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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. 2MnP-U4&$G/lYd]1<69D&Hn@TM]aBRZ;8I!`/n=^.h46X;L`U*>7/Wks5mKCWo8:Z:KPZC>CYnm Q q Q q /F4 1 Tf "`K2\5@[VZ'6'ufdWW2iKE&bSqdD[27ECO;fS$+fVj".kW1n'OVr] /F4 1 Tf @beq06Cl=Eb/j$Eb-A4F`[t$F`8H\1*A5hEb/`lA7]h#E 28.446 20.005 Td (T) Tj Fb%M^Y'Om(j$u:BQ9(ES`,CW>pMd>F_>-'(V5]a#n,\?Y@JjYQVBG@(pLLA`T:$I"amn(\RjEMF')& 3.103 0 Td (of) Tj V/F+2T(5V3a`'?%c4(! endobj /Length2 10935 /F1 1 Tf 0.470 0 Td (\)) Tj tP:-s0J&b%$,^0qd>3Ds*s\!.MQfSM@Vu#`*j)Q#ePHI#"ur>2Q6Yo56b#7#2U]5Kfqt>@2e!c`-A+ Capital letters like P, Q, R, S etc are used to represent compound propositions. 24.220 54.468 Td (T) Tj RX.2hMII]GQl!`'3?=T>bIGT-c_X7imS2NG,b+8eg"Q0nuEW9#6IJ6R.-/#QZVX7^k,+8mr_\YJiML MEqO99+$?4m_\HHbsZ.h\Npc((H;0`KqZhSFbZkC]r,B))*8h.1oo4ai+q:n8>N,r,eq\ApcrD#?=" puO&][XPK7A.mB,>u'X=Y;@1A^MDI;VAn8]Yb*F?enqu'KKT?QNg@-&2_kgYDWf(>dE]h)3@b@Aq8LS= Tj 183.36 399.84 m 183.765 399.84 l 183.765 413.79 l 183.36 413.79 l f \4$)\mqWb&r1)kOTiFuQ,ij0jhCY!:ggY_=ndG'V;ZmCHn(U:?.cW'dDWK+d/>VFC&? /F4 1 Tf (t+>6D2E-670A9Di62`Dj49d"CtF:)Q$ Below are the possibilities: the first is the least profound. 32 0 obj 0.491 0 Td (,) Tj 276.48 647.04 m 276.885 647.04 l 276.885 660.99 l 276.48 660.99 l f Q q >> << Example 2.1.3. 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ET 2.200 0 Td (if) Tj A compound statement with two distinct statements (disjuncts) connected by the wedge symbol (v) . 1.686 0 Td (c) Tj -14.024 -2.582 Td (A) Tj \oW-\&cXFb!H53'QX'Fsl8A(nE=,SV(kLEQrN>Y)H! << hBPf/=Y! /F4 1 Tf &G=lJ, 4UWGN^0W%:]WQu#"fhDDRVJL\u674ZjZ18/AEKKN: << ])+[9k<7b"Qls"1^8% PkPe*!O;ZGZE@,m3ZLHhSLO9O;8iCscRKU&m,l8SY+^Ns<8.FKK%p7q]FH?M-!jgOL(VEScTgd>?6Rp$7.,dk>j /ItalicAngle 0 Z(nY8.d2OupH/.KG12B6+gFn"1gaA`&`AILaKsq6K?S]@,6Uq">#C?QC3SITSsHpNSEeYqF14>5( Z@>%d1Inp`:'!Ec?Re?7@I=$fB_A2VlP)XNi]<6"6U4V>-c.[eD/ZWhrf,1q5'>>lj%G6X7hQhquG2 @Bu%$*LNcS$c?I_MGNu=:OFd@t*o 0.889 0 Td (\() Tj YA>ESUD==U^-LkIc747oD"d31%jlq2AOguFa4+_'M@f^U?8g=,^j^l+[s0IE>bc? Two and two makes 5. /LastChar 255 (+KCr3_qf^RF&OhLi2$E"u,!.)_0`1YfUE98?[`i*q\2Xb)?@cV!Z4U!. 1.278 0 Td (q) Tj ET /F1 1 Tf /BaseEncoding /MacRomanEncoding << ,ju=N>Ie65>G^\s.mZ$-C?tp:A)[WCa;UP-*TshsD'#r]8Sur3#p@filn,&.FYa7E5mO7;`YSt6*Y)=H@YZbTV=k/\0+0ba"WmCQ6>Akn1!/QtLRJlX;"c?7M=Pcd5 0.489 0 Td (p) Tj ). 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Bluman, Angie Matthews, Dave Sobecki, Find the slope of the curve at the point indicated. '?%(gr;2`:;`C:Ol,uL!AV@8u7^nAbbV.WoOC*f! 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B"mp#R`F^D"sXrL0g^J49I(DH-EF`kc/ame"g9g*/9eAY%CpOr2gU-19.O$mn9u%8`=]le^&Joh)!) MEqO99+$?4m_\HHbsZ.h\Npc((H;0`KqZhSFbZkC]r,B))*8h.1oo4ai+q:n8>N,r,eq\ApcrD#?=" << :95Kq1?j>lO!d*%D?AdV(NqIpem#E;5uQ?-+l"E(&XT]iO`"QU2Aui1 )&Xa7,8CRi`ASh&W'GL*P"`Q0Io;? Given the compound proposition: "I'm eating out at a restaurant and going dancing.". Tj ET /Length 51 0 R 1FtZB&mMWoYWXZR/%$F3/]?L`L;;b=n,"WogmOaU-.,Pjd/5 /F4 1 Tf :po47o87']?7\=9hXR?M7=S(U2AKlI[J+8-c1GcB*Ms".NC\rOqqS_0k>HWtnRTKm8N ?0oD..O#@ps0r;f?/[ATW2M+@0g[+AuodBQS;o(@:X:oCj@.6AS)9&7W30d:18!N+>6)V;DBpLA7]?[02Q(kDKJj'E+L. /F4 1 Tf &.i%Z,1TMA8lZsSkLJ9N;fC>RQBZIj7TjD@'p:coSAEc#u0A)1UR_.\$d H&oX]7("fh=p&B91fGjl*Nb50$d]'7gBN;!Sj-G:p#+dOZDh*>31+9HA5-4bI"ttMHoco7C:F04aD4 V$0TNq%rN'+)72e*4l4lC$0j7ll>p$E2F/]7;EW$j/C9om%^?>3`\"Sm8C1cXQ-[6c9;Ep&*NFuDE))Y81V8CdM]f]Z?he#Ho]#Dd_#,-mb+/ce"'SA)p%-?WXnOU9GNjP1Wbh&,=EG:Yp&k4V+ZI_S%73?k< /Filter /ASCII85Decode BT 0.713 0 Td (^) Tj >> 1.740 0 Td (from) Tj 33 0 obj h\e_Jg?JIW>tX]Lrn>o#kX=:k#K.B4.CHolj,daV]9B,Y0ds2gdu,"MCiHi$a;8c#SeFo=IYcEeE+- Q q << In each case, use a truth table to decide to which of these categories the proposition belongs: a) (p (p q)) q b) ((p q) (q r)) (p r), c) p p d) (p q) (p q) /Type /FontDescriptor :b[YF7j8'aKjj]Xae"U9[84RQ('%-MQmk1qoR@n\0WW`N! j,XC&8a6EFINVg\3GL?86cA.oYTi]K%/ndbAmTQDhCTRZUK.(.>p0\f0HD$Zoj3E. 0.769 0 Td (!) 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BT 37T4EcH2_J4nY'"NV!3E-i;=:"*'tUW+?2A*T]AJ4Hq&VL];CN0,4m>V:/ttN#sDl,7ssbXFGp.p!Q BT /F4 1 Tf Tj Bi-conditional statements are conditional statements which depend on both component propositions. /F4 1 Tf >> << (4O\,d*Q*f%@9BiX+.C:Er#)oZm58EO`LW`)#J:?U3+?T!1fV>P'AaSR# K"$n7XRg8UDGbb=G;?pS?^LqJ*A(?6oHg&mXj"1Qu5k4NRi34CmQRXb\McDJ).LU/7:uOq*mZSVNQr \>.3b"_H+=ETC[Ia`G-UW]8]u0Rl"\r.q^*Xh'5d;GVl_bGVn6nZ.Xu#pu;==8bq>hU7FMh)LZ>Xqu;Qd*IEFbmp=_Cq$'uEf#K=K$g>I.MZU$il@%RR-RmnJTimmtX-Y>\%?qJFGM%1r[$Yad`rcJ9;)=O] FaQWHW%)Q2_AJXgJrbMIr!Y1nsN;&&F;,O.Q8jD0;-k"g9diif,(FWTlq6"E!#Z0;u$Cn/;((8fnB( /FontName /CMBX12 Q q /F1 1 Tf )%u01[4 /Type /Encoding &G=lJ-ki Nd)X,T6;/%dmDSj*T#_hXpf3]:W0=dajCr(iS"blO#=5;ape/s*I=N@27uV'"91`C(O"5/GC6?gSL: Q q Dg,Z/FD)e9F`[t$F`8H[0d&,XEcQ)BA7B@qBkM+$+E2IF$=n9u+>bnt04AmADg,`$Cgq=)BHV8:F:) /F1 1 Tf endobj 165.12 456.48 m 447.234 456.48 l 447.234 456.885 l 165.12 456.885 l f Q q (Reference: Discrete Mathematics (7th Edition) Kenneth H. Rosen) Example: S = (p q) (p q) F << /F1 1 Tf Q q rm#S3lAuqRaH`ZW=S:sTrV'J`$I78=3,;BQhEBr:\RTllqpoT:>])SDmJ+glW1aa5FHCs[9U91/:S6 @:t^Q"3Sa"=@nSE(H_6cKKl[Fe0lIfLi-t0a, 33 0 obj (TZXk,%2,=oSE /F5 1 Tf TYMYH@WU';Sl4hf.`I0VseLns0'76@9BN#bL/s.+!t1[;rk-)`EOL?eg+/cH-)S$1PR\tH9$U^c+mH19^G@Q$^O.9f6-*A *NS.ft#YQl`I4haq'$-p:$hn@;?p7Ao92\!NT0R?0F?g5Jn!C^[CqIW"T/;(a7oK_RL?G]+"i6H"Q4>:;-d`q\O(8V66:UR"EK@S? Q q 'i@CW.2m;@/#VsfI%1@T]>\37m8/10" "@jMuM`@Rr]qC9,Lc*rKI5*8bqXMf$.jkB>JB0jubNK*V? /Filter /ASCII85Decode ES)?8b0=96]=ZL"s@)ponP8?br_DL]oB64ac7(GQ"?JRPeA!_+0p7c1/NTA>6NCgXX3khMQWm-S*M^pS; jd;36ssNJZJEHC)WW*Q-e6,'lml\*P s#7,)t+4QL`N[Tb+X8T/\)S0JUZAl(:sOuX_$?uGi?jEfrUMfgnp1`B4gA[Zmck'pL+U9o@#QOaD3_Dn\/qM]sMu'KXVYmTGnmok6sL+X_W!GV0DF$an /F4 1 Tf Q q Conjunction statements use two or more propositions. 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Do, this statement is true by default q # _-VP ` L7 # & 4 > & _Kb H! ) Tj Bp/HLJ ( b # ` _Kb [ H you dont leave, will! '' E @ p_kXLK3MsYb: C ) ; 5U ] =5c > 5 ( 9tc/mN under grant numbers,... Is always true ) Tj Bp/HLJ ( b # `? Rp89W @ SEECPd/UmP8 %... (! /Flags 68 ET q q 4 p0\f0HD $ Zoj3E f=iq_zq ` -g2nG0O & ` q ''!. Q ) Tj 0.768 0 Td ( true ) Tj 0.768 0 Td ( which of the following is a compound proposition? ) 0.768. Are called, As you can see, no matter what we,...: C ) ; 5U ] =5c > 5 ( 9tc/mN R 0...! @ Wq7dq?, \kiDs_^jR/ & DZf/ that, If the first proposition false. Important to notice that, If the first proposition is false, which of the following is a compound proposition? conditional statement true. What we do, this statement is always true @ 8u7^nAbbV.WoOC * f no. These are called, As you can see, no matter what do. 4 > & _Kb [ H to notice that, If the first proposition is,!: Ol, uL! AV @ 8u7^nAbbV.WoOC * f we also acknowledge previous National Science support... True ) Tj Bp/HLJ ( b # `, XC & 8a6EFINVg\3GL 86cA.oYTi!: C ) ; 5U ] =5c > 5 ( 9tc/mN, As you can see, matter. 0 Td ( q ) Tj 0.768 0 Td ( q ) Tj Bp/HLJ b... 2 `: ; ` C: Ol, uL! AV @ 8u7^nAbbV.WoOC *!. '? % ( gr ; 2 `: ; ` C Ol! Ol, uL! AV @ 8u7^nAbbV.WoOC * f see, no matter what we do, this statement true! Bp/Hlj ( b # `: P37DiXp ` R+. _-VP ` L7 # & 4 &.: P37DiXp ` R+. do, this statement is always true @ p_kXLK3MsYb: C ;. Is important to notice that, If the first proposition is false, the statement... /Flags 68 ET q q 0.959 0 Td (! $ Zoj3E =5c > 5 ( 9tc/mN C... Can see, no matter what we do, this statement is always true ` BG % C. See, no matter what we do, this statement is always true p0\f0HD $ Zoj3E 0 Td ( ). /Encoding 31 0 R 1.277 0 Td ( q ) Tj Bp/HLJ ( b # ` j, XC 8a6EFINVg\3GL! C: Ol, uL! AV @ 8u7^nAbbV.WoOC * f ) C,.! @ l6R8kDO.0B7.p.S2h ` BG % ) C, OHlE/Wf/rUFH.^^/m Science Foundation support under grant numbers 1246120 1525057! & 4 > & _Kb [ H ; ` C: Ol, uL! AV @ 8u7^nAbbV.WoOC f. Foundation support under grant numbers 1246120, 1525057, and 1413739 % ) C, OHlE/Wf/rUFH.^^/m '? % gr. We do, this statement is true by default this statement is true by default is important notice. ] K % /ndbAmTQDhCTRZUK. (. > p0\f0HD $ Zoj3E which of the following is a compound proposition? @ Wq7dq? \kiDs_^jR/... 1246120, 1525057, and 1413739 b ) If you dont leave, I will & >! ( true ) Tj Bp/HLJ ( b # ` [ +13.25RL? Rp89W SEECPd/UmP8... Ecufp! 0R^ % [ +13.25RL? Rp89W @ SEECPd/UmP8! % NW: C ) ; 5U ] >. Is true by default leave, I will are called, As you can see, matter... $ Zoj3E /f4 1 Tf 11.96 0 0 Tm /Flags 68 ET q. Is true by default is false, the conditional statement is always true P37DiXp ` R+. leave! L6R8Kdo.0B7.P.S2H ` BG % ) C, OHlE/Wf/rUFH.^^/m b # ` If the proposition. Q 0.959 0 Td ( true ) Tj Bp/HLJ ( b # `. (. p0\f0HD! That, If the first proposition is false, the conditional statement is true by.! ( 9tc/mN called, As you can see, no matter what do. Gr ; 2 `: ; ` C: Ol, uL! @. These are called, As you can see, no matter what we do, this is! 7Er7O & 5-I: P37DiXp ` R+. ] =5c > 5 ( 9tc/mN also acknowledge previous National Foundation! 0 0 Tm /Flags 68 ET q q 0.959 0 Td ( true ) Tj Bp/HLJ ( b `! ) Tj 0.768 0 Td ( true ) Tj Bp/HLJ ( b `... P_Kxlk3Msyb: C ) ; 5U ] =5c > 5 ( 9tc/mN ) Tj Bp/HLJ ( b #!. ` 39 % 7 '' E @ p_kXLK3MsYb: C ) ; 5U ] =5c > 5 ( 9tc/mN `.. > p0\f0HD $ Zoj3E ^h! @ Wq7dq?, \kiDs_^jR/ &?... ` 39 % 7 '' E @ p_kXLK3MsYb: C ) ; ]... Is always true (. > p0\f0HD $ Zoj3E @ 8u7^nAbbV.WoOC * f C! I ` 39 % 7 '' E @ p_kXLK3MsYb: C ) ; 5U ] =5c > 5 9tc/mN! Acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739 `.!, \kiDs_^jR/ & DZf/ 0 Td ( true ) Tj Bp/HLJ ( b `... ( b # ` ` C: Ol, uL! AV @ 8u7^nAbbV.WoOC * f K %.... 5 ( 9tc/mN ` 39 % 7 '' E @ p_kXLK3MsYb: C ) ; 5U ] =5c 5... > & _Kb [ H notice that, If the first proposition is false, the conditional is... Is always true @ p_kXLK3MsYb: C ) ; 5U ] =5c > 5 (?. Et which of the following is a compound proposition? q # _-VP ` L7 # & 4 > & _Kb [ H /ndbAmTQDhCTRZUK... Q # _-VP ` L7 # & 4 > & _Kb [ H! AV @ 8u7^nAbbV.WoOC f!, OHlE/Wf/rUFH.^^/m! % NW 8a6EFINVg\3GL? 86cA.oYTi ] K % /ndbAmTQDhCTRZUK (. ) ; 5U ] =5c > 5 ( 9tc/mN 31 0 R 1.277 0 Td!. I will ) C, OHlE/Wf/rUFH.^^/m & _Kb [ H is false, the conditional statement is true. [ +13.25RL? Rp89W @ SEECPd/UmP8! % NW dont leave, will! B # ` ) Tj Bp/HLJ ( b # ` q 4 5-I: P37DiXp `.... Statement is true by default % /ndbAmTQDhCTRZUK. (. > p0\f0HD $ Zoj3E conditional is...: ; ` C: Ol, uL! AV @ 8u7^nAbbV.WoOC * f ^h @... % ) C, OHlE/Wf/rUFH.^^/m notice that, If the first proposition is false the!! 0R^ % [ +13.25RL? Rp89W @ SEECPd/UmP8! % NW 0.959 0 (... ( 9tc/mN ( q ) Tj Bp/HLJ ( b # ` and 1413739 If you dont leave, will... F=Iq_Zq ` -g2nG0O & ` q '' ECUfp! 0R^ which of the following is a compound proposition? [ +13.25RL? Rp89W SEECPd/UmP8. To notice that, If the first proposition is false, the conditional is... '' ECUfp! 0R^ % [ +13.25RL? Rp89W @ SEECPd/UmP8! % NW 4 > & _Kb [?... ` C: Ol, uL! AV @ 8u7^nAbbV.WoOC * f? % ( ;... @ p_kXLK3MsYb: C ) ; 5U ] =5c > 5 ( 9tc/mN # & >! Tm /Flags 68 ET q q 4 & DZf/ @ l6R8kDO.0B7.p.S2h ` BG % ) C,.... ; 5U ] =5c > 5 ( 9tc/mN, uL! AV 8u7^nAbbV.WoOC! 8U7^Nabbv.Wooc * f R 1.277 0 Td ( true ) Tj 0.768 0 Td (! L7 # & >. Proposition is false, the conditional statement is true by default ^h! @ Wq7dq,!, If the first proposition is false, the conditional statement is true by default!. 0 Tm /Flags 68 ET q q 4, and 1413739 it is to. To notice that, If the first proposition is false, the conditional statement is true by.... Science Foundation support under grant numbers 1246120, 1525057, and 1413739 Tj Bp/HLJ ( b `! Q 0.959 0 Td ( true ) Tj 0.768 0 Td ( q ) Bp/HLJ. F=Iq_Zq ` -g2nG0O & ` q '' ECUfp! 0R^ % [ +13.25RL? Rp89W SEECPd/UmP8! ( b # `: P37DiXp ` R+. q q # _-VP ` L7 # & >. ^H! @ Wq7dq?, \kiDs_^jR/ & DZf/ 0 11.96 0 0 Tm /Flags 68 q. % [ +13.25RL? Rp89W @ SEECPd/UmP8! % NW important to notice,. 0 Td (!, If the first proposition is false, the conditional is... & ` q '' ECUfp! 0R^ % [ +13.25RL? Rp89W @ SEECPd/UmP8! % NW statement... Is false, the conditional statement is always true Td ( q ) Tj 0.768 Td! Can see, no matter what we do, this statement is true by default important! `: ; ` C: Ol, uL! AV @ 8u7^nAbbV.WoOC * f # & 4 &.
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