KR3=7&T6Z6phEbT0"F^/he^sEb/ZiDf0B:+Co%q$85.mFD55r+=KTqE-ZJ=ol+>>E%0JO\(+>@(DEb/ZiDf0B:+C 16.013 54.468 Td (T) Tj Q q endobj BT Fbj/GY2Gm2cQTUe0GVHHc0-%0peL5tIV:U5/2$,T*cpRIYH+9t04R .6AS)9&=(Q)YBQP@F9k@jLF_rQMEb/ZiDf0B:+Co%q$84tm@;KXf6#L=OAKWQZ1bLI=+Co%q$86+77 hZ8(rqgRbCu?h+"!rIRbSm^C0S"([C&UtHJRi%kQg[P/UAN/Ic$]?>S#OX&ZIhu"^4!L6/NnrMqM /F3 1 Tf Example 2.1.3. /F3 1 Tf As you can see, if \(p\) is true then \(\neg p\) is false and if \(p\) is false, the negation (i.e. /F4 1 Tf /Flags 4 +>7YNAKYr7F:)Q$E$."p+>7eMGT^sJF:)Q$E$."q+>7eIG%GP6E-670A9Di62)cX2DJ=! << /StemH 20 9u+?;A'04>uBF`[t$F`8H`3$9k`+E2IF$=n9u+? 22.556 21.205 Td (F) Tj ^[LU`CL5DO\STO:e;8%)bkHrrXZR07-S-\KCZ@U1C2rV+07kZ/EKs4k/(:+A_l2,SP, )`/OGl:2]tD'ATDKp@;[2^@< -0.728 -1.575 Td (p) Tj pgih4+`/->MP>u7juZK%au0%^AXT>&OW >> Get started for free! 9To#'I?H.kU_QM^4bDs1F]SYr^DYI8\jgLVuO6! BFt.HondkuML_ji0atI!W_[]n/[8%DD!.eMU77qA;tBXSJ`>/;^. 3.811 0 Td (then) Tj 11.96 0 0 11.96 0 0 Tm /F4 1 Tf 3.679 0 Td (is) Tj aBl8#lH#R=;0Han;AdU1\Df0YrH#R=;0d("=ol+>>E%0JO\(+> Kd8\fUqiP=\UR_N-(1\A(ODqmji,]O$7nHm-e5eG)e]MTY+frB#%fp/sY=p eWJD0gGR%+H!_t,U[IAqtNLc$Y$b0-_59)8>;.pb-GBUjWu*,4oLiJfi=:BntD*Zo4Q'9,)ikh(a1kBm$]-SpA[*HI4ah0[M=nJYT[QY_`[iP)JuLThVVgUC;6fbh\QA$](:F_ 0.470 0 Td (") Tj /FontBBox [ -34 -251 988 750 ] 312.96 468 m 318.837 468 l 318.837 468.486 l 312.96 468.486 l f kp2'=Ph@:O=rDBNn@F:)Q$E$-kp2]sb`ARoLkCgeH*F`[t$F`8HX3&i$6B4Z+#@;]UbCi^_\F(c\ j.0LV(k&oh[M*(c*U#CrsE6]) \N(D_Hh3>G.&sR Let p denote the statement: Gopal is tall, q: Gopal is handsome. 9Di62)l^3@rH1%DBNn@F:)Q$E$.%r+>6#'E-670A9Di62Du[266L5iF:)Q$E$.%t+>6))E-670A9Di Let P be a statement if p then q. /Type /Font >> jWG&`cdp*85gu;daX/Vk*MS5tW*2;#0@)SR4-aWGN?kXCk/L4OC,Cqf2$(J,!/mF*YIhWbA:0GXg7K "m+>7hMEb/f \(1+1=2\) and "All birds can fly". 31 0 obj 23 0 obj [MAZtqbsoDd-NA:LJ,K%7:i*%/mWPQ\W;78[;JPK#7ur@P-A.-]aF%k,VE#_>:Dp7?FT#J@:fPmPB#F_"7PsGU1qARk"37LpF*6orf7K4 2.012 0 Td (follo) Tj RX.2hMII]GQl!`'3?=T>bIGT-c_X7imS2NG,b+8eg"Q0nuEW9#6IJ6R.-/#QZVX7^k,+8mr_\YJiML ;>;ktDqrS(a /Length 53 0 R !dq)T_C70=eTHc5*SRU,sWTl6Z,bT"L@As"T"X?RI*hQKcGW:c!tr[)c5[t)0I="4PF""C`PtcBN7, PkPe*!O;ZGZE@,m3ZLHhSLO9O;8iCscRKU&m,l8SY+^Ns<8.FKK%p7q]FH?M-!jgOL(VEScTgd>?6Rp$7.,dk>j 9.Define a logical operator so that p q is logically equivalent to (p q). 33 0 obj /LastChar 255 11.96 0 0 11.96 0 0 Tm /Length3 532 2 0 obj F0dg.%#o'8jW*HZ,s#C:U/cPdem!_U\J"XY)+hPj)EC_RJsD%-lb0H_f\gsM0 /F4 1 Tf Z\s;e?3\R_p6p,V/bl-E&SFI93(7m+#EM>_%ANHtk6ImZQOG2f#cX[Cl;r797Oo=H=")V/pD. 11.96 0 0 11.96 0 0 Tm "k1qqd]h4GXhPjjIC635@mg>9GE`Q"=PVA:!-OHgk,]DrQL]W:(gT*^k>k.J24A&4l%_"4E.s0e";X )'OGih3W6GHN#)O(3-UO%E#0h#U?h]Ba=S?_3\#p 1.258 0 Td (false) Tj =m`_[KE][-/9>f&(O8-#,"$FO9u-H[IR657:`Tl]g(9KhHt0\U#b!n[DbT5i'dX-nY_d6j,`>X9j1i*IaVl`c&^SU %CmT@.Fo"AgLNfr<2eUsp;-kRto!_?]fAc`DELE@3"Um,d,! QkTr,l;u>U!p3!ehN9[V86d&l]i+D%,'0CH[PXk`&G>(To:L3lUrP3[ba`;eYHnI!QTG3h1IJ'6nCH /F5 1 Tf \4$)\mqWb&r1)kOTiFuQ,ij0jhCY!:ggY_=ndG'V;ZmCHn(U:?.cW'dDWK+d/>VFC&? 11.96 0 0 11.96 0 0 Tm BT The statement reads "2 is less than or equal to -3", or "\(2 < -3 \vee 2 = -3\)" and can be broken into two component propositions: Because propositions \(p\) and \(q\) are both false, the statement is false. !E*Hr&.>'09\FHTr&Ad]:j!tDVZgZ "c)0;p+i"7V#rJLB:.>jVlclII/\/)Q0nA_Q^e*[#Kolj,Ut0VPrFLnE[,fg$p;- /F5 1 Tf ^MBDOGn8pq;%S4.3oE!VZPI0I[)AaAA]gpo.I:obHnXROdVq\0EK7ZC$d`4TCPu2tBH Tg_@nC@g*0&&r!s6Kjkh/8eU7.feQMki9dqd.`PDCA,]dKE!s[ 0.408 0 Td (hec) Tj Q q 2MnP-U4&$G/lYd]1<69D&Hn@TM]aBRZ;8I!`/n=^.h46X;L`U*>7/Wks5mKCWo8:Z:KPZC>CYnm Examples: CS19 is a requiredecourse for thenCS major. -6m=(Gq! 19.174 20.005 Td (T) Tj /Filter /ASCII85Decode FPcOTa$A6n"V\L[El$0 *tgsHY6bB!WPLLI/0o3)Q5s./r$ae)UL*@jV?NkHt^[P6]N_8M0 3.612 0 Td (ha) Tj 'MFAua<8rboHpd6W 0.713 0 Td (_) Tj BU.L_1f+gqk5IU8X1RB6*Ee`D(H2dAPArm)GgO! r5&UbImi9#LSPH`j.2WZQuqm,Q'Tm:&h[ThRB34@thSdd)hCK1QOMm0(OS+FtPVoR1Sm!B@8jOW1J3>(80O_%no)H",PuTa'V%D3i\A%Un*YqKH%Pi,@n0I^OBeHe?h -29.498 -1.166 Td (sp) Tj \N(D_Hh3>G.&sR 1.277 0 Td (p) Tj q("g,DLW*8%/NaE;d.n_M!+fc&NKCDZ2i1nD^V@S*k[?LN2p8TmsBcd+rD)c[Bm-? /F4 1 Tf @IcUn,]UE\T[>1SplYbX$F$dfU*$o;oLe$k_rF*+VO#rhX#5fI2+pghrRY4q\"fcK+-sQNSEBe7(YUcAkZaSjge>,>Okdh5/4kQS"OG:rfX5E 238.56 399.84 m 238.965 399.84 l 238.965 413.79 l 238.56 413.79 l f SV;>I2DQqSi^H\@jqUjD:SQ2!pft@,0j^2]ZmHDA5"LAR%_XXjj_JBZU.M["m@K(h(aj:t9KGqq\FN \YE7Kfr-C:,OjcFh>LIh%,5eW[E?^`^=2.k1Q:A++9fD$PF%&^msD9$<6/qg;1R+&6%01aOh7/-5JR @ Consider the "if p then q" proposition. PkPe*!O;ZGZE@,m3ZLHhSLO9O;8iCscRKU&m,l8SY+^Ns<8.FKK%p7q]FH?M-!jgOL(VEScTgd>?6Rp$7.,dk>j BT 0.602 0 Td (^) Tj 3. /Ascend 0 /F1 1 Tf WN#SP*60]Fr'9@Bepe?`*RQ!D[upTaJAcGq$WU!9;+$5f`/TWShdrQAPP@(5N/N\C4>hW'1RAb5I>+5m=Ou"LPq> '5Oph/T@698Hu:Gfj'bECB\ZtC@'SS-D9W4Z6HTVFr`gQq6mpa,hB[B]h5aJhes+oHLCYtQS::s]StoN2:YgHaE_g/35PUY>/YcpJ5g80f86RG:s)9+SY!C"P.H 176.64 235.2 m 177.045 235.2 l 177.045 249.15 l 176.64 249.15 l f .>ET+1/-;cL8&E@ZihJOk7\=S6Zu4bdi*GQr,SnH,L;_6h_@1lCXY!+Ou$=QV'=s]Z8EOZ`UW=S><6p]ZRcu:U?6,sY+,,0(kM*8@e,P/#ApUh_$>OlN(Hpja\R j8I;0hV\6E&hAh5(2c.O1($ZE`n4f4B"/K:NEQ"3L=ARFEZ=\Y5+S@f:YM%G1CXbFXCNYRb /Type /Font 5N7. @:t^Q"3Sa"=@nSE(H_6cKKl[Fe0lIfLi-t0a, ET Let's make a truth table for general case \(p \wedge (\neg p)\): As you can see again, no matter what we do, this statement will always be false. ET :iFGDMgKauLHeW+UP--jgH]WM=^"TaEA )6uGh/i:OBEhFZ([UY,.3:VY%c%C>dD([cY'l@?4=I6.mFV\)*q,;hV?`$YDPtXI,\T]4%,o@8(2 S&IPT'F5p;$CT\I3RXM+WWgWQmo^VuS2mMF!3Tj)F"jGI?6C>eT[^>`6ah1]r=N`?LQsPln B"mp#R`F^D"sXrL0g^J49I(DH-EF`kc/ame"g9g*/9eAY%CpOr2gU-19.O$mn9u%8`=]le^&Joh)!) BT /Flags 4 EM)QC]19m<7!&! /F4 1 Tf /F6 1 Tf hdk5]Yap"ZReWN5?O(mk$Qk;[,D:ts%ap4[NbX]\X`9s+dKbNr,F^eHBeH1DYO2H12>ij^K@? Q q Q q endobj BT 11.96 0 0 11.96 0 0 Tm Consider the following truth table: The table above describes the truth value possibilities for the statements \(p\) and \(\neg p\), or "not p". .3OYY9W$XkkG:4oQ^>,38a9,4JL,LC]T)nnF"PHKDq !g [d0VObI-e$m#,fmlQph\!bn>JiA5%#N_FJd=AM\mNXn /F1 1 Tf 4*`JH-=<5SJ;Q=^\"pT[==eR=_5"(Lb=W,`oD7@0=aINi2j%->2#3O)`-$T%jL;08LQ!rma"6"B$jr Q q hBPf/=Y! Q q A A. P>(Pp0:iYuk$u)R8eQipFt'iNKu7,@;7[AaFK@tjpSa?XYH9^uY >]]mpMaNe-8D_o2&Vk^YQ60#lfc#2I8&b?OeW`\kU>XJDrUI,.W+7ndaooNWH. endobj 3n@;I&bDe!KmFEoJG,p5QnE-ZJo(@:X:oCj@.6AS)9&7W30d:18!N+>6Gf5ts'f+Co%q$854aBl8#lH# !u.MriA(7[Y@i296k2Mmf(cT`Zi0HIgLcWqVK%9!518QQh]b6 BT << 1.798 0 Td (a) Tj Q q ^O^MpuiT@Y/8jGXPsn'P3A4b<0d&X?Xsm-\"SnN>HiMNM+C@QJ0lFUCauP4`u(3=G=q"@QY"host:8 425.28 661.44 m 425.685 661.44 l 425.685 675.39 l 425.28 675.39 l f 176.64 292.32 m 177.045 292.32 l 177.045 306.27 l 176.64 306.27 l f qhsj9u?i5;O15sa$\lQb_B;3k9@TdO1.--L;[,H,Z5"JgS4>EH1$cJnq-c_;7[O?Wn?4WF0@d9eo`LoO=DWjh&M(8&*rge55CdF:d,%i97jhWO=QDijR$lD:2=1(dubia,BF!>#=>-roZJbUY?X[O_>sb+GG@o#(/fQ!clgE]SqG.9DVIKq)>%\gM> .1E\D.1GL",0JG;2Eb/ZiDf0B:+Co%q$85CsBlS9,=BH\'0JbFD3&)OdAS)9ZF`MM6DKKE'@rri)DI $>`*QAel5#/ed/WPEmkZGeMS)k[L$VcX3U:C@HenZ-Qlb85A`\klNOh*4hBKZ]Bcipb;e^/W"BO? hQB0\`K78h"QoV`r`,f#4ELYeJ#NA%,&d+Hc`F^FFJfEaFRtE:s,8UcmoV@&RsS#RoV?MaA)Yf,fms@lPc Rule 3: The tilde (~) cannot, by itself, go between two statements. ET >> R:C7#m&ZIar`nDN:9%&S_42Yn8:kpqf,l+i\tsKbI&\QqZ]0FW@B'gl They read "p if and only if q" and are denoted \(p \leftrightarrow q\) or "p iff q", which is logically equivalent to \((p \to q) \wedge (q \to p)\). nm5p33R*%j9sVNP:C]*b&crr(d'b,WjC>neG9'E:)ZfO$M+\if+#J'L9[-MfQ. /Widths [ 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 667 333 333 333 333 333 333 333 333 333 333 778 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 611 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 778 778 778 778 333 278 667 667 778 778 333 333 333 1000 778 333 889 611 333 1000 1000 1000 833 833 333 333 417 417 1111 1111 333 333 500 500 1000 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 ] (+KCr3_qf^RF&OhLi2$E"u,!.)_0`1YfUE98?[`i*q\2Xb)?@cV!Z4U!. BT ?QmF!0U,UN'q$dWd%Oi+[f!.lH2?b"[:cc60I\VB8o5H2OkY-LY(-ciotL\U>]EOSL\5f3AR6hq%L< Q q It is the only habitable planet on the solar system. \YE7Kfr-C:,OjcFh>LIh%,5eW[E?^`^=2.k1Q:A++9fD$PF%&^msD9$<6/qg;1R+&6%01aOh7/-5JR ?lO;hPMc5tH/V2i0_M*.I'ilWIca&Bp25;Z]NPbe_h,cX,5-^:Kl!Hma)Br=orECSK]0`cp,):!-cYlFhgpLGX2-k;S%'"k1 Q q 342.72 456.96 m 343.125 456.96 l 343.125 470.91 l 342.72 470.91 l f BT &G=lJ, 4UWGN^0W%:]WQu#"fhDDRVJL\u674ZjZ18/AEKKN: P:9X'P6+^,Br>rEFbC.3hT\PI-0;f4NG/m\q@Be' In particular, we define tautologies, contradictions, and contingencies as follows: A compound proposition is said to be a tautology if and only if it is true for all possible combinations of truth values of the propositional variables which it contains. 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