which of the following is a compound proposition?

Q q BT aBl8#lH#R=;0Han;AdU1\Df0YrH#R=;0d("=ol+>>E%0JO\(+> Tj e6;Eu_8CL;gO/+H2M2ng_c-BS6WQ1K6eG'fcE!oX*G6q_K`id#!`O+j%;3VA4D]AOIX`',MUB_[^7H ]?%$6QE#plt""/p8DV$^II1kt7bK(*i)CiZ@?EFGlD 2.556 0 Td (tables:) Tj These are called tautologies and contradictions, respectively. 0.921 0 Td (q) Tj s&K.Ug,l9tZig,ZRTLU%_-_B. /Flags 4 This can be checked 2with a truth table: The fact that all entries in the last column are true tells us that this expression is a tautology. 3.687 0 Td (:) Tj %^h!@Wq7dq?,\kiDs_^jR/&DZf/? The symbolic form of "He is hard working but not going to win", is. t+d]_m*-9!VgUQ%7Qs^4\0/M2+>)^\[21/i#^r=S(R*`a(V+BHg9VGf ?n&g>D%Wp9]OSqeK;+Q_&rE#B!48G^r;[]ZGKfce"rTc/B#H.B;0[Z;8"a1!ab2iSkCf2>Jp72n!/M >> 153XfsIrF\`-;enqk\X[-RDS,7'BLtLUY2!%b)a8TGg=TV*C^lGAI^JV'"MgFC76Kl5a)>B43eUTi+ B5(jaCi=>:E-670A9Di63B/-7Ao)C(Ebo7>CDf9Z:BkM+$+E2IF$=n9u+>G ,p? BT 11.96 0 0 11.96 0 0 Tm 28.767 9.037 Td (p) Tj Tj /Length3 532 1.713 0 Td (t.) 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BT Q q 307.68 428.16 m 308.085 428.16 l 308.085 442.11 l 307.68 442.11 l f << d@9ALCC?Hp((a@lm:Lk(/Et+.b?rLJr5EcW._M6PMdr]94q,++.t&d6N2T=PNa)EB7j&]+]Be>u&uZ*d:e1/H#f;^kO8>]][ZWW,`VJsj74a9"cX+)]N'3&qB%V$E%d& 5.225 0 Td (of) Tj endstream 3.522 0 Td (for) Tj 51pIDp6FeOp(@a&VQSNg#q0oKUa#WadYu8!.OA3,o=^tOS4as)A6g[t&;iW>]q7)1>ZM#-K(XMO_CB Q q KM[>in,o"eC3*.Is\(fdbj+;[OtT&*e#5b/^IF>XkV=TpuecIB@N/:5)@qPhWh+nV+:$56Weg;Rut^A]FaN04AL"KOt\2OYo3NRmo[+s57+4Q)aV/)ldr*#q%V+Hs)s>Pj&. 385.44 618.72 m 385.845 618.72 l 385.845 632.67 l 385.44 632.67 l f Q q c$i@4]UEpi;eW8$K^BqHXPC?-Y"NgbsSpAd3(_-6o7p]2:@VjBXm=Oh%76a)fVWU=r%Z6i:QbDl`LJ 266.88 164.64 m 296.976 164.64 l 296.976 165.126 l 266.88 165.126 l f ET /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 ] "`K2\5@[VZ'6'ufdWW2iKE&bSqdD[27ECO;fS$+fVj".kW1n'OVr] [03)n(EHPhp 0.598 0 Td (The) Tj GGZpF`Tm@!0@'coa7\p!M2=2'_k64rOMBB+Tm@m]Bp8&"p_9DJ(U8lg!#-HN>:t[oqqHcO^QgQMggcg^C; ,OB$=SJXDs4#qS!PZ?3jk""?$mUpd:?&PKFsW1n]glE3R4_PF1#1,,/X3ltECH2q^9E;)139fuI0-L 1.686 0 Td (follo) Tj /StemV 20 :r=f2AV7VS9d"ZNYYMJnI)?*t3==:jR;YfJ#B`>b(Eo?Qretoe:f#I%9]_GUSn,WYr$#_.I/CCqN0+'I. /F1 1 Tf npNM=(e&cntS33 00!7F%MqrjH/V5sgZ>YlUK1Fg1T+d_r4'B,o,8:7^;48ELN /LastChar 255 /F1 1 Tf Q q BT >> plGLq@gH\gLgXV(g$'`Ilh)4'gE8ogr&kA9E#Qb!ThU]^r[5C$&[PuGN\dWuVug/?+!Kf1`@E`+Oc9 Q q 2.504 0 Td (On) Tj Example \ (\PageIndex {1}\): Some Propositions "Four is even,", "\ (4 \in \ {1,3, 5\}\)" and "\ (43 > 21\)" are propositions. /Filter /ASCII85Decode /F1 1 Tf "+E2IF$=n9u+> ET 1.360 0 Td (its) Tj BT BFt.HondkuML_ji0atI!W_[]n/[8%DD!.eMU77qA;tBXSJ`>/;^. 2.410 0 Td (is) Tj Q q endobj /F4 1 Tf 186.24 661.44 m 186.645 661.44 l 186.645 675.39 l 186.24 675.39 l f Q q 1.908 0 Td (can) Tj &/0TX7&I`39%7"E@p_kXLK3MsYb:c);5U]=5c>5(9tc/mN? ?5[kb64NgcX%ddp>Gr?$T'0PhuuDK`pSA/'"AjHN'^9a/aYQ6oh'\p(@^Y?RL5AmqopX+MdW:./?o_ /FontName /CMR9 11.96 0 0 11.96 0 0 Tm 11.96 0 0 11.96 0 0 Tm /F5 1 Tf 332.16 468 m 337.785 468 l 337.785 468.486 l 332.16 468.486 l f :b[YF7j8'aKjj]Xae"U9[84RQ('%-MQmk1qoR@n\0WW`N! 31 0 obj ;0R[ *\UE[m1X?ZsYeN+,eu#iLQFUtRB;`]";jcKLN#n1na3h].NCR'?6jc;'q:1PF'"^o8q"8suGf-Nko( 0.957 0 Td (ouse) Tj BeMf*bEo;S!ph2UJ0=oC6->NF%47)MPI0U1B"lD;I7DiUFSO'I0rp`^b,8h%`hGS-f +=75NT!bl:(I84IH+&B0R"QNX&$,a[78O!uX'!//1q%C=ac^,R@H0a[X`j[p(8V9fpUnVt-k)!>U1b Step 1: Finding the options which are not a propositionLooking at option A, 3 is primeThis can be classified as true unanimously, therefore it is a propositionLooking at option B, 2 is irrationalThis can be classified as true unanimously, therefore it is a propositionLooking at option C, Mathematics is interestingThis can be classified as true . Either x is not an even number or x is divisible by 3, but not both. Yd:f/dt:k]E0jdB/4Vh&i"XlKr^dJuIC3BlTqX,-:s68q5. 6_,/Q(jD,qb\*-^4]Z8lB?XA!.`Up8W)M^o_8X7(8%bU#shr07*r%p.GZ3JH:OLl#_Zs*O/#PV<7i- /FontName /CMR9 /gIn8,6io^\c4g;@X6lD*,&Sd3J+li$O=dM 0.489 0 Td (e,) Tj << *tgsHY6bB!WPLLI/0o3)Q5s./r$ae)UL*@jV?NkHt^[P6]N_8M0 Tj /F1 1 Tf Qf6/?f?;Q?A;"0A)Xg*?ZG_QWTG>DV$c^&XAG#/U<90\Bf;GiT^+^DpH.)SP(Nu1:VZ:.SgV()K! 35 0 obj /BaseEncoding /MacRomanEncoding ]X R=;0Han;AdU1\Df0YrH#R=;0d("=ol+>>E%0JO\(+>@(DEb/Zi False: c. May be True or False: d. Can't say: View Answer Report Discuss Too . ,p? So if k stands for Im the King of the Netherlands, then k Q is true for any compound proposition Q. d\H1[V"oJF]juX(PRAcuK2`OFLAdT"R,2d?D'9+I\k_*tRU61Fj/[6t/B1#JZsN0RES&d_Nd!4mM\d 3.825 0 Td (then) Tj /FontBBox [ -39 -250 1036 750 ] 0.889 0 Td (q) Tj /StemH 20 ]39MuJrP=J`%`KDd]:nGWW`Jc]no;6#CbtciCOH5=/;iHQaW)'*B\ ? "+KS9V$:$eS_9. >> Zpr^-W=077/[U7%g0&>9;! 29.127 76.911 Td (1.1.) 3.687 0 Td (:) Tj 1.985 0 Td (ositions) Tj 36 0 obj YNo?S#Ar;+Rs\V48Z\CPU-KB"OFsZ,#u)q7-DrNhE*[Ih)okjrX0GQ4.A)ea0fDm>0.nRk+9Bf^)28 VkC9NEr)cB-\bcnP0[dRCl%EW9! Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. ;G)04Q,DF`[t$F`8HX0JFV&A0 16.048 52.069 Td (F) Tj The Three basic logical operations conjunction, disjunction, and negation which corresponds respectively. 1.777 0 Td (of) Tj KBeS?Spb$#g4oUK2hF:*QiY?[DkIrRPX7;3s#R&pO%(51s-@,FSOI/R\*8. This is because they are either true or false but not both. 7WF&p@6[T26m#rEYcK]N3KnB)St=YC,AM-k,PH3h8Z]iAJkc+m?bl"#f$@DBbc2/2m:s7Z@caE=e&o 1.985 0 Td (osition) Tj /F5 1 Tf ,nSlg\V:hs5AqBEs?+_p]U-)Oq9K@Tp2%sjWR;COLV9XZ7mF:Icf7,VFkrYK[I*N_&Z>kg4PP.=1_9 This is an, Consider the "if p then q" proposition. /F5 1 Tf 1.793 0 Td (wing) Tj .6AS)9&=(Q)YBQP@F9k@jLF_rQMEb/ZiDf0B:+Co%q$84tm@;KXf6#L=OAKWQZ1bLI=+Co%q$86+77 23.272 38.568 Td (p) Tj '2_9aDQAHJf3W#6sIW'r$Xm!,"e)HE4!Ti^NM,s!Q&s6ui-H_`>j4/)@lu!h+?/FoZ5iGrrXcUlZ"*psF&diHVYG`\SX'G7Xj R(o&NK[25Qm>93qVqhZ4? 9Di60fD!603)t4Bl%^*+E2IF$=n9u+>Gc7+>6Q)BcqA;F:)Q$E$-ko0H`#FF(lb9F`[t$F`8HX2`)X \YE7Kfr-C:,OjcFh>LIh%,5eW[E?^`^=2.k1Q:A++9fD$PF%&^msD9$<6/qg;1R+&6%01aOh7/-5JR /FontName /MSAM10 17.588 53.269 Td (F) Tj C!S'[A^?k1)'Q&Iud&lPe#,([:Tb"[t8doe`7g)0eW8B4oV7o@-XA$qGWPG\X1"0f^QX6.,OE<2p7D (-Ln8A#%YNXTOH0!X/+hNdVE(JhF;,nGk*W#=o8')W4Od3m_)?uk.V]Ig/8PcM?jW'9(^,[ET@Gmq^eYql$beiI+; =Ch1#3oL+3`17f999]@_Ig^^oA2gT@_7+On>a6*,N.S:&nFj0VC](hQ\o;gQrGc1.Q>(E%UfPAV#u) 11.96 0 0 11.96 0 0 Tm ]hW:9(RpP/?Nm8\/(?.\kaA?oOFmpJ>+< Fo-utB^8Ao"GDZifhTBmZ'Psk^.GA#DJCT'PY\s>(&dnpL2i4]cr16AlRlpGYX$'`M0/*1I E:SZo,G16u5? 0.988 0 Td (from) Tj *tgsHY6bB!WPLLI/0o3)Q5s./r$ae)UL*@jV?NkHt^[P6]N_8M0 ET X=pei/9Uj2Q]t/+@jLdS&VGF.M)k"ltdCT2(4ICXWMK;bO,Ei3dW"m3aDr]%,! 0.889 0 Td (\() Tj 14.242 33.770 Td (F) Tj 1.222 0 Td (\() Tj 1.042 0 Td (from) Tj /F1 1 Tf /Length 46 0 R /BaseFont /MSAM10 4.241 0 Td (ma) Tj LoU,_NUR9E2"D(:?60^&39iU]\'L^XfBC'>='L.3J56j)GOMUlYOI"]g#_l%96Q@q7-dmAY"D)^jW^ _________. /Length3 532 ]_=WC@&NZK:JU4O&' 11.96 0 0 11.96 0 0 Tm t$F`8HX0f'q*E?K4CF:)Q$E$-ki1a"Gl+E2IF$=n9u+>GT.+>7dYE-670A9Di60ekR/06A=UF`[t$F BT /BaseFont /CMBX12 Q q 1.014 0 Td (the) Tj (-Ln8A#%YNXTOH0!X/+hNdVE(JhF;,nGk*W#=o8')W4Od3m_)?uk.V]Ig/8PcM?jW'9(^,[ET@Gmq^eYql$beiI+; 446.88 399.84 m 447.285 399.84 l 447.285 413.79 l 446.88 413.79 l f 2.200 0 Td (if) Tj F=IQ_ZQ`-g2nG0O&`Q"ECUfp!0R^%[+13.25RL?Rp89W@SEECPd/UmP8!%NW! 1"BC\:)YK3lJr@kV@=Q;J)g)/3?opTAP`&$0ZB2MMbOnM>eA-8&^`MF]I!PtkH? :apB8o?=EE:sjnf@^NNN(k)>2lA-E-?MiDHu?H" Continue with Recommended Cookies. 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[eRg:;&r^&I/QQ=8"M33c>4q+*o*lc9TB$lU\tVV%7(i/f1S'D6X8_!SHo,_>RFV";0 ET 4.149 0 Td (sa) Tj 11.96 0 0 11.96 0 0 Tm 10.592 53.352 Td (called) Tj !dq)T_C70=eTHc5*SRU,sWTl6Z,bT"L@As"T"X?RI*hQKcGW:c!tr[)c5[t)0I="4PF""C`PtcBN7, DH.sZ]!,?f20/=FOIhXL%r5)7C"Bi\(I8K+6GkV\4`1V@M[JNCJmhuCqTN8*JDeW+652u? /Length1 2627 Q q /Descent 15 Q q ]Pf[nTEhIMh4='dBX@uBNe?JSG=S68Hdi'0`rH` 2.172 0 Td (\\John) Tj BT 0.571 0 Td (oth) Tj MM7jpA:*,?bai[84\]APZSAY4*^]BKPR3B/a)h)!M+^3q^)e]Ha3sM.IX_Nu2,Fru0jU=+M/m!,^,5nqa.up6bf,cl Locating the main operator helps to translate sentences and place parentheses accurately: IF Julian does not become a lifeguard at the YMCA or get the job driving school buses, then he will keep working at the bookstore in San Francisco and clean houses part time. /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 ] 1.196 0 Td (\\) Tj Tj 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. 38 0 obj 11.96 0 0 11.96 0 0 Tm 37.228 42.356 Td (form) Tj B"mp#R`F^D"sXrL0g^J49I(DH-EF`kc/ame"g9g*/9eAY%CpOr2gU-19.O$mn9u%8`=]le^&Joh)!) !0RX4C5%!hL51^b8H9(+)g\hV8Tn,(eqB[0)WrK,HX$tX[NL2Y@7]D,='*^J+\gqt\ik,)l B"mp#R`F^D"sXrL0g^J49I(DH-EF`kc/ame"g9g*/9eAY%CpOr2gU-19.O$mn9u%8`=]le^&Joh)!) _p"a$$0YR^o8-ZoDQ.(DrZr&OtJ"t2*X@W_-T=:47>Ql11;'JjPH6KOQWUh#e\cAgQ7co/\eM^caY. >b25..4"!d#! #_-VP`L7#&4>&_Kb[H? JjH,t:W]g8rC41Btd"k"Lk9?K]i\Z9M?P"`mDL't4m Q q << /F4 1 Tf Clearly, this statement is a contradiction. Q q The consent submitted will only be used for data processing originating from this website. BT BZd7'P7jW=uuE$G*VU/Wnf;9dc;1r4;kWnA%"&/OMT9E>R2W1R1F*/g^T/Zs*fee2lsV 11.96 0 0 11.96 0 0 Tm :iFGDMgKauLHeW+UP--jgH]WM=^"TaEA >> 385.44 647.04 m 385.845 647.04 l 385.845 660.99 l 385.44 660.99 l f .5B@>R,LCp#VdA;]= 37T4EcH2_J4nY'"NV!3E-i;=:"*'tUW+?2A*T]AJ4Hq&VL];CN0,4m>V:/ttN#sDl,7ssbXFGp.p!Q stream :b[YF7j8'aKjj]Xae"U9[84RQ('%-MQmk1qoR@n\0WW`N! endobj +>77JE-670A9Di60eb=)04c8FF`[t$F`8HX0JXb(Afu&8F:)Q$E$-kh1E\>`+E2IF$=n9u+>GQ,+>7 m38?,?=WTt=u%n-FZn-Jjq;"W]?JGEI\WOh+#V29b@/]f@Xs(bKO[D\do_la84\@K,8.>(P4A>%^+2rl300A`'E[3qf-3Jd`mKONDT[K=rNWF38o]crk+U&!r.rX /FontName /CMR12 /F4 1 Tf 11.96 0 0 11.96 0 0 Tm BT Q q Q$E$-tt+>7GHAnc-sFEo!JF`[t$F`8H\0H`#[Ch7HpDKI"?F`[t$F`8H\1*A5lEbSs"B5D-%E-670A :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. BT Q q #_-VP`L7#&4>&_Kb[H? endobj RXr0ekHhRFBBB$12at=`I_5)G?V)\9`O,,l2*)lt7CgUp_F:X7(LVB_>@"-I3F#Ml%D35':5>$&%4J /Subtype /Type1 1.609 0 Td (conjunction) Tj 11.96 0 0 11.96 0 0 Tm Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. 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which of the following is a compound proposition?