/F3 1 Tf 8F`[t$F`8HX3Ahp4F*2G@Eb0<5+E2IF$=n9u+>Yer04fB7\FEarZuF7_WDffQ2BkM+$+E2IF$=n9u+>bbp05t?? 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No, because every possible proposition is expressed in at least one sentence. @C9? 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"DE.FA9T``XAM_:YZEbQ0,8=f$G,9H+Uda^#CRJ0A$npb!delo3+PMY_oY;QL`QffulO?BaC<0N;lS ET \4$)\mqWb&r1)kOTiFuQ,ij0jhCY!:ggY_=ndG'V;ZmCHn(U:?.cW'dDWK+d/>VFC&? /F1 1 Tf /Type /Encoding ;*e`f8lo"_'(&H2h& 53c["IS7`!D_lL;L! 1.305 0 Td (with) Tj << MEqO99+$?4m_\HHbsZ.h\Npc((H;0`KqZhSFbZkC]r,B))*8h.1oo4ai+q:n8>N,r,eq\ApcrD#?=" R=;0Han;AdU1\Df0YrH#R=;0d("=ol+>>E%0JO\(+>@(DEb/Zi Neither x is an even number nor x is divisible by 3. x is not an even number and x is divisible by 3. cmKWl10.\:n3`^Slq=Yc"FMG";>MMTNY>r$aT>R,-]I/pXAi`mZBB!3ca_;7N^$Ig-KD2Y]ES7%P9chjrr4e"s9X'5g?b6L^MJqK7O3O-p$FQMnZT>b)lH /F1 1 Tf /Flags 4 &,p+E2IF$=n9u+>GW,+>8$`E-670A9Di60etO-04f$3@<6*)E-670A9Di60etR.04f!2@<6*)E-670 /F1 1 Tf >> Q q /F4 1 Tf BT ET BT << =Ch1#3oL+3`17f999]@_Ig^^oA2gT@_7+On>a6*,N.S:&nFj0VC](hQ\o;gQrGc1.Q>(E%UfPAV#u) << Medium. HkJX?B)\%cl?2ieKY-<5%BM6^O-Zn\kSG4"S+6[,SBW%RRe(Hi2R*aVc"HXI@7ejA6,.Zhj>6= !1'n0;Z03+O5d7X3u*AOM\tB:Vg:)n9ZE_8JK_T'TX*,%XEt`fcpD/G?Ru=,_/t5cn3A=[f%&2]7q? 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EM)QC]19m<7!&! hQB0\`K78h"QoV`r`,f#4ELYeJ#NA%,&d+Hc`F^FFJfEaFRtE:s,8UcmoV@&RsS#RoV?MaA)Yf,fms@lPc /BaseEncoding /MacRomanEncoding ^id]3@J0LTru4(4+LX4n\FLHE@ihdIG)(EM@tKMLDOM0m9i_VK!lklS@ko]>\-<0Wg`U$r\ jc]n`/(dIbKm:O.9]q(g)/o%kPcd38GPD#l!kI#k /FirstChar 0 11.96 0 0 11.96 0 0 Tm 20.546 40.836 Td (examining) Tj F`[t$F`8HX0KC7/D'3e?F:)Q$E$-ki0H`#d+E2IF$=n9u+>GT*+>7XUE-670A9Di60ekF+05r%QF`[ /F1 1 Tf Q q This is called an, If a person is asked whether they would like a Coke or a Pepsi, they are expected to choose between the two options. LoU,_NUR9E2"D(:?60^&39iU]\'L^XfBC'>='L.3J56j)GOMUlYOI"]g#_l%96Q@q7-dmAY"D)^jW^ [eRg:;&r^&I/QQ=8"M33c>4q+*o*lc9TB$lU\tVV%7(i/f1S'D6X8_!SHo,_>RFV";0 7!Uhh2.E#+9(/>5O5>@M1e5P;i&GB`Unr(nK7n_SPih/FXBXL1XQT/3"#+71\\8HWQ!>:e:AOVAL[j2/Bk/hM@gfQ"NE_aKqbrU,pHli*7i/q'113MM$6'YqjnacJ*,M! @d\PO%N-^kI>=:ECeRP29S5Q)SD.Eh3#Y!5"u#OT'*qg_C!S@0u4kY,=t/>j_LVK#\\t9/DQ4`#q"4 Tj 0.771 0 Td (is) Tj /F4 1 Tf BT Example 2.1.3. Q q 1.387 0 Td (k) Tj 'kqWVG1Qq>,GL-4];P?U_1=1oRL^0;1j'(ie]`RGi&kK-9_c*%f O0670+;rE'c!1G%nD"?->%n5nsE8QI=8^Oj@,6AC>X=ML%8Q$?coPF_+/cl1lddnn['%t.i)ZoPVD3Ee8WtEmj) Q q oa9/7ls@;*Es;W;8F3OuPbdec5mg.Y\-:9fXYT%B)rMk5Q=H2!$lo0EC>M.hZk2'+NLsbQAm.Th\@> 11.96 0 0 11.96 0 0 Tm 28.103 16.119 Td (q) Tj 1.033 0 Td (represen) Tj 2.578 0 Td (are) Tj /CapHeight 0 /FontBBox [ -39 -250 1036 750 ] /BaseEncoding /MacRomanEncoding BT p@n:kpib6utY@QpN@&jC-&V[T*lVp#i[+ChPD"LEV)TPJ6nMN' W#"SCjS[P6?SYsa6p$? (p_q)is a tautology. >> << ,OB$=SJXDs4#qS!PZ?3jk""?$mUpd:?&PKFsW1n]glE3R4_PF1#1,,/X3ltECH2q^9E;)139fuI0-L In other words, compound propositions are those propositions that contain some connective. Q q /F4 1 Tf /F4 1 Tf !X1/=5(knM6kX#G#FL`eq5SZNK 11.96 0 0 11.96 0 0 Tm 0.747 0 Td (!) gE#!i-79a0Q`V+>O#4V!e`LEJQVgq5h5IdOsEDk);UVPB-8*XMp^1K2>!r:iqA5f'Mfu Q q 3.766 0 Td (then) Tj 0.779 0 Td (is) Tj 3/S[[gr3fr`!ja4J!9MX,.Mu@g[^A5e2tLU.R*-VLh1,E<0Y1N&Q2R_uILS;GTu,d( _>UMJF5*#k4+Ef5^K7d]$b*n7=DHXMXEeVf^4r`q_<>gf>+c[d%m*_-6&3RE>:RF=HBtWCFn*RmBI8 Then the converse of Q is "If it is not a square then it is not a closed figure with four sides". 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Q q /F4 1 Tf 11.96 0 0 11.96 0 0 Tm '?%(gr;2`:;`C:Ol,uL!AV@8u7^nAbbV.WoOC*f! -27.644 -1.166 Td (and) Tj ET It is neither true nor false. ,WnXaMq#cBR[0pfrICr(d#4/r#_INfU6dCoP`8t:jsR8-LE(AT*mSo"pe2TL:(m*gp]PP:9kbN:)UDIlo4I:Z;b+N-_NW'h[c7S3tA,YfJ\L[RFGS/smUo`HDdFPT`*Na4E$aH`"fT\sV)jm_94TH!NVd?HJm9IiZo897NGiB'i/PHa5nThSA9SqndGJ-O57,_A";m,+>WH:I.mDklm8N7]SUE*U Q q @beq06Cl=Eb/j$Eb-A4F`[t$F`8H\1*A5hEb/`lA7]h#E 00!7F%MqrjH/V5sgZ>YlUK1Fg1T+d_r4'B,o,8:7^;48ELN BU.L_1f+gqk5IU8X1RB6*Ee`D(H2dAPArm)GgO! endobj >> PkPe*!O;ZGZE@,m3ZLHhSLO9O;8iCscRKU&m,l8SY+^Ns<8.FKK%p7q]FH?M-!jgOL(VEScTgd>?6Rp$7.,dk>j /F4 1 Tf >> rm#S3lAuqRaH`ZW=S:sTrV'J`$I78=3,;BQhEBr:\RTllqpoT:>])SDmJ+glW1aa5FHCs[9U91/:S6 Q q 'E^aNd6Tj)h)0qJ!hK/$o$YHV"*'Z\MGmH^p#$83/`=kaau$aYl[=sgf*ldl=J-"[rFUqJWq$9T^WF?/1L?L We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. 22 0 obj A A: b. (lJR2X&q=Z/"&f9&\+)=kI7E0gX3STi9$JJ;'*Nsj:>h=_ /Type /FontDescriptor TD5ch.@@#,hBhZf8/+h#00!EY$kE@Qo4P5oI9h8(.%0Wk+?*B:PaH!TLHe-rCo.XsanVQEI3H_5Del-"jX2cfBl@tA.q1>8g?m+\-U#t. The compound propositions p ~ q and p ^A -q have the same truth value when (a) P is true and q is true P is true and q is false: (c) p is false and q is true. 9Di60fD!603)t4Bl%^*+E2IF$=n9u+>Gc7+>6Q)BcqA;F:)Q$E$-ko0H`#FF(lb9F`[t$F`8HX2`)X ;rCjmDh":idJTsb1tppft*3LR[-mnODZD&E-dA5CCI$&22LofWKHQKSVNTgT 20.018 40.856 Td (Logic) Tj /Length 43 0 R >E41k`"L_]\Rj,ArM2IN/8_b?Qr!j*hmui1rIu:C8)8T+brrYE'0t=0pk!Dc6* E9uB]?2&X.IY\PMZ%0[Sc'A^,6TeJ@S#q^$uDXT,I:7r?+_`J,i%A,"NqpFi BT 5&x_1+ &3 &x_2 + & &x_3+ &2 &x_4 && =-&1 a)aF`MM6DKKK)Ch4`#AU%cr$@jsYJu^mufeD>uV[X)+0cb7qqCN@A=>;f0AsHAVHKd8G2p0-O(^A? Below is the truth table for "and," otherwise known as a conjunction. 12.091 48.626 Td (Note) Tj 9Di62)l^3@rH1%DBNn@F:)Q$E$.%r+>6#'E-670A9Di62Du[266L5iF:)Q$E$.%t+>6))E-670A9Di %2pF.@YrCLLki+*u3J:3?HJ!Tn`mhB)YgU! 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WN#SP*60]Fr'9@Bepe?`*RQ!D[upTaJAcGq$WU!9;+$5f`/TWShdrQAPP@(5N/N\C4>hW'1RAb5I>+5m=Ou"LPq> @>d@Lac\II$0s,n_8('R[/C@MX$AcIPiVU]G6]J/. /Length 44 0 R Q q gt;sqiD@uYpXiZ3q,1bSkVFOVQ@&^B"[PQZskHjT1d(Gmp8g?#L"!DX\fn6]sNe@7#_1S]);\%@Ghe A proposition that is neither a tautology nor contradiction is called a contingency. 0.713 0 Td (_) Tj 11.96 0 0 11.96 0 0 Tm puO&][XPK7A.mB,>u'X=Y;@1A^MDI;VAn8]Yb*F?enqu'KKT?QNg@-&2_kgYDWf(>dE]h)3@b@Aq8LS= _B%K$LDD9&q,;V?X,r0(b6s7?canlKJ8B'BmG&N#26/E*u&h7`n2&*Y/LfChE'!ga_j%sOQ7CO6AB#RlE :'`rh3Ys99%6Yp#PMl@j>L+$dp^H8a7AI]V3S27!+ZF*"]GJ-bVYY=9[%`NA` Here, All these statements are propositions. 1.204 0 Td (except) Tj Q q "+E2IF$=n ?A\$[TJ;S%.mE+O*7`t`0NM:ieo#JPGP2ta2d>0kCA]WrL9WQh[k^S>_b&Am(2cY'%+L+OjG\=Gf'L Then the converse of Q is "If it is a square then it is a closed figure with four sides". 201.6 414.24 m 202.005 414.24 l 202.005 428.19 l 201.6 428.19 l f H[B?BGIpq>^nV&.r:J:_E!V]AckS_0!7'Je. /F1 1 Tf /StemH 20 14.208 37.368 Td (T) Tj ? BT /StemH 20 QkTr,l;u>U!p3!ehN9[V86d&l]i+D%,'0CH[PXk`&G>(To:L3lUrP3[ba`;eYHnI!QTG3h1IJ'6nCH Q q 1.967 0 Td (\\if) Tj /ItalicAngle -14 >GW,+>8$`E-670A9Di60f_-704\d.Eb0#? endobj )`/OGK&$7-ucEb/[$Bl@lW@'A+>Gl:0d&c)B-8r`+>Gc81H@ stream 6^J4%K'nM;bE8ElLptXLU3;c+=-!+Qudqq^%W7S7fE/2T8TPBfa;lU;")IHFKor9E@ST,1Jr#(65). << /F7 1 Tf All rights reserved. 0.921 0 Td (q) Tj /FontDescriptor 24 0 R Tj 22.319 16.119 Td (p) Tj >> .QP4tOdMKN53dH>X693cu'h /Subtype /Type1 /Differences [ 32 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /therefore /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /precedesorcurly /.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 /orunderscore /.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 /.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 /squaredot /squareplus /squaremultiply /square /.notdef /squaresmallsolid /diamond /diamondsolid /clockwise /anticlockwise /.notdef /.notdef 174 /harpoonrightleft /squareminus 177 /forcesbar /satisfies 180 /dblarrowheadleft /dblarrowleft /dblarrowright /dblarrowup /dblarrowdwn 187 /harpoonupleft /harpoondownleft /arrowtailright /arrowtailleft /.notdef /.notdef /shiftleft /shiftright /squiggleright /.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 ] 11.96 0 0 11.96 0 0 Tm ::m>[q[mlKC1PCeC$/F(7;(!qj0R<2"^W* '2bVa&jr Q q 1.277 0 Td (q) Tj In logic, this is also the case, but we can make that clear by displaying the truth value possibilities. d. It is a proposition that requires the use of prepositions to form logic. r%:%PB(uC^!`1K(g2(iY7Q? O$Bi+PZp1-?-be4Yoi2,)Y_D!sr@!;MBNY8! /CapHeight 705 /F1 1 Tf /Widths [ 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 313 438 438 375 375 313 375 313 375 375 563 563 563 563 563 563 563 563 563 313 375 375 375 375 375 375 850 800 813 862 738 707 884 880 419 375 881 676 1067 880 845 769 845 839 625 782 865 850 1162 375 375 375 375 375 375 375 375 375 547 625 500 625 513 344 563 625 313 344 594 313 938 625 563 625 594 459 444 438 625 594 813 594 594 500 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 676 938 875 787 375 880 813 875 813 875 375 375 375 656 625 375 938 938 375 344 563 563 563 563 375 375 500 574 813 875 375 375 1144 875 313 563 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 375 ] 5Oo>?bT@j&>,^bKSrs'brD4+TW1@#2)6']KJ^;SL\3.%R*V_p$sW26ggK1Ak%"9s*jO\asS#Z=#.33g+69R(Z6JI /F1 1 Tf 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. -16.152 -2.615 Td (Example) Tj )#L0SN@OJ!qGd$2rM(B^18F)f:tb$G5&Wm#=;B"X2#o$$I&[TX`& 1.985 0 Td (ositions) Tj 1.967 0 Td (v) Tj Find and create gamified quizzes, lessons, presentations, and flashcards for students, employees, and everyone else. 11.96 0 0 11.96 0 0 Tm BT So the proposition "not p or (p and q)" is only false if p is true and q is false. :a_-A&4,FbA#0+ ?.bo1O-3t\C/tQ%gB 213.12 277.92 m 213.525 277.92 l 213.525 291.87 l 213.12 291.87 l f 11.96 0 0 11.96 0 0 Tm 21.551 47.038 Td (\\if) Tj /Type /Encoding 0.713 0 Td (_) Tj =V^+W.kjM6SrJW@rMlUsM0Mpdt3nhXZq[S0$mh /ItalicAngle -14 DJjim%.#_N0Ir!LZY9uce-m"0IZHaqPcb? P4iT*! :'`rh3Ys99%6Yp#PMl@j>L+$dp^H8a7AI]V3S27!+ZF*"]GJ-bVYY=9[%`NA` ]?%$6QE#plt""/p8DV$^II1kt7bK(*i)CiZ@?EFGlD A contradiction is a proposition that is always false. Q q 11.96 0 0 11.96 0 0 Tm 24.470 42.773 Td (e) Tj BT /FirstChar 0 /BaseFont /CMR9 So there! Q1: Which of the following is a compound proposition? /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 ] 0.602 0 Td (^) Tj 1.146 0 Td (sa) Tj /F4 1 Tf /StemH 20 endstream :apB8o?=EE:sjnf@^NNN(k)>2lA-E-?MiDHu?H" ET @:t^Q"3Sa"=@nSE(H_6cKKl[Fe0lIfLi-t0a, Q q /F1 1 Tf \N(D_Hh3>G.&sR ET hZ8(rqgRbCu?h+"!rIRbSm^C0S"([C&UtHJRi%kQg[P/UAN/Ic$]?>S#OX&ZIhu"^4!L6/NnrMqM \N(D_Hh3>G.&sR Consider the statement "If a closed figure has four sides, then it is a square." ?Tl%E")uOUmHu!.%tR@M@Wep0C!m.CLgjA6Wu.g&Dk&K5ohsKU>e0A9 ET 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. 1.716 0 Td (bi-) Tj $$. 39 0 obj +E2IF$=n9u+?2/"02j!4F`[t$F`8H_1a"GN+E2IF$=n9u+?25$03'-6F`[t$F`8H_2]sbT+E2IF$=n ?5[kb64NgcX%ddp>Gr?$T'0PhuuDK`pSA/'"AjHN'^9a/aYQ6oh'\p(@^Y?RL5AmqopX+MdW:./?o_ 0.862 0 Td (op) Tj 24.255 50.870 Td (F) Tj 2.229 0 Td (alen) Tj a) Either you leave or I do. @Hq^jkiB2keSp7O5cf+iqeu+q0bZ./NI*GcQV(blnpW=M\$G>[jP$>3kj3UH]5(8!ppWHcZZ8`l6tf 2.233 0 Td (John) Tj F`[t$F`8H\0d&,kBQ@Zq+E2IF$=n9u+>kns04o-FEZf=DF:)Q$E$. ET [01BbgDe*R"B-8re2BZFKEaa5#0H`(m1,Up3HSHXWDId='+D#e-BHSr\DJsW- MM7jpA:*,?bai[84\]APZSAY4*^]BKPR3B/a)h)!M+^3q^)e]Ha3sM.IX_Nu2,Fru0jU=+M/m!,^,5nqa.up6bf,cl 23.955 33.770 Td (T) Tj Was he right? '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 c$Fld'\pQ"X\Lm(8qg)G\e#V;Orh;9S8@l_eJ\Fo!.8hBW9n^D\HaGntG'J0UJl^b-XK 273.12 442.56 m 273.525 442.56 l 273.525 456.51 l 273.12 456.51 l f S(#]CO?ia&Yh^frG)/Gqs#4kN:8)a#BAYMLS_*5.3j"meGT!Z40!o+ E9uB]?2&X.IY\PMZ%0[Sc'A^,6TeJ@S#q^$uDXT,I:7r?+_`J,i%A,"NqpFi 2.464 0 Td (is) Tj 3.522 0 Td (for) Tj /F5 1 Tf B5(jaCi=>:E-670A9Di63B/-7Ao)C(Ebo7>CDf9Z:BkM+$+E2IF$=n9u+>G Q q For example, the proposition ((p q) q) p is a tautology. *&"@:X:oCj@.6AS)9&7W30d6:,U8HS%!5+>#T30H`))0fC.52`6#(ART+fDJXS@A7]? "@jMuM`@Rr]qC9,Lc*rKI5*8bqXMf$.jkB>JB0jubNK*V? /F1 1 Tf 385.44 633.12 m 385.845 633.12 l 385.845 647.07 l 385.44 647.07 l f @VFYq#Wl=gH]8 4.149 0 Td (sa) Tj 11.96 0 0 11.96 0 0 Tm 1.046 0 Td (a) Tj In propositional logic a statement (or proposition) is represented by a symbol (or letter) whose relationship with other statements is defined via a set of symbols (or connectives).The statement is described by its truth value which is either true or false.. Propositions \color{#D61F06} \textbf{Propositions} Propositions. /F4 1 Tf Tj /F4 1 Tf >MBOu9o+E2IF$=n9u+>Gc2+>6DsD.7$a+E2IF$=n9u+>Gc4+>6Q*+E2IF$=n9u+>Gc5+>6Z-B5M'"E nn+AN=kc:W-\k3RoOuZ*I(sb]/FO /BaseFont /CMR12 Match the fully parenthesized expression on the left to that on the right. -c*De"tE/$X>bAt_j]52'EmJV[+-`g!T\PIjS+g4&$t!^6:m;^]!3Gshe=Q3`:p)Otq3Yq@`7Xk\=N[s1.oRFh"3e"T[8L_$,P:X[-l:Un?Ht72]GHGho^lMCXU [fj%858sr/XpLebEAJn\J5+e0gdfY2c`K@\sBA-XB$0,:#J$F33hrp>VkQ,UFIO4okUf,mYk E^jI'>i)XnMfK2B'g"(=\! /F3 1 Tf -17.795 -1.782 Td (p) Tj 11.96 0 0 11.96 0 0 Tm 1.735 0 Td (con) Tj DH.sZ]!,?f20/=FOIhXL%r5)7C"Bi\(I8K+6GkV\4`1V@M[JNCJmhuCqTN8*JDeW+652u? >> 1.277 0 Td (q) Tj a5Z?".T6FWu2XXhf#NLjNHA[`YBEJ=E\QL+4(/IZj^"NWQ@0[4a6. 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HcgL^LhRJDnTLG4+FdWM^GZt5EWm$71TY@H``\>r/h2t#O=+;7#D,n%":,P /BaseEncoding /MacRomanEncoding /F3 1 Tf Tj ]5H W.<7)[BEScLRM0K)"[,!2OWh1t'.O/GNM8Jpo"[DYRop)WppZMrZROC"ZcX!=&Hg`>Mn-Yd((L;7Wf k!>8,KN?ZPLoRhDASAMLF+Bh/0/m;p>s,$_Sp0BZjBP*!$g/#NZ=fpa+s8WmFIUNiZ)dO*6O`4K/kg3BnO40egesfBkn>QCbWU^ /Type /Encoding 2.315 0 Td (Illinois) Tj ?Tl%E")uOUmHu!.%tR@M@Wep0C!m.CLgjA6Wu.g&Dk&K5ohsKU>e0A9 -&x_1+ &3 &x_2 + &4 &x_3+ &3 &x_4 &&= &4\\ [V)]4de8k:f&bE>atN5=f3]X/@P8=?fij$NcC9?bc:Hb92\K%2l\68=JB*0@"UFZ,lmPH^ndg5B$NK5#%`7G>Cd+X6=L_$k?ShpW9 11.96 0 0 11.96 0 0 Tm Q q ZNb30:qBif^J>*8]6.Z\IcaY&.if*cid"R&pKk40J%q:sNFnX=70ZM[YFdH-HVb-T9XO@c7;otBpm] True. 11.96 0 0 11.96 0 0 Tm endobj +0d&2*0Jah12)'H!ART+fDJXS@A7]?[03)n(EHPhp=qZf'aKc;P? 241.44 647.04 m 241.845 647.04 l 241.845 660.99 l 241.44 660.99 l f Tj /Length2 17614 BT 1.277 0 Td (q) Tj ET ET 395.04 292.32 m 395.445 292.32 l 395.445 306.27 l 395.04 306.27 l f 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. [eN8,.B>\La"KR68f[/J^@V3E1RN;_:_s,8$0cH_Me9Q@GN%0*2MSp+ It is common to use a table to capture the possibilities for truth values of compound statements. >^fU`nI_VB]iO(&-7`_bS%YJLfG9jrl7r17%%!P^o@,ucR,Bb6ONc@Ie1$4+OG',W/#S6Bh'j^CL0! H=K]EOt>HU",L\,C3\fmPlSm6VT?0oV5EXp7:s;pc6@d%Q(#>N*a59uupb1EuhCUYQ`p(l=,_)PLUqe& BT 0.571 0 Td (ositiv) Tj 22.523 20.005 Td (T) Tj Q q 0.489 0 Td (ed) Tj :r Discussion One of the important techniques used in proving theorems is to replace, or sub-stitute, one proposition by another one that is equivalent to it. /FontFile 37 0 R 9.Define a logical operator so that p q is logically equivalent to (p q). endobj /F4 1 Tf "S,W(Koqr,Bi\U8#u#I+sXt^tKR#g BT ]=ptWiN\'5mW(dAE8gN5&T-dh?WZT5(>)ieU@c?_QSTbGHhD*M1THI)+oDT`\TOt/hQehFgl6r5]rj60O\^$>eZ=ImEL8(emJ&ua#`+f /Descent 15 "^8I2pIgV+sbp^o /gDWDqR'K(<7LNlRCYRkTAGQ^Br[n6dQ%i^/qY'#!63PM_ZBh?a1$K8]Dqns7HmYVSH)$07+hKRhI: QhTk&s&^%gJj8Z?fK]8.Y;4c2+JE`jS-s94j7agNh)eiJIB@7Y,rAC@8@O%V(a;WPo4uu.8n%Eol%. :qmg,"/MmG*3/MVSkjPL%>([^2=\t4hItu;fe7H;`mouV3HlT,$4LpTh5]nfhG?t[1\sK%tQ5m1-VA3kd'*qo3g"srPAKP)_4eYs$5t[:B24pQCipMO:Q8WFSOKRqCZYFRL>0%IB!D@ /Differences [ 32 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef 45 /.notdef 47 /.notdef 58 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef 74 /.notdef /.notdef 86 /.notdef 88 /.notdef 91 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef 128 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /Gamma /Delta /Theta /Lambda /.notdef /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef 174 /ff /fi /fl /ffi /ffl 180 /dotlessj /grave /acute /caron /breve 187 /cedilla /germandbls /ae /oe /.notdef /.notdef /OE /Oslash /suppress /dieresis 199 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef 216 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef 241 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] >> F)OLdBD(o\VGghMnL%eO0Y:K.LB998FP.+bfj?Cj-]O0!VS#CCg,2kF&!q17i"+\`i=3P;udM7t75? [MAZtqbsoDd-NA:LJ,K%7:i*%/mWPQ\W;78[;JPK#7ur@P-A.-]aF%k,VE#_>:Dp7?FT#J@:fPmPB#F_"7PsGU1qARk"37LpF*6orf7K4 >> ZbAt_j]52'EmJV[+-`g!T\PIjS+g4&$t!^6:m;^]!3Gshe=Q3`:p)Otq3Yq@`7Xk\=N[s1.oRFh"3e"T[8L_$,P:X[-l:Un?Ht72]GHGho^lMCXU 'fp5E5,`pBC'G)n&mWH\?mg(3?m-0:%KtE=FhI"XAK'mEib@qO=+9HI9;u"P7Cjl@6A2s 0.950 0 Td (osition) Tj Q q 1.387 0 Td (k) Tj Q q /F7 1 Tf )JIF`[t$F`8HX0Jst+BcqA;F:)Q$E$-kh2BXYf+E2IF$=n9u+>GQ/+>7LQE-670A9Di60ebR005MbM "EW4CA(IqpG\YA43^,GuF@-MH(Bm3M 0.713 0 Td (_) Tj 11.96 0 0 11.96 0 0 Tm The dual of a compound proposition that contains only the logical operators , , and is the compound proposition obtained by replacing each by , each by , each T by F, and each F by T. The dual of s is denoted by s. 1.899 0 Td (p) Tj ET @rj[1PCe3ng,A2%DIQ&J!QG_.R^e+RC(*U#4RSTh#29o%22*jnGg*drpnK.:hbhX.DgIN*@.N9qQT? M3)! 0.816 0 Td (sp) Tj Q q @UiFfA6NiVgUMB$4:VodNMslclG8^qagN30.oZq_Ui]ggbM(7,UOetb(QDIlVUBWT3>Dh=E5BB9jp8 1.631 0 Td (true:) Tj To the English words "and", "or" and "not". 4.241 0 Td (ma) Tj endstream 2.519 0 Td (no) Tj ^[LU`CL5DO\STO:e;8%)bkHrrXZR07-S-\KCZ@U1C2rV+07kZ/EKs4k/(:+A_l2,SP, 11.96 0 0 11.96 0 0 Tm W.U,,qsC=]Hqf:I!DJli2^gZ=^f(i2?CE4Q_P&g5JGfVE>*V4paXkUep\AoQNo+c;2%_QI\r+Rr]@T QBrJ%62*rAH7O]i$JiN,l`%)eS1]\.1O%+*iTiX9R$Ha[K:;pe7maDGRG;N2hbA!VQdd0YF@l@g[5?. (Here the connector "and" was used to create a new statement). 1.327 0 Td (q) Tj /CapHeight 705 It is important to notice that, if the first proposition is false, the conditional statement is true by default. -0.728 -1.575 Td (p) Tj /F1 1 Tf 3.082 0 Td (is) Tj /F1 1 Tf Q. )`/O..3NYB@:X:oCj@.6AS)9&:2b5g@q?c26Z6phEbT0"Fo(@:X:oCj@.6AS)9&7W30d:18!N+>6)V;cth/+Co%q$854 Find parametrizations to model the motion of a particle that starts at (-a, 0) and traces the ellipse (x/a) + (y/b) = 1, a > 0, b > 0, twice clockwise. stream %f:+f`1O`&nq-i 153XfsIrF\`-;enqk\X[-RDS,7'BLtLUY2!%b)a8TGg=TV*C^lGAI^JV'"MgFC76Kl5a)>B43eUTi+ kA#17i__?>7MW!pd\t/pTd@-q2X]X7lX7`"r(4:0c,VJKC^.mi!K`fSBSots3_qO656&u)J@re6[9^ Typically precedes the antecedent of a conditional statement Rr ] qC9, Lc * rKI5 8bqXMf... 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