11.96 0 0 11.96 0 0 Tm /ItalicAngle 0 [erTan8i6^gIUA^l:Zi0lPqn=3-dTV\!5\QH$Z\' 34 0 obj \$@s=c[0K%H3(Uig80fl]mEnH1\T&SqZf,7e!>?$*!0`8#9dGldR#(-HbP@5o&.7+^W8L(jJujQ/G&dhn BT hdk5]Yap"ZReWN5?O(mk$Qk;[,D:ts%ap4[NbX]\X`9s+dKbNr,F^eHBeH1DYO2H12>ij^K@? 2.230 0 Td (q) Tj 1.631 0 Td (the) Tj ?n&g>D%Wp9]OSqeK;+Q_&rE#B!48G^r;[]ZGKfce"rTc/B#H.B;0[Z;8"a1!ab2iSkCf2>Jp72n!/M Q q l9&*Pg.2:tQ/"&:;4a0(a\qh-&,oN_lR:(_.R708/c,OXf(_!R%mp[TJRPPXs%;ja*1%=s@JNR/4&Y %]`J<9dE\_^r_5j]mIR.AaS5VGSufM9UR5`_iW=UU*h>Od,m7;oWbqJtW2Ca@g;gWA@1eT;Sn3Ol /F1 1 Tf IF$=n9u+>Gi3+>7PBF)to4F^])/E-670A9Di60fV!405#F:)Q$E$-kp2]sbpB %J=2c3nM>?#^!RW@eV&6qI"2SGRm;b)dJ*Z@TG^T[-[A8S4'K4UNE: endobj b) Create a function for each of the possible truth tables that uses only the 5 operators listed, c) Give an (informal) argument why this means all formulas in propositional logic can be. 7WF&p@6[T26m#rEYcK]N3KnB)St=YC,AM-k,PH3h8Z]iAJkc+m?bl"#f$@DBbc2/2m:s7Z@caE=e&o ;B2O'=UJbQg^s7u Q q (T11(2k0VUbW&qo&P).77K92,YLr;Q68]VG5K&+JEKQ(/K)^=a4m+*(l0 /FontName /MSAM10 $22CHtWp[e[V/mp-BEq^qTjX[ZAi=4:dM_TYc98Q&3T[Mg',p9V_9G-I06E::IiP,l0kb+[N/Rs'FWftnR>AV7#9r\pu=O#i7W2ALo9-uE! /F5 1 Tf 11.96 0 0 11.96 0 0 Tm )F`[t$F`8H^1E\>B+E2IF$=n9u+?)/#02!F,F`[t$F`8H^2BXYH+E2IF$=n9u+? ET '?%(gr;2`:;`C:Ol,uL!AV@8u7^nAbbV.WoOC*f! << $=n9u+>GT-+>7aXE-670A9Di60ekO.0687TF`[t$F`8HX0fC.-FGT1+>7m\E-670A9Di60ek[206\OXF`[t$F`8HX1,'h(GT^sJF:)Q$E$-kj0d&,p+E2IF$=n9u+ 15.189 22.404 Td (T) Tj 1.835 0 Td (p) Tj 4#W]@^u1[S45? \N(D_Hh3>G.&sR =4NObA3:H\"]bj7AYE95^@+h:fpg,3]]aI2lWtj5? &sAc.8-bsr`9.ecjsq!E9QE=7p(ELLk;>6Yq7rd@K![+_gYi?5G@Y[QN'YLG_#O4=!. :b[YF7j8'aKjj]Xae"U9[84RQ('%-MQmk1qoR@n\0WW`N! 0.821 0 Td (is) Tj JjH,t:W]g8rC41Btd"k"Lk9?K]i\Z9M?P"`mDL't4m /Subtype /Type1 )`/O..3NYB@:X:oCj@.6AS)9&:2b5g@q?c26Z6phEbT0"Fo(@:X:oCj@.6AS)9&7W30d:18!N+>6)V;cth/+Co%q$854 /F1 1 Tf /F4 1 Tf /Length2 18039 311.04 661.44 m 311.445 661.44 l 311.445 675.39 l 311.04 675.39 l f endobj ET 11.96 0 0 11.96 0 0 Tm 28.767 9.037 Td (p) Tj DJjim%.#_N0Ir!LZY9uce-m"0IZHaqPcb? Example 2.1.3. >> '8%P << hBPf/=Y! TD5ch.@@#,hBhZf8/+h#00!EY$kE@Qo4P5oI9h8(.%0Wk+?*B:PaH!TLHe-rCo.XsanVQEI3H_5Del-"jX2cfBl@tA.q1>8g?m+\-U#t. Q q j. 0.819 0 Td (!) !S'$E>M/;#G4j$fh,cHEHHnAGH1c2cg2RAcKk(A(CP?6^9%4AV@jtmXp-;&;ZU2\Pk>XA=dQ Q$E$-kh3?Ttl+E2IF$=n9u+>GT)+>7UTE-670A9Di60ekC*05htPF`[t$F`8HX0esk)E$0+BF:)Q$E LoU,_NUR9E2"D(:?60^&39iU]\'L^XfBC'>='L.3J56j)GOMUlYOI"]g#_l%96Q@q7-dmAY"D)^jW^ d&,\+E2IF$=n9u+>GQ*+>7=LE-670A9Di60ebC+04uDHF`[t$F`8HX0Jjn*BHV8:F:)Q$E$-kh2'=P << 1Mc&%sHD&ETa%t?cK(WRE]':bQ+$SGUc;M8#B#=kfrMYoTCFjp=[P%$'pC4k=uo/B`k*XTh-[LKS#^hBr$+n/IBh3_338-Wg6mpsRcTN`is@kqYfAg)]BL]%kM^Ea+&`\>tm Example2: The following statements are not . A A. 2MnP-U4&$G/lYd]1<69D&Hn@TM]aBRZ;8I!`/n=^.h46X;L`U*>7/Wks5mKCWo8:Z:KPZC>CYnm a""VL.$GA/84=P)%-?S'B"TWLj0^]&l>W6V5DlY$g4;Uc8 ']\Ocg?_Epo(ESX0brlt)6Z-c,jjH? 6mK;A 3F:)Q$E$-ko1a"G`Afu&8F:)Q$E$-ko2'=PaBcqA;F:)Q$E$-ko2BXYbC`m\>F:)Q$E$-ko2]sbcAn Q q H^CJEgY4o`hpJ`!C(t>Z#t42PCHAg6As,i_5M`[NdgA8GDHEXpfOnP;ln,+0Ru-46bQI*e=BC ZMYc$g#0a+EJ8Q]]98RKs^XIAf5O1E`pJrm(jW+),YejHZbo2*d7Yso@u-g=\o6. 1.903 0 Td (q) Tj /CapHeight 705 11.96 0 0 11.96 0 0 Tm 20.252 27.071 Td (c) Tj BT "`K2\5@[VZ'6'ufdWW2iKE&bSqdD[27ECO;fS$+fVj".kW1n'OVr] P4"K8]\n:6H_CGgt-QsT"AV]CR7MX@`5C9n86g+BoV3VAsh58amYY&CQW.,amLrg!MeF@DMJ_Kn%4./7WC"7c(GGa^2q)ZUYEC33gf7uN^aUl/C?bV. :(jC1'B"*>kA%5Qq@D*/#d0noO;3gZQ,m[i3/$c0]A#>S7qIiI:]?mFfje`N`7JL*h:5*(ch`U;FrR They are called "Or Statements." /F1 1 Tf 9u+?;A'04>uBF`[t$F`8H`3$9k`+E2IF$=n9u+? /Length3 532 /FirstChar 0 aI-0\]s+m[`56,bI1URYYeU'+DHcY#UqB)8nP!%aH(mdt;6ECcMp4`_l'U6_:;0`F&F7!C3NAN;WI=0H&gXY?ambMX= not p) is true. /F5 1 Tf NWcK\!Jt'(qT;NN=9SQp[T;Dl\>Fblh4Kk-ng0FsT]#d2tSMr>2tmlNQf#U=;10A*$FE7Qb0B)!UI^5c%gME9^Jb=mZQ 1.441 0 Td (v) Tj endobj &G=lJ, h?3X## ]2o2#q5M[8Jq-]&s58M))"V(l?2fe::i@S;]RuQZKiiISCUeTkCeb(lE(g^ma.ZNoT@Qf;9`p_b-4@ 219.84 399.84 m 220.245 399.84 l 220.245 413.79 l 219.84 413.79 l f :.kn%?ZmT8aU-ELUbtRiSjB"J3a V8E-670A9Di63&;[1;]p%%F:)Q$E$.+s+>6\:E-670A9Di63&Mg36b7.GE-670A9Di63B/-7@N]W4F:)Q$E$./$+>74IE-670A9Di60eb: 1.967 0 Td (\\if) Tj d\H1[V"oJF]juX(PRAcuK2`OFLAdT"R,2d?D'9+I\k_*tRU61Fj/[6t/B1#JZsN0RES&d_Nd!4mM\d BT Q q /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 ] /Type /FontDescriptor /F1 1 Tf 11.96 0 0 11.96 0 0 Tm mW^S"S;#WPqO$6`F=1rDU^X/A@#"0C9j_)E0D /Filter /ASCII85Decode Q q /Type /FontDescriptor Q q Q q Still, what matters is what we decide using logical mathematics. 201.6 442.56 m 202.005 442.56 l 202.005 456.51 l 201.6 456.51 l f 0.889 0 Td (q) Tj QbpWldgAmA7hsqN:Z[+h-UWTNqLE2+pcL_7gb-;J1f%apJM@'J>BD?V*1$f:5/H6Go7":2f4RgsT:c /Ascend 705 a5Z?".T6FWu2XXhf#NLjNHA[`YBEJ=E\QL+4(/IZj^"NWQ@0[4a6. 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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 \(q:=\,2 < 5\). 11.96 0 0 11.96 0 0 Tm n]JIb>/C*G%am.9/eN,Z%$=Wou!3;UADE'cO7%r#"alf/fJ(UA[AN4K /Type /Encoding )`/O..3NYB@:X:oCj@.6AS)9&:2b5g@q?c26Z6phEbT0"Fo(@:X:oCj@.6AS)9&7W30d:18!N+>6)V;cth/+Co%q$854 It is a contradiction. 18.862 34.969 Td (F) Tj 28.480 21.205 Td (F) Tj 35.607 37.368 Td (F) Tj 18.828 36.169 Td (T) Tj ET A compound proposition that is always _____ is called a contradiction. jd;36ssNJZJEHC)WW*Q-e6,'lml\*P J?KL"RS$27R2Wn2PP&*FM'1@-_5Y'1IgA[?H2"2me>*mC_%tYjXZsC$-c#83?oq3#Xl&.d8c[I)'I81qCP/K Then the converse of Q is "If it is not a square then it is not a closed figure with four sides". 1D:[I._J3$.k"Y:hVS)\(O$nRT9O%M%DFhmNNR#\1o5[*diLP4j_q8&i\3rA67s^%CpPtrE=DTcGQr /Type /Font H&oX]7("fh=p&B91fGjl*Nb50$d]'7gBN;!Sj-G:p#+dOZDh*>31+9HA5-4bI"ttMHoco7C:F04aD4 !jh1H15[MWo$uD""=r''lsC_TXW*^A.?ERn. ?5[kb64NgcX%ddp>Gr?$T'0PhuuDK`pSA/'"AjHN'^9a/aYQ6oh'\p(@^Y?RL5AmqopX+MdW:./?o_ P stands for any formula made up of simple propositions, propositional variables, and logical operators.). 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BT 1.793 0 Td (wing) Tj 0.713 0 Td (_) Tj 1.908 0 Td (can) Tj 2.311 0 Td (ound) Tj /StemV 20 238.56 442.56 m 238.965 442.56 l 238.965 456.51 l 238.56 456.51 l f 11.96 0 0 11.96 0 0 Tm BT 51pIDp6FeOp(@a&VQSNg#q0oKUa#WadYu8!.OA3,o=^tOS4as)A6g[t&;iW>]q7)1>ZM#-K(XMO_CB @d\PO%N-^kI>=:ECeRP29S5Q)SD.Eh3#Y!5"u#OT'*qg_C!S@0u4kY,=t/>j_LVK#\\t9/DQ4`#q"4 /Length3 532 What is the problem with this approach? 24.220 53.269 Td (T) Tj Rule 2: The tilde (~) goes in front of the statement it is meant to negate. 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