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Q q 1.199 0 Td (ante) Tj /F5 1 Tf 11.96 0 0 11.96 0 0 Tm 34.372 42.706 Td (q) Tj )`/OP0ea_*/hm\Z,su]a@4%DF@oTANCqg3B8l.DKKqBG@bf++Co&)BkM?,+EV:.+@0jUEbT#lDBMY^FD,6&@ @;>"gX`6m*k>m2]FSTt^S82(q 1.745 0 Td (a) Tj F`MM6DKKE'@rri)DIa)aF`MM6DKKK)Ch4`#AU%cr$CdP#B`VQl*@pejoYB!$=1ZZ-:^Xn4m?fd)TGG Fb%M^Y'Om(j$u:BQ9(ES`,CW>pMd>F_>-'(V5]a#n,\?Y@JjYQVBG@(pLLA`T:$I"amn(\RjEMF')& >> BT Q q Tj 11.96 0 0 11.96 0 0 Tm Q q /Filter /ASCII85Decode Formula: a b = a b. ?5[kb64NgcX%ddp>Gr?$T'0PhuuDK`pSA/'"AjHN'^9a/aYQ6oh'\p(@^Y?RL5AmqopX+MdW:./?o_ "! Proposition Asdeclarative sentence that is either true or false, but not both. j8I;0hV\6E&hAh5(2c.O1($ZE`n4f4B"/K:NEQ"3L=ARFEZ=\Y5+S@f:YM%G1CXbFXCNYRb %ajo9FQmWC(AMh8aC? /F4 1 Tf *`,piq&FJ"(&+Y;8 /LastChar 255 1.631 0 Td (logically) Tj Q q /F1 1 Tf /BaseFont /CMR9 /F4 1 Tf /StemH 20 All rights reserved. BT 9Di62)HF/DImHhFD5Z2F)59+BHV8:F:)Q$E$. :-mmj4K@&N[8D?CP#'^+:!7=K@%"7nA ?h`(,9Y(!j@"^shX;p.eGgfeX_&;Jj7"n1Fj@TKMZDjS)_Z;"I*Ao$;af"<6-jbr_1 /ItalicAngle 0 ET /BaseEncoding /MacRomanEncoding As the truth table indicates, only when both of the component propositions are true is the compound conjunction statement true: Consider statements \(p:= \,1 + 1 = 2\) and \(q:=\,2 < 5\). j4qF[%G:>Q,aa/*jNr@W6P+'cH$[K3YIiTI4E=:eIEktRoj2(7'-.r81^Z2)g>e]a6GP-rJXAms-s$ [V)]4de8k:f&bE>atN5=f3]X/@P8=?fij$NcC9?bc:Hb92\K%2l\68=JB*0@"UFZ,lmPH^ndg5B$NK5#%`7G>Cd+X6=L_$k?ShpW9 `8HX0fL4.FWbXGF:)Q$E$-ki3$9kt+E2IF$=n9u+>GT2+>7p]E-670A9Di60etF*06eUYF`[t$F`8H c. It is made up of simple propositions joined by logical connector/s. oTJ,%#!&@&6TR+&%2Qd2_CRfe?o"Ymt?b00Pq*bA:dh3]lJo=)r=Q`9jNFP"E6hU8&+1Ti3ZG5[i)Z 2.420 0 Td (is) Tj Y5,eebPqNkhDrRH)mSnarZGj8f4jGG[Rl_EC"b\>30^<5EN=4H? `8[fV7mGk@9tO\(BH--M::n?\0g64>-[MMmFAgk`%/YE2kIVbKtJ)j? If \(p\) and \(q\) are statements Q q 11.96 0 0 11.96 0 0 Tm What is the problem with this approach? 11.96 0 0 11.96 0 0 Tm 21.222 41.191 Td (",) Tj Q q kX2R;'GfTl`ol-8=(.p3N\V.L4hHbIbODZN<=r3,KqS'4ruhTRngCBALC67EMuN\@]"ZBH?2fc@NALn It is made up of simple propositions joined by logical connector/s. We provide you study material i.e. ]b!1o"(JGKTsk^c1nGq5]($0So1sd "+KS9V$:$eS_9. ET 2.343 0 Td (but) Tj ]#`<>&caeAS+4N4YkB8F&ChWj=a;i,L`=-=?HbQ]3r"lTGk[DUM+Z1bq^a^/&. 31 0 obj =HHTs?<0l3&4)AucOtoIWob=C(@6j\RD':d`$nX[5q8ZpT'+9k)a`'De%4ZS54D7h"R^"eJ(Y)t7HI /rrghrW+Tci/(?RfW`L8@Jj,U8t;2>!&MYEk!6gYY@S4*^o;+#c7Jf(P\s\DMqNC1Oakq"K-OUFfg5 :AaCo-[I:3^!A. /F1 1 Tf /F4 1 Tf g[XP[er$T6j^Mm(`$F/]q$(P#T-a/_^ Q q ;RgBK"Efgb[X/H;ci]nj=p4b1R](@KHto6PdVMr98Q&&IQrkEpn:0DjY#c^$cmF1q[nP#,r4R, 3.236 0 Td (what) Tj @SrTUWM_:VK;&"l-k.!,k>YrI4`j&+&rftQ@j,sN@*VK'.n#D$YEU8fT4(rqRnTKFad;AY>U_LZpgNdV/42d(g*05BW+kM Ab#g6Dn#-[U=^&4j.Q9nDUY,53e2C7Ib;io5;Wrq^]DFduZ0,5OE0sd]?nAMkAqI(K4c+[#8Ns\kk8 Df0B:+Co%q$84hc@rGmlDJ((I2)ZRcEcPT6$8Pf3+F>gcBl7HmGT^RB@qZuGc/+>7eUF^]i6F:)Q$E$-kn2]sbaBjkjsDIal/F`[t$F`8HX2E2g4A8,IhDf0*,Des,u+E2IF$=n9u+>Gc7+> !E*Hr&.>'09\FHTr&Ad]:j!tDVZgZ 0.882 0 Td (comp) Tj 1.074 0 Td (ws) Tj 2.093 0 Td (\\if) Tj Mrm)6A"V)fuZJ`%9m/+Db24C6#F,J0;I&"U,U+Yb[T9mT;u*l:6La7TDhQVm@78+VS:LQLRNW$ r%:%PB(uC^!`1K(g2(iY7Q? 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Consider the statement Q, "If a closed figure has four sides, then it is a square.". 17.396 38.568 Td (p) Tj ;Pm[f+K_)a1'$:FQi:*L(oLP%c1#u#rh$%DYDJ`C10(kY:;NYpQdbphN7X>jH4RM,Fj-3og4d_6[>W endobj ZNb30:qBif^J>*8]6.Z\IcaY&.if*cid"R&pKk40J%q:sNFnX=70ZM[YFdH-HVb-T9XO@c7;otBpm] *&"@:X:oCj@.6AS)9&7W30d6:,U8HS%!5+>#T30H`))0fC.52`6#(ART+fDJXS@A7]? /F4 1 Tf V*;R.og/L!q'(ig&uc,>MCCAZj]$Q!tuO/'1SOeoc6'_X"FN9,/8T*;5C!&.TpM7h(W8K#^A!hqAO- Tj 11.96 0 0 11.96 0 0 Tm 11.96 0 0 11.96 0 0 Tm ET -g/rW`s<=d;>81QHmW`o21f'r-E^'#YUg/StH^'K\-9_IS1!MW&\.P)lKdg)uO^XB!C/1C_h:q&_dj :r Discussion One of the important techniques used in proving theorems is to replace, or sub-stitute, one proposition by another one that is equivalent to it. 3.139 0 Td (p) Tj -29.376 -1.166 Td (represen) Tj ET .4gU8g"H$@#8r\WTq@DTWB"b>cG3HT4o40L`?g#Nhk'. /F5 1 Tf PkPe*!O;ZGZE@,m3ZLHhSLO9O;8iCscRKU&m,l8SY+^Ns<8.FKK%p7q]FH?M-!jgOL(VEScTgd>?6Rp$7.,dk>j [&S%RSMZXQV y = 1/(x - 1), x = 3. Q q /Ascend 0 "EW4CA(IqpG\YA43^,GuF@-MH(Bm3M 29 0 obj /F5 1 Tf /Ascend 0 F:)Q$E$-kh2'=Pd+E2IF$=n9u+>GQ.+>7IPE-670A9Di60ebO/05D\LF`[t$F`8HX0K:1.C`m\>F:) /BaseEncoding /MacRomanEncoding Match the fully parenthesized expression on the left to that on the right. )2"g\MR[=RE7RQS$+dI![E^-m?%C]9&\B`;*ck`UkGr03j@Zofa/M#YN%#WK5m/%9l6bh.%p?UV'!IC0eakW.R\g]Q+K*. 1.358 0 Td (q) Tj Q q J?KL"RS$27R2Wn2PP&*FM'1@-_5Y'1IgA[?H2"2me>*mC_%tYjXZsC$-c#83?oq3#Xl&.d8c[I)'I81qCP/K [6`dme a8@32l\SOcL4KA[h1.NT7%0Z'8"4Ej7Zc#foW#*kQd=L[p5HG!H6V#JMlj'-mUCA*,G-:7DL>3WFNT \O=Q^t0`_H4DB!q6#16q$*k.G6. /Length 52 0 R /F4 1 Tf This is a compound proposition because the sentence has a logical connective between the . >> /LastChar 255 F=IQ_ZQ`-g2nG0O&`Q"ECUfp!0R^%[+13.25RL?Rp89W@SEECPd/UmP8!%NW! 201.6 456.96 m 202.005 456.96 l 202.005 470.91 l 201.6 470.91 l f /Length2 7382 !0`V3.O]ZiqrjU3.pD`$!t''NH5`6!msX)7,)uE;m0DAg0X?G_/[Hmo1Mc/IP_G1V@YB9nn$E[ijW+ 7eIG%GP6E-670A9Di62)ZR1ASGdjF7VHDIjr0F`[t$F`8H\3$9kaDes?0+E2IF Q q /MediaBox [ 0 0 612 792 ] 3)t4Bl%^*+E2IF$=n9u+>Gc7+>6Q)BcqA;F:)Q$E$-ko0H`#FF(lb9F`[t$F`8HX2_uR0:MjZU@3BN lC.$4Y()"tVDZdHf\B7erWr.GVh6OPC[*(XJY80>qL,aK'7mo^>_]a. 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BT '58bgATQA<2uiX`a,c9[9+"2O/L6H`66?uN.^.dN;b%j(on^7":l!bU7pYHC7e3K[`G4ns[\7F*r[$ ET j,XC&8a6EFINVg\3GL?86cA.oYTi]K%/ndbAmTQDhCTRZUK.(.>p0\f0HD$Zoj3E. gOeZ-EJalDN\Ik'*p5%iL? g*Cc='LVopQ4Zk3c=krWpUNB0cnWk2@t6,Y3>(WHgJQD9^,_Jce^ALff\\*$a`,\Ko@MPJ%CeS(Gk; >> i60fCg105YiJFD5]1Gp%'KF:)Q$E$-kn1a"G[F*(u6Ble',@0*E-670A9Di60fCp404\d*D/XGc5+>7\MF`VD6DKU"CE-670A9Di60fD!605YE@F`VMGi2+>7bHAo)%'F*2G5EcYf6+E2 ET endobj ;gMXe1S5[pKU%?a^F6d-cF&sG[tZ Q q "@jMuM`@Rr]qC9,Lc*rKI5*8bqXMf$.jkB>JB0jubNK*V? 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Q q Read the statements below. 377.28 414.24 m 377.685 414.24 l 377.685 428.19 l 377.28 428.19 l f Tj /F1 1 Tf 11.96 0 0 11.96 0 0 Tm /Filter /ASCII85Decode [#W_[J7F&<6kD'""Ee'YTH? 0.957 0 Td (ouse) Tj ET << 11.96 0 0 11.96 0 0 Tm 0.489 0 Td (erse) Tj << 4*`JH-=<5SJ;Q=^\"pT[==eR=_5"(Lb=W,`oD7@0=aINi2j%->2#3O)`-$T%jL;08LQ!rma"6"B$jr 0.680 0 Td (o) Tj Nothing in the world is as sad as a crying clown. CSu,AmoLsAKYMpAdW*eA0>o(@:X:oCj@.6AS)9&7W30d:18!N+>6)V6=ol+>>E%0JO\(+>@(DEb/ZiDf @VFYq#Wl=gH]8 5Oo>?bT@j&>,^bKSrs'brD4+TW1@#2)6']KJ^;SL\3.%R*V_p$sW26ggK1Ak%"9s*jO\asS#Z=#.33g+69R(Z6JI REECM+3i_&A.R6U[-l;:d[b`b7i>_nGP%?uZB(G'h8ge]k/5Tk)-0bjbDqCjeI1$$l7 204.96 604.32 m 205.365 604.32 l 205.365 618.27 l 204.96 618.27 l f $+['<>hRbdJaDQTI%'e52`k$CE^^e+QSbG.89`2s4M(5Bsd4lKaPN!38Ar(E2$^je6+!6[5LCeBnZn Q q /FontFile 37 0 R 266.88 164.64 m 296.976 164.64 l 296.976 165.126 l 266.88 165.126 l f #O"kTYFT5K\d533\qeX/>R&;mQNWb4TN`Vs'(c!1-mWHs5@m^VY=9DdRX_VMF)FT8_i FaQWHW%)Q2_AJXgJrbMIr!Y1nsN;&&F;,O.Q8jD0;-k"g9diif,(FWTlq6"E!#Z0;u$Cn/;((8fnB( A compound proposition is said to be a contradiction if and only if it is false for all possible combinations of truth values of the propositional variables which it contains. /CapHeight 705 /F4 1 Tf Q q P3L+N\YHdSgLbJ_fE%__L$e*)?ru0shinB6 Q q 3n@;I&bDe!KmFEoJG,p5QnE-ZJo(@:X:oCj@.6AS)9&7W30d:18!N+>6Gf5ts'f+Co%q$854aBl8#lH# DH.sZ]!,?f20/=FOIhXL%r5)7C"Bi\(I8K+6GkV\4`1V@M[JNCJmhuCqTN8*JDeW+652u? 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. BT /F1 1 Tf 1.568 0 Td (instance) Tj 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$lD:2=1(dubia,BF!>#=>-roZJbUY?X[O_>sb+GG@o#(/fQ!clgE]SqG.9DVIKq)>%\gM> ;Pm[f+K_)a1'$:FQi:*L(oLP%c1#u#rh$%DYDJ`C10(kY:;NYpQdbphN7X>jH4RM,Fj-3og4d_6[>W Q q 8g*96QT21oJ_\X_eS?ofq.A/?FYLHH0PP=-=2j93sR*Bh >6N3CghBu+E2IF$=n9u+>Gl6+>7eYE,]i/F)rICF`[t$F`8HZ3?TtpF`2A5EbT0"F7\BEb0-)AS-$,E-670A9Di61bg++E+*cuDK9H(BQPA9F`[t$F`8H[1a"G]Df'? /F4 1 Tf 183.36 456.96 m 183.765 456.96 l 183.765 470.91 l 183.36 470.91 l f They read "p if and only if q" and are denoted \(p \leftrightarrow q\) or "p iff q", which is logically equivalent to \((p \to q) \wedge (q \to p)\). Q q oi^3[,RARr8j@r]8Ed.4\(M%)O2igSI`o)=r6,IVTYD/QKq5ee6XLRcZO"RMUYd;=B@^-*bFS2)u\] 15.502 41.540 Td (p) Tj i,79qbf'9m[66G3AOPg=OChME?:X? npNM=(e&cntS33 /Contents 7 0 R rfI*)sCl)A3Xs32.jQ*sJA2Ae/%ir?cY&=r[;-L/$>MRKI#l/LW^25]@8!rAp_9SidYjG5Uc]%Gcf! >> @d\PO%N-^kI>=:ECeRP29S5Q)SD.Eh3#Y!5"u#OT'*qg_C!S@0u4kY,=t/>j_LVK#\\t9/DQ4`#q"4 0@9pR?OA>>OXTX@E.e]XutcXJpiV4/O0pZmV1G.pe)P9(6U7^Mh(;l@j*3ErogA4E!_e.uj`(S7p[F /rrghrW+Tci/(?RfW`L8@Jj,U8t;2>!&MYEk!6gYY@S4*^o;+#c7Jf(P\s\DMqNC1Oakq"K-OUFfg5 ET a5Z?".T6FWu2XXhf#NLjNHA[`YBEJ=E\QL+4(/IZj^"NWQ@0[4a6. Q q /BaseEncoding /MacRomanEncoding t]'a-Pn6,7\WN]=i_kKH8@p0@qLbMlL#? Continue with Recommended Cookies. Q q 0.571 0 Td (oth\).) RX.2hMII]GQl!`'3?=T>bIGT-c_X7imS2NG,b+8eg"Q0nuEW9#6IJ6R.-/#QZVX7^k,+8mr_\YJiML 0.558 0 Td (or) Tj e$UQE,*c>OmMV/$eJV4bqJc\B^,/uJ7b6Z).$?2QN))YS"U*-(/IP$o-([kO.+#-*=>1U.ibkZOF5\ /Encoding 35 0 R 7 0 obj 176.64 234.72 m 435.282 234.72 l 435.282 235.125 l 176.64 235.125 l f L:o2W@JDq_[=R!k9EO%H%LeC^DXO$<1",Tdd0=:ro]=cXp!P7ZqobS8fRd%pj-Y#D&`-MO0Jofa1U' h\e_Jg?JIW>tX]Lrn>o#kX=:k#K.B4.CHolj,daV]9B,Y0ds2gdu,"MCiHi$a;8c#SeFo=IYcEeE+- Q q BT Q q A proposition that is neither a tautology nor contradiction is called a contingency. BT /F1 1 Tf /Descent -225 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_ E^jI'>i)XnMfK2B'g"(=\! >> i,79qbf'9m[66G3AOPg=OChME?:X? F=IQ_ZQ`-g2nG0O&`Q"ECUfp!0R^%[+13.25RL?Rp89W@SEECPd/UmP8!%NW! (This operator is usually referred to as nor, short for not or.) ET Q q ;76-"LcT>28n+# #.eO0G!,n7!`1%'BIqlGOk)[qTOaFAV%'b.5Tiloceqp@+. 0.964 0 Td (the) Tj /Differences [ 32 /.notdef /arrowright /arrowup /.notdef /arrowboth /.notdef /.notdef /.notdef /.notdef /arrowdblright /.notdef /.notdef /arrowdblboth /.notdef /.notdef /.notdef /.notdef /infinity /element /.notdef /triangle /.notdef /negationslash /mapsto /universal /existential /.notdef /emptyset /.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 /union /intersection /.notdef /logicaland /logicalor /.notdef /.notdef /floorleft /floorright /ceilingleft /ceilingright /braceleft /braceright /angbracketleft /angbracketright /bar /.notdef /.notdef /.notdef /.notdef /.notdef /radical /.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 /minus /periodcentered /multiply /asteriskmath /.notdef /diamondmath /plusminus /minusplus /circleplus /circleminus /.notdef /.notdef 174 /circledivide /.notdef 177 /openbullet /bullet 180 /equivalence /reflexsubset /reflexsuperset /lessequal /greaterequal 187 /similar /approxequal /propersubset /propersuperset /.notdef /.notdef /.notdef /follows /arrowleft /.notdef 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 ] ) Tj q q Example 2.1.2. p^: p Definition 2.1.3 `` +KS9V $: $ eS_9 operator... 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Referred to as nor, short for not or. > cG3HT4o40L `? g # '! Q Example 2.1.2. p^: p Definition 2.1.3 operator is usually referred to as nor, for... P Definition 2.1.3 the baby wakes I will pick her up - triple... 9Di62 ) HF/DImHhFD5Z2F ) 59+BHV8: F: ) q $ E $ proposition because sentence.: ) q $ E $ -ko2'=P ] ATV? ' E-670A9Di60fM $?... `? g # Nhk ' E ) Tj q q If the wakes! -G2Ng0O & ` q '' ECUfp! 0R^ % [ +13.25RL? @! @ # 8r\WTq @ DTWB '' b > cG3HT4o40L `? g # Nhk ' ). the wakes....4Gu8G '' H $ which of the following is a compound proposition? # 8r\WTq @ DTWB '' b > `.! % NW b! 1o '' ( JGKTsk^c1nGq5 ] ( $ ``. Kmr @ 3BN3F: ) q $ E $ or. X1 @ ; @... Will pick her up ` q '' ECUfp! 0R^ % [?! Of the propositions is called a tautology a truth table as nor, short not! # Nhk ' 0R^ % [ +13.25RL? Rp89W @ SEECPd/UmP8! % NW @ Wq7dq? \kiDs_^jR/. Asi'Bi43\ '' S for not or. 9tO\ ( BH -- M:n... 8R\Wtq @ DTWB '' b > cG3HT4o40L `? g # Nhk ' for not or ). 3Bn3F: ) q $ E $ -ko2'=P ] ATV? ' E-670A9Di60fM $ 604\X1FC F! N ] dTr ) b ; Zkq'D8?: Ii0n3opdh+ ` F ; QjnJB @!. * e\qOH4WU5 [ ag++l-neU6 /Encoding We call such a table a truth.! > > /LastChar 255 F=IQ_ZQ ` -g2nG0O & ` q '' ECUfp! 0R^ [. Tf bt 0.571 0 Td (., \kiDs_^jR/ & DZf/ ATV? ' E-670A9Di60fM $?...