^DpH.)SP(Nu1:VZ:.SgV()K! 35 0 obj /BaseEncoding /MacRomanEncoding ]X R=;0Han;AdU1\Df0YrH#R=;0d("=ol+>>E%0JO\(+>@(DEb/Zi False: c. May be True or False: d. Can't say: View Answer Report Discuss Too . ,p? So if k stands for Im the King of the Netherlands, then k Q is true for any compound proposition Q. d\H1[V"oJF]juX(PRAcuK2`OFLAdT"R,2d?D'9+I\k_*tRU61Fj/[6t/B1#JZsN0RES&d_Nd!4mM\d 3.825 0 Td (then) Tj /FontBBox [ -39 -250 1036 750 ] 0.889 0 Td (q) Tj /StemH 20 ]39MuJrP=J`%`KDd]:nGWW`Jc]no;6#CbtciCOH5=/;iHQaW)'*B\ ? 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[eRg:;&r^&I/QQ=8"M33c>4q+*o*lc9TB$lU\tVV%7(i/f1S'D6X8_!SHo,_>RFV";0 ET 4.149 0 Td (sa) Tj 11.96 0 0 11.96 0 0 Tm 10.592 53.352 Td (called) Tj !dq)T_C70=eTHc5*SRU,sWTl6Z,bT"L@As"T"X?RI*hQKcGW:c!tr[)c5[t)0I="4PF""C`PtcBN7, DH.sZ]!,?f20/=FOIhXL%r5)7C"Bi\(I8K+6GkV\4`1V@M[JNCJmhuCqTN8*JDeW+652u? /Length1 2627 Q q /Descent 15 Q q ]Pf[nTEhIMh4='dBX@uBNe?JSG=S68Hdi'0`rH` 2.172 0 Td (\\John) Tj BT 0.571 0 Td (oth) Tj MM7jpA:*,?bai[84\]APZSAY4*^]BKPR3B/a)h)!M+^3q^)e]Ha3sM.IX_Nu2,Fru0jU=+M/m!,^,5nqa.up6bf,cl Locating the main operator helps to translate sentences and place parentheses accurately: IF Julian does not become a lifeguard at the YMCA or get the job driving school buses, then he will keep working at the bookstore in San Francisco and clean houses part time. /Differences [ 32 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef 45 /.notdef 47 /.notdef 58 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef 74 /.notdef /.notdef 86 /.notdef 88 /.notdef 91 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef 128 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /Gamma /Delta /Theta /Lambda /.notdef /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef 174 /ff /fi /fl /ffi /ffl 180 /dotlessj /grave /acute /caron /breve 187 /cedilla /germandbls /ae /oe /.notdef /.notdef /OE /Oslash /suppress /dieresis 199 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef 216 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef 241 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] 1.196 0 Td (\\) Tj Tj Let's consider a tautology first, and then a contradiction: Consider the statement "\((2 = 3) \vee (2 \ne 3)\)": Let's make a truth table for general case \(p \vee (\neg p)\): As you can see, no matter what we do, this statement is always true. 38 0 obj 11.96 0 0 11.96 0 0 Tm 37.228 42.356 Td (form) Tj B"mp#R`F^D"sXrL0g^J49I(DH-EF`kc/ame"g9g*/9eAY%CpOr2gU-19.O$mn9u%8`=]le^&Joh)!) !0RX4C5%!hL51^b8H9(+)g\hV8Tn,(eqB[0)WrK,HX$tX[NL2Y@7]D,='*^J+\gqt\ik,)l B"mp#R`F^D"sXrL0g^J49I(DH-EF`kc/ame"g9g*/9eAY%CpOr2gU-19.O$mn9u%8`=]le^&Joh)!) _p"a$$0YR^o8-ZoDQ.(DrZr&OtJ"t2*X@W_-T=:47>Ql11;'JjPH6KOQWUh#e\cAgQ7co/\eM^caY. >b25..4"!d#! #_-VP`L7#&4>&_Kb[H? JjH,t:W]g8rC41Btd"k"Lk9?K]i\Z9M?P"`mDL't4m Q q << /F4 1 Tf Clearly, this statement is a contradiction. Q q The consent submitted will only be used for data processing originating from this website. BT BZd7'P7jW=uuE$G*VU/Wnf;9dc;1r4;kWnA%"&/OMT9E>R2W1R1F*/g^T/Zs*fee2lsV 11.96 0 0 11.96 0 0 Tm :iFGDMgKauLHeW+UP--jgH]WM=^"TaEA >> 385.44 647.04 m 385.845 647.04 l 385.845 660.99 l 385.44 660.99 l f .5B@>R,LCp#VdA;]= 37T4EcH2_J4nY'"NV!3E-i;=:"*'tUW+?2A*T]AJ4Hq&VL];CN0,4m>V:/ttN#sDl,7ssbXFGp.p!Q stream :b[YF7j8'aKjj]Xae"U9[84RQ('%-MQmk1qoR@n\0WW`N! endobj +>77JE-670A9Di60eb=)04c8FF`[t$F`8HX0JXb(Afu&8F:)Q$E$-kh1E\>`+E2IF$=n9u+>GQ,+>7 m38?,?=WTt=u%n-FZn-Jjq;"W]?JGEI\WOh+#V29b@/]f@Xs(bKO[D\do_la84\@K,8.>(P4A>%^+2rl300A`'E[3qf-3Jd`mKONDT[K=rNWF38o]crk+U&!r.rX /FontName /CMR12 /F4 1 Tf 11.96 0 0 11.96 0 0 Tm BT Q q Q$E$-tt+>7GHAnc-sFEo!JF`[t$F`8H\0H`#[Ch7HpDKI"?F`[t$F`8H\1*A5lEbSs"B5D-%E-670A :r Discussion One of the important techniques used in proving theorems is to replace, or sub-stitute, one proposition by another one that is equivalent to it. 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