21
2+2=4 29 < 12

SMtuba1€¦ · .*/(0+01.'/'(+'(2&2 #0+0& 034 "5$ 60 7&'()'+'(.'& #2(#2(+'(2&'.'&1 *+,&+,(-s≡t 82)9'82:+' # '+2(;2&234".$ 60 7&'()'+'(.'& #2(#2(2&'.'&1 *+),:,&+,(-s ≡ t 82)9'82:+

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Page 1: SMtuba1€¦ · .*/(0+01.'/'(+'(2&2 #0+0& 034 "5$ 60 7&'()'+'(.'& #2(#2(+'(2&'.'&1 *+,&+,(-s≡t 82)9'82:+' # '+2(;2&234".$ 60 7&'()'+'(.'& #2(#2(2&'.'&1 *+),:,&+,(-s ≡ t 82)9'82:+

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

Page 2: SMtuba1€¦ · .*/(0+01.'/'(+'(2&2 #0+0& 034 "5$ 60 7&'()'+'(.'& #2(#2(+'(2&'.'&1 *+,&+,(-s≡t 82)9'82:+' # '+2(;2&234".$ 60 7&'()'+'(.'& #2(#2(2&'.'&1 *+),:,&+,(-s ≡ t 82)9'82:+

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p, q, r, . . .J$-$ /&*N(*)(2 7+(*1()(*$+ '3>*#)#, '(>(*)(*$+$ '(:

D = 1 = :!"#$ Y = 0 = ;&'()*6$1J()(*$4)( J76B(*( (>$D2 4&+$: -$* p 7+(*1(6$+$+ '3>*# 3)1&65 <(4& !: 4&+)5.3)1&65 " <(4& # $)( J76B(*$)$*8 "#*&'& 1 $)( 0 /(* '$)( #4&-$)(+ J76B(*$1)(*'$*26&456&) '(>(*)(*$ 43,B#*8 O&)+5D &: 7+(*1(+$+ '3>*# 4& '& 4&+)5. 3)'#>#+# -(A)$*B(+ -$*(* 6$1J('$*8"$%&' (')*+)%)*O&)+5D & -$* /0,01 $F(*(+ 7+(*1(4(: %&()' 01# <'-#.-2 '(+$)$*8(*'),%)*p : =&# %&")%!#>q : ?'6&#&+ & %&@ &%(&#)' & A&B& C!6 7!"$65$#>7+(*1()(*$ 4&)5+'5*8 p 7+(*1(6$+$+ '3>*#)#, '(>(*$ 12 q 7+(*1(6$+$+ '3>*#)#,'(>(*$ 0 '5*8 P&/& ,56& 3)&*&,: p 7+(*1(6$+( !"#$: q 7+(*1(6$+( %&'()* '$4( (>$D8-.%)/., (')*+)%)*"$*'(+ F3, /0,01 $F(*(+ 7+(*1(4(: 01(-*16 01# <'-#.-2 '(+$)$*8(*'),%)*p : ?'5&(%&2 ?6 -'1@ 0<(9-71' - 1# B- D1@-2 =&#& -'1@ 0<(9-71' - 1#>q : 3$9E' A&B& 7) &6 -"1( 1# B-%& 0-' E*E%!#$.>7+(*1()(*$ -$)(.$,B$*8 p 7+(*1(6$+$+ '3>*#)#, '(>(*$: 3+# 3)#.B#*&+:

Page 3: SMtuba1€¦ · .*/(0+01.'/'(+'(2&2 #0+0& 034 "5$ 60 7&'()'+'(.'& #2(#2(+'(2&'.'&1 *+,&+,(-s≡t 82)9'82:+' # '+2(;2&234".$ 60 7&'()'+'(.'& #2(#2(2&'.'&1 *+),:,&+,(-s ≡ t 82)9'82:+

!"! #$%&'%(%& p1 : !"#$%#& '()!*+ ,-$.)/*!()(*01p2 : 2*+)& 3#0#()!*+ ,-$.)/*!()(*01!"#$%#&#$'"'" ()*$+&+, (#*#$&#$'"# -.*&/(/$0 1#"2#$ )&.$.,3 q !"#$%#4'"'" ()*$+5&+, (#*#$'3 q 6+ )&+78+$."3q1 : 45.6! 7#8# /9 #' ();*$(*01q2 : 4)! 6<6%=05>1!"#$%#&#$'"'" ()*$+&+, (#*#$&#$'"# -.*&/(/$01'&#7', !"#$%#&#$'" ()*$+&+, (#*#$&#$'"'" ".4/& -#&'$&#"# #*'"'3 -+"(." 4)"5$.,' ,#4'%&#$(# '" #&#6# #*'20 !"# $"!%&!'!%:)*$+&+, (#*#$&#$' .6"/ )&." !"#$%#&#$#3 ()!' -!)0>)$)0 6. (. )<();)0 -!)0?>)$)0 (#"'&'$0 p '&# q !"#$%#&#$'"'" (#",&'*'3 p ≡ q 4'%;#4'6&# ;!48#$'&'$ <# =p()!' q= ('6# ),+"+$0>"#$%#&#$'" (#",&'*' .$.78/$/&/$,#"3 6.&"/2 . )"&.$/" ()*$+&+, (#*#$&#$'"#-.,/&/$0 >"#$%#&#$'" ."&.%&.$/ (',,.8# .&/"%.20 >$"#*'"3

p : (23 = 8) '&# q : 4*0 7#@"#(# A .6! 8#0(90 !"#$%#&#$'"'" ()*$+&+, (#*#$&#$' 1('$0 :)&.6/4/6&.3 -'$-'$&#$'"# (#",8'$&#$? (p ≡ q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a : J#$F.-. N b : @(.". @,(#"'2 -!&;#4'"(#('$0c : O#" #$#6' .G0 d : 5 + 7 = 12e : 5 + 7 < 5 p : P#(' #< '& (#*'&('$0q : P!E#, #< '&('$0 r : Q 8#, -'$ 4.6/(/$0s : R .4.& -'$ 4.6/(/$0 t : S ., BTTT'A.(#&#$' <#$'&'6)$0H.I U#$'&#" -+ 'A.(#&#$ .$.4/"(." !"#$%# )&."&.$/ <# )&%.6."&.$/ .6/$/"/20

Page 4: SMtuba1€¦ · .*/(0+01.'/'(+'(2&2 #0+0& 034 "5$ 60 7&'()'+'(.'& #2(#2(+'(2&'.'&1 *+,&+,(-s≡t 82)9'82:+' # '+2(;2&234".$ 60 7&'()'+'(.'& #2(#2(2&'.'&1 *+),:,&+,(-s ≡ t 82)9'82:+

! !"#$ %& !'()$("() *( +)+"#$ %&'()' *+,&+,(-& .*/(0+01 .'/'(+'(2&2 #0+0&034" $ 60 7&'()'+'(.'& #2(#2(+'(2&' .'&1 *+,&+,(- s ≡ t 82)9'82:+' #'+2(;2&234".$ 60 7&'()'+'(.'& #2(#2(2&' .'&1 *+),:,&+,(- s 6≡ t 82)9'82:+' #'+2(<;2&234=4 >?,/-.,12 7&'()'+'(2& .*/(0+01 .'/'(+'(2&2 #'+2(+':2&234(a) (−5)3 < 125 (b) ! "#$ %&'&%&()#$*(c) 53 = 125 (d) +,(!- "#$ %&'&%&( )&.#/)#$*@4 A01,(-.,12 7&'()'+'( ,(,8-&., #2(#2(+'(2&' .'&1 *+,&+,(- #'+2(;2&234B4 >?,/-.,12 7&'()'+'( ,(,8-&., #2(#2(+'(2&' .'&1 *+,&+,(- #'+2(;2&234

(a) (−3)2 < 9 (b) 012!13 4-51(!-6(7( "-82&(9#)#$*(c) 33 = 27 (d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p #/& q "#$&$ >(&$?& #:&3 p ∧ q "#/&8#2 >(&$?&:#3

1 ∧ 1 = 1, 1 ∧ 0 = 0, 0 ∧ 1 = 0, 0 ∧ 0 = 0 "!4 $2=$-//-$7!/- 9-(7?/-(7$@ p ve q )#!& 12=(=$* 60 10(,++,(-G ,?,/-.,12 82)9'+'(+'.' ;,&-)+,:,#2+2(234D ∧D = D, D ∧ Y = Y, Y ∧D = Y, Y ∧ Y = Y "!4!$

Page 5: SMtuba1€¦ · .*/(0+01.'/'(+'(2&2 #0+0& 034 "5$ 60 7&'()'+'(.'& #2(#2(+'(2&'.'&1 *+,&+,(-s≡t 82)9'82:+' # '+2(;2&234".$ 60 7&'()'+'(.'& #2(#2(2&'.'&1 *+),:,&+,(-s ≡ t 82)9'82:+

! ! "#$%&$'$% ($)*%* !p "#$ q %&$'($#$'") p ∧ q *"#$+", %&$'($-"&"&"& !"#$#%"#&!./"'0

p∧q !"#$!' (%#&)#*!%!% +,-&. ,")/*0 !1!%2 p !"# q !"#$#%"#&!%!% 3#& !'!*!%!%+# +,-&. ,")/*0 4#&#'"! 5# 6#7#&"!+!&8p q p ∧ q 1 11 11 1 1 p ∧ q *"#$+", %&$'($-"&"& /23'45#4, 67*#2-4) 87&/7 9%-6$'"#("+6"'0 !"#$%#! : −8 < 0 5# 0 < +8;: 9 !& 1!:7 */60+0& 5# ; !& 7#' */60+0&8!: ; !& /*/" */60+0& 5# 4 < 5 +!&8<: 4 !& /*/" */60+0& 5# 5 < 4 +<&8=4 *"#$+", %&$'($#$'/$& *"'"& "-"&"& /23'4) %6$,"#$'"& 87&#?+ 2#/434&4 -$@$5*"#"82' (4-4&4@A=4&#7'?& /23'4#4, /$3$'#$'"&") 67&?(7 B$ 67*#287 9%'$) 78'? 78'? *$#"'#$8"&"@0&#'!#()

p, q, r <1 (%#&)# !*#2 /$/-0+/'! (=#"!'"#& */-"/%0&> ! p ∧ p ≡ p (∧ ! "#$%&'%'%( )*+'!,!-"! p ∧ q ≡ q ∧ p (∧ ! .+/ 0+,!#!1 )*+'!,!-#! (p ∧ q) ∧ r ≡ p ∧ (q ∧ r) (∧ ! 2!/'+#1+ )*+'!,!-?*@/7> =4&#7'?& "-C76#7'?) -?'7-?8#7) 7+73?/7," /23'4#4, 67*#2#7'?&/7& 9%'D#$*"#"'0=4 67*#2#7'? "& $#$8"&"@ B$ "-C76#7'? 7E?,#78?&?@0C C p ∧ p 1 1 1↑ ↑

p ≡ p ∧ p

p q p ∧ q q ∧ p 1 1 11 1 11 1 1 1↑ ↑

p ∧ q ≡ q ∧ p

Page 6: SMtuba1€¦ · .*/(0+01.'/'(+'(2&2 #0+0& 034 "5$ 60 7&'()'+'(.'& #2(#2(+'(2&'.'&1 *+,&+,(-s≡t 82)9'82:+' # '+2(;2&234".$ 60 7&'()'+'(.'& #2(#2(2&'.'&1 *+),:,&+,(-s ≡ t 82)9'82:+

! !"#$ %& !'()$("() *( +)+p q r p ∧ q q ∧ r (p ∧ q) ∧ r p ∧ (q ∧ r) " " " " " " " " " " " " " " "" " " "" " " " " "" " " " " "" " " " " " "

↑ ↑

(p ∧ q) ∧ r ≡ p ∧ (q ∧ r) !"#$% ” ∧ ” #$# %$&'()*( +,('$-$#$ ./0/ 123/ 4+35(&*(1 $6$#7 8/&/#5(,'(&$1/'.2&287(p ∧ q) ∧ r = p ∧ (q ∧ r) = p ∧ q ∧ r%$6$*$#.( 9/,/%$'$&$,:&'!" ()*'+, -."/*" $12"3$+% p $'( q %$&(& +#(&*( $3(7 p ∨ q %$'()$1 +#(&*(3$7

1 ∨ 1 = 1, 1 ∨ 0 = 1, 0 ∨ 1 = 1, 0 ∨ 0 = 0 ;<:=>1?&/''/&29'/ 5/#2*'/#2&@ p A(9/ q” .$9( B1?#?&:C? 1?&/''/&7 /)/-2./1$ 3$*4('(&'( .( 4+35(&$'(%$'$&DD ∨D = D, D ∨ Y = D, Y ∨D = D, Y ∨ Y = Y

p $'( q +#(&*('(&$7 p ∨ q %$'()$1 +#(&*(3$#$#$# !"#$#%"#&!E.$&:p ∨ q %$'()$1 +#(&*(3$#$# 9/#'2) B'*/32 $6$#7 p $'( q %$'()(#'(&$#$# 0(& $1$3$#$#.( 9/#'2) B'*/32 4(&(1'$ A( 9(5(&'$.$&:p q p ∨ q " " " " " p∨q %$'()$1 +#(&*(3$#$# .B-&?'?15/%'B3?7 9/#./ 4+35(&$'*$)5$&:4#3'5*'# : (2 < 7) ∨ (1 + 5 = 6)<: (2 + 2 = 4) ∨ '()&!*+ ,-#&!.) ./0)*/%1)1/&23=: (2 + 2 = 5) ∨ '()&!*+ ,4&56) ./0)*/%1)1/&23!: (4 + 2 = 7) ∨ (3 + 1 = 6)

Page 7: SMtuba1€¦ · .*/(0+01.'/'(+'(2&2 #0+0& 034 "5$ 60 7&'()'+'(.'& #2(#2(+'(2&'.'&1 *+,&+,(-s≡t 82)9'82:+' # '+2(;2&234".$ 60 7&'()'+'(.'& #2(#2(2&'.'&1 *+),:,&+,(-s ≡ t 82)9'82:+

! ! "#$%&$'$% ($)*%* !"# $%&'(%) *+',-'&',.'+ %&) /0/+/+ .12,#3 .*,./+ /+/+ 56+&7( 1&.#2#+# 8'9':$%&%51, -#8#+#9;"# $%&'(%) *+',-'&',%+ .12,#&#) .'2',&',%+%3 <6+7-6 =' <6$&156 >*,'3 65,7 65,7$'&%,&'5%+%9? !"#!$% p, q, r ! "#$%&$ '($) *+*,-.*/' "0$1'/1$% (*,1*#-%2 ! p ∨ p ≡ p (∨ ! "#$%&'%'%( )*+',-,."! p ∨ q ≡ q ∨ p (∨ ! /+0 1+-,#,2 )*+',-,.#! (p ∨ q) ∨ r ≡ p ∨ (q ∨ r) (∨ ! 3,0'+#2+ )*+',-,.3(4*52 "# *9'&%)&',%+ %8@6<&6,73 87,6875&63 6(627.6)% .12,#&#) <6$&1&6,7+.6 >*,/&:-')<'.%,? "# <6$&1&6,7 %+ '&'5%+%9 =' %8@6<&6,7 607)&657+79?$ $ p ∨ p % % %↑ ↑

p ∨ p ≡ p

$ & p ∨ q q ∨ p % % % % % %↑ ↑

p ∨ q ≡ q ∨ p$ & ' p ∨ q q ∨ r (p ∨ q) ∨ r p ∨ (q ∨ r) % % % % % % % % % % % % % % % % % %↑ ↑

(p ∨ q) ∨ r ≡ p ∨ (q ∨ r)&'(#)% ” ∨ ” +%+ $%,&'(-' *9'&%2%+% .6A6 )786 >*8<',-') %0%+3 @6,6+<'9&',%)6&.7,7@3 6(627.6)% $%0%-.' 5696$%&%,%9?(p ∨ q) ∨ r = p ∨ (q ∨ r) = p ∨ q ∨ r ! !" #$%&'($ )*+',-'+., !"#!$% p, q, r ! "#$%&$ '($ *+*,-.*/' "0$1'/1$% (*,1*#-%2 ? ∧ +%+ ∨ 0$%'#$ (61.*# .*,-1&* "0$1','2

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)B? ∨ #-# ∧ 0$%'#$ (61.*# .*,-1&* "0$1','2p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

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! !"#$ %& !'()$("() *( +)+"# ∧ ! ∨ "#$%! $ &'()' )'(*+,' -#$+!(!.(p ∨ q) ∧ r ≡ (p ∧ r) ∨ (q ∧ r)$# ∨ * ∧ "#$%! $ &'()' )'(*+,' -#$+!(!.(p ∧ q) ∨ r ≡ (p ∨ r) ∧ (q ∨ r)/&0'1. # %&'()*)+ ),-./01 .2.*03.4) /.5(63.+ 789:(:9# ! " q ∨ r p ∧ q p ∧ r p ∧ (q ∨ r) (p ∧ q) ∨ (p ∧ r)# # # # # # # ## # $ # # $ # ## $ # # $ # # ## $ $ $ $ $ $ $$ # # # $ $ $ $$ # $ # $ $ $ $$ $ # # $ $ $ $$ $ $ $ $ $ $ $

↑ ↑

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r);# %&'()*)+ ),-./01 .2.*03.4) /.5(63.+ 789:(:9<- = 9 q ∧ r p ∨ q p ∨ r p ∨ (q ∧ r) (p ∨ q) ∧ (p ∨ r) > > > > > > > > > > > > > >> > > > > >> > > > > > > >↑ ↑

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) !"#$$!%!&#' ()!'%!)#) *$+%,+-+ . !"#$#/01)2%3 2!% p - $%,$&! ! )3(%4+45 )$($%! ! )$(!61!%!+,$&!7+$ $+)$ $)!+$ 7$ !- $%,$7$8 p ! 6(?@,?&? A3'*)()B )$ !+!%9 p′ 8 ∼ p 7' )' ¬p &!,:$+$%! )$ ;!%!&! !+$ :-&1$%!+!% <$ Cp +)+ 6(?@,?&? A3'*)()BC1 )!7$ 354 4%=

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! ! "#$%&$'$% ($)*%* ! !"#$%#!p : !"#$%$ &'()*%+ ⇒ p′ : !"#$%$ &'(%'!*%+q : ,$- &*. /0 '&!*-+ ⇒ q′ : ,$- &*. /0 '& !"1$.!$-+r : // '2'. 2'&* !"1$.!$-+ ⇒ r′ : // '2'. 2'&*!*-+

p p′ (p′)′ " " "↑ ↑

&#'!#() ,$- 34"-%" !51-6 $2"7 !"1$.$&'4.*8)*-+ 4"-%"4$4 9"4!$2$ &'4.*8 $2"7 !"1$.$!51-6!6-+,$- 34"-%"4$4 !"1$.$4$4 !"1$.$7 9"4!$:2$4" !"49)$-; &'4$7 p = (p′)′ !<-+#$ %&'()*('+), )-./0(/+1 0/2(34/, 5%+6(7'*0'4)+8 !"#$%#!9:/;14/*) %,'+7'('+) 4';)(('<),)&8p : =$&'4'7 >2&' 9*)'2*4!'!*-+q : 2× 3 6= 6

p : ,"-.$47 ?'89"4) !"1$.!$-+r : 7 6< 15@3A<%Bp′ : =$&'4'7 >2&' 9*)'2*4!' !"1$.!$-+q′ : 2× 3 = 6

p : ,"-.$47 ?'89"4))$-+r′ : 7 < 15#$+/4/ :$,/ 4)**/0 '4)(7'()4)+8 p : C=$&'4'7 >2&' 9*)'2*4!'!*-C %,'+7'-),),4';)()= C=$&'4'7 >2&' 9*)'2*4!' !"1$.!$-+C %,'+7'-)4)+8 p1 : C=$&'4'7 >#-6('9*)'2*4!'!*-+C %,'+7'-) p ,), 4';)() 3(/7/&> 3 2/:*/ 2)+ %,'+7'4)+8 #$ 3(5$=%,'+7'('+), 4',*();) ?' 3($7-$&(/,7/-1 *3,$(/+1,4/= /,(/7(/+1<(/ )(5)(',7'7'*5'+'*0);),) /@1*(1*(/ 3+0/</ *3</+8 ! ! "#$%&'('&') 8 A, '*) /(1:01+7/(/+4/ ?'+)(', %,'+7'('+) 4';)(('<),)&8C8 #$ %,'+7'('+ )(' 4';)(('+),), 43;+$($* 4';'+('+),) */+:1(/:01+1,1&8D8 p, q, s, t %,'+7'('+) )(' 4';)(('+),), 43;+$($* 4';'+('+),) /<,1 0/2(34/ 5%-E0'+),)&8

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! !"#$ %& !'()$("() *( +)+"# $%& p '()&*)+% ,-./(/.# $0(01 +/&-,2-1 13141"151# # # 6)&) 7)8%22),%(%.# 9)& 7):;-+/(7- <027080(0. '()&*)2)&%( 7=8&0206 7)8)&2)&%(% p >) p′ (?( 7=8&02067)8)&2)&% %2) 6-&@/2-@A/&/(/.# $%& 60&-2 B/6-&-<%27%(%. *%C5# p, q, r, s 7'&A '()&*) %+)1 <0 '()&*)2)&%(DE-F $%&)& <%&)&1E<F G6%@)& %6%@)&1E F IB)& ?B)&1E7F J'&7)& 7'&7)&7=8&0206 A-<2=2-&/(/ ,-K/(/.#L# J=8&0 >) ,-(2/@ '()&*)2)& ,-./(/.# M-.7/8/(/. '()&*)2)&% 7)8%22),%(%.# ! !" #$%&'$( )*&+,&%&+$* -&.$%%&*,&/$ ! "#$%&' ()$&**&$+p %2) q %6% '()&*) %+)1 -@-8/7-6% )@%A2%62)& >-&7/&#

(p ∨ q)′ = (p′ ∧ q′)

(p ∧ q)′ = (p′ ∨ q′) !"#$% G26 )@%A2%8%( %+K-A/1 -@-8/7-6% A-<2=2-&7-( N'&?2?&#p q p′ q′ p ∨ q (p ∨ q)′ p′ ∧ q′ O O O O O O O OO O O OO O O

↑ ↑

(p ∨ q)′ ≡ p′ ∧ q′

p q p′ q′ p ∧ q (p ∧ q)′ p′ ∨ q′ O O O O O O O O O O O O O ↑ ↑

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! ! "#$%&$'$% ($)*%* !(p ∧ q)′ ≡ p′ ∨ q′ !"#$%#! " !" #$% &'(' )* ',-*. (/0' 0'1 2/()323%4/03156′ ≡ !" #$% &'(' )* ', 4/73843. (/ 0'1 2/()323%4/ 4/7383156#"

[p ∨ (q ∧ r)′]′ ≡ p′ ∧ (q ∧ r) 9/ :;.#'% <".'8*6 ! !" #$%$&$'( )* +*&(,-($%&' ()*+,)- ./+01+*+0 2%)1% 234056 (%&'*%0' )7+ 2%)1% 8%/*', 3*%()*)0"!38/=/%8/.3%3% >$-$% 4;7."8", 4/7/.8/.3 3?3%@ 4'32' A 4;7."6 4/7/.3%3 '8'%>38/=3, B%/.2/0/ >3. 9393*3:) &/C4;7."6 4/%383.5!38/=/%8/.3%3% >$-$% 4;7."8", 4/7/.8/.3 3?3%@ 4'32' D 0'%8*=6 4/7/.3%3 '8'%>38/=3, B%/.2/0/ >3. ;+*),-) &/C0'%8*=6 4/%383.5 !"#$%#!& <,%4'2%-) )=%2+*+0) )7>%9*%8'/'&" " p ()0 ./+01+ )7+6 p ∨ p′ ()*+,)- ./+01+7) ()0 9393*3:)2)0"#" p ()0 ./+01+ )7+6 p ∧ p′ ()*+,)- ./+01+7) ()0 ;+*),-)2)0"?" p ()0 ./+01+ )7+6 p ∨ 1 ()0 9393*3:)2)0"@" p ()0 ./+01+ )7+6 p ∧ 0 ()0 ;+*),-)2)0"A" (p ∧ q)′ ∨ (p ∨ q) ()*+,)- ./+01+7) ()0 9393*3:)2)0"EB1$2F $)0)/ ) C+ )-)/ ) )=%2+*+0)/ )7>%9*%0' %,%4'2%-) 9%(*3*%02%/ ;'-%0"p p′ p ∨ p′ D D p p′ p ∧ p′ D DD D$50%2%/6 p ∨ p′ ≡ 1 C+ p ∧ p′ ≡ 0 ;'-%0"$+,)/ ) )=%2+/)/ )7>%9'/'6 %,%4'2%-) 23405*5- 9%(*375/2%/ E.0+()*)0)&"> F p ∧ q (p ∧ q)′ p ∨ q (p ∧ q)′ ∨ (p ∨ q) D D D D D D D D D

Page 12: SMtuba1€¦ · .*/(0+01.'/'(+'(2&2 #0+0& 034 "5$ 60 7&'()'+'(.'& #2(#2(+'(2&'.'&1 *+,&+,(-s≡t 82)9'82:+' # '+2(;2&234".$ 60 7&'()'+'(.'& #2(#2(2&'.'&1 *+),:,&+,(-s ≡ t 82)9'82:+

! !"#$ %& !'()$("() *( +)+"#$#% &'()*+*, &-(-)+-). / '+&*(*%&0%1 (p ∧ q)′ ∨ (p ∨ q) 2.+-3., 4%-)5-6.2.) ! !"!#$%$&'"070%1 &'()*+*, $02+'6*%* &#7-%+-5-, 8-).%-1 4%-)5-+-) -2.).%.% *8:*%,*)0++0);%; ,*++0%0)0, &0 <)'2+-5+-). =47-2.+.).7>(p ∧ q)′ ∨ (p ∨ q) ≡ (p′ ∨ q′) ∨ (p ∨ q) ()* +!&,-./

≡ p′ ∨ q′ ∨ p ∨ q (0$&"*12*/≡ p′ ∨ p ∨ q′ ∨ q ()*3$12*/≡ (p′ ∨ p) ∨ (q′ ∨ q) (0$&"*12*/≡ 1 ∨ 1 (p′ ∨ p ≡ 1)≡ 1 (1 ∨ 1 ≡ 1)

(p ∧ q)′ ∨ (p ∨ q) ≡ 1'+&*(*%&0%1 2* 2.+-3., 4%-)5- 2.) ! !"!#$%$&'"* 84%$-5- 4.*&2*"*& *6$&$ .+- =475- 84%$-5. &-%.+.)? ! !" #$% #&'%()p : 07 89,*.$. $:$ :*.-&; *1$ $&'

q : 07 6$& $:$<:*.-& 89,*.%$&'4%-)5-+-).%. @$=*A 20(+0 ;8+0 2.)2.).%- 20(+0&;(;5;7&01p $=* q : >07 89,*.$. $:$ :*.-&; *1$ =*? $:$<:*.-& 6$& 89,*.%$&' 2.+-3., 4%-)B5-6.%. -+&- -&-).7? !"#$% p $"* q 6$&*& 4.*&2* $=*? p⇒ q 6$"*1$: 4.*&2*=$?

(1⇒ 1) ≡ 1, (1⇒ 0) ≡ 0, (0⇒ 1) ≡ 1, (0⇒ 0) ≡ 1 C ?DE:7&-""-&;@"- -.;2"-.;& A* ”p ise q” %$@* !:7.7&' "* ,*)0++0)1 030(;&0,. 6.5:-+B-)+- &- :46$-).+-2.+.)>(D ⇒ D) ≡ D, (D ⇒ Y ) ≡ Y, (Y ⇒ D) ≡ D, (Y ⇒ Y ) ≡ D C ?FE

p .+- q 4%-)5-+-).1 p⇒ q 2.+-3., 4%-)5-6.%.%.% 6$"*1*."*&$G&.)?p⇒ q 4%-)5-6.%- :!17""7 4.*&2*? ,*&*: $&2* 0&+0); &0 H-).+.) H- p ,*&*: $&$&

q &.8- &- ',*%*)? "* ,'3*++* 4%-)5-&-1 p 8- 4. 8" (A-&=-@;2? B$C! *</I q 80 .6-=!.79 (A-&,;? B8:82/ &-%.+.)?p ⇒ q :!17""7 4.*&2*=$.$. @-.";1 !"2-=; $9$.? q A-&,;=;.;. @-.";1 !"2-=;,*&*:"$ A* @* *&"$%$&'< J p⇒ q/ / // ! !! / /! ! / p⇒ q 2.+-3., 4%-)5-6.%.% &'()*+*,$02+'6*1 80%&0 :46$-).+5.3$.)?&'"()*('

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! ! "#$%&$'$% ($)*%* !!" !"#$ %&'#(') *!"#$ %&'#('+, -'#'./,#,#0p : 1,+2/32# 4' 5 #'/3'# +5.7'3,+!#8q : 9&:27+!& 2#/;+!#8#$%&'%(%&) )*)$+p⇒ q : 1,+2/32# 4' 5 #'/3'# +5.7'3,+!# ,7') '&:27+!& 2#/;+!#8,-./((/ #$%&'%0)$) 12324)()&)3" " !"#$ %&'#(') +2&3;< %&'#('+, -'#'./,#('=0p : >&.2#2?*2 @242 7!"$./$#8 q : >&.2#2 .$='+ .$/A$&*2*;#8#$%&'%(%&)$5%$ -(/.2$+p⇒ q : >&.2#2?*2 @242 7!"$. ,7') >&.2#2 .$='+ .$/A$&*2*;#8,-./((/ #$%&'%0) 12$(6.76&" 8 92(5%+ p 6⇒ q 5/&":" B2&3;< %&'#(') *!"#$ %&'#('+, -'#'./,#'A,3,#0p : (6 = 9) ≡ 0, )(% q : (6 < 9) ≡ 1 #$%&'%(%&)$) 5;.;$%()'"

6 6= 9 -('206 6 < 9 -('206$2 %$<%( 5%=)(5)&" 8 92(5% (6 = 9) ⇒ (6 < 9)<%&%,7)&'%0) >2& -(24)()&" ?%'%, ,) 12$(6. 4)& #$%&'%5%$+ 5-=&/ 4)& #$%&'%*6,2&6(24)()&"@" B2&3;< %&'#(') +2&3;< A,# %&'#('+, -'#'./,#'A,3,#0p : (6 6= 6) ≡ 0, )(% q : (7 6= 7) ≡ 0 #$%&'%(%&)$) 5;.;$%()'"

6 6= 6 -('206 6 + 1 6= 6 + 1 -('206$2 %$<%( 5%=)(5)&" 8 92(5% (6 6= 6) ⇒(7 6= 7) <%&%,7)&'%0) >2& -(24)()&"

p ⇒ q ,-./((/ #$%&'%0)$5%+ p #$%&'%0) +'/'#3, ,-./(+ q #$%&'%0) -'#'.3,,-./(5/&"p⇒ q ,-./((/ #$%&'%0)$5% p #$%&'%0)$% @,C!/'=+ q #$%&'%0)$% @5.5( 5%$)&"q ⇒ p ,-./((/ #$%&'%0)$%+ p⇒ q #$%&'%0)$)$ .2#<;/; 5%$)()&"A)& ,-./((/ #$%&'%$)$ ,2&.676 4232$ 5-=&/+ 4232$ 12$(6. -(24)()&" !"#$%#!!" p : x = 3 )(% q : x+ 4 = 7#$%&'%(%&)$5%$ -(/.7/&/(2$+

p⇒ q : [(x = 3)⇒ (x+ 4 = 7)],-./((/ #$%&'%0)$)$ ,2&.676+q ⇒ p : [(x+ 4 = 7)⇒ (x = 3)]5;&" 8 92(5%+ 4/ #&$%,7%+ 9%' (p⇒ q) ,-./((/ #$%&'%0)+ 9%' 5% (q ⇒ p),2&.676 5-=&/5/&"

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!"#$ %& !'()$("() *( +)+ ! p : x = −4 "#$ q : x2 = 16%&$'($#$'"&)$& *#+,-+'+#.&/p⇒ q : [(x = −4)⇒ (x2 = 16)]0*,+##+ %&$'($1"&"& 0.',2-2/q ⇒ p : [(x2 = 16)⇒ (x = −4)])3'! 4& .0/ 6$' 7.(.& (x2 = 16) ⇒ (x = −4) *#(.7! (x2 = 16) ⇒

(x = +4) )$ *#.8"#"'! 9 6.#)$/ 8+ %'&$0-$/ (p ⇒ q) 0*,+##+ %&$'($1"&"&)*:'+#+:+/ (q ⇒ p) 0.',2-2&2& )*:'+#+:+&+ ;$'$0-"'($<$8"#"'!=! p : △ABC ! "!#$%& ' & ( ! )*+#$( &,q : △ABC ( ! )*+#$ $ !#$%&-%&. %&%/.$(% a2 + b2 = c2 '%0.$1./. 2%&(.&,%&$'($#$'" ">"&p ⇒ q : △ABC ! "!#$%& ' & ( ! )*+#$ /#3 △ABC ( ! )*+#$ $ $ !#4$%&-%&. %&%/.$(% a2 + b2 = c2 '%0.$1./. 2%&(.&,q ⇒ p : △ABC ( ! )*+#$ $ !#$%&-%&. %&%/.$(% a2 + b2 = c2 '%0.$1./. 2%& /#3 △ABC ! "!#$%& ' & ( ! )*+#$( &,*#+'! 4(./ 8+ %&$'($ 6$' 7.(.& )*:'+ )$:"#)"'! !"#$% p⇒ q !567--7 8$#&9#/ $ $ 0.',2- -$'1"/ q′ ⇒ p′ 8$#&9#/ ( &, &'(&$% :567--7 ' & 8$#&9# !%&6.1 1#&/ $# (#$!1 &;

p⇒ q ≡ q′ ⇒ p′</=%1; 4,.:2).0" )*:'+#+0 -.8#*1+ "1-$&$&" ?$'$ $0-"'!p q p′ q′ p⇒ q q′ ⇒ p′@ @ A A @ @@ A A @ A AA @ @ A @ @A A @ @ @ @

↑ ↑

p⇒ q ≡ q′ ⇒ p′ &'(&$% p -# q ! 8$#&9# /#3 %6%0.(%! (#$!- ! /%0-%$.&,p⇒ q ≡ p′ ∨ q

p q p′ p⇒ q p′ ∨ q@ @ A @ @@ A A A AA @ @ @ @A A @ @ @↑ ↑

</=%1; B.&).0" )*:'+#+0 -.8#*1+&).&>20.'!

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! ! "#$%&$'$% ($)*%* !p⇒ q ≡ p′ ∨ q !"#$%#!"# p : △ABC !"#$% #&'#$()*+),

q : △ABC !"#$%$%$ (!+-()+ #&.%),p⇒ q : △ABC !"#$% #&'#$() %/#0 △ABC !"#$%$%$ (!+-()+ #&.%),q′ ⇒ p′ : △ABC !"#$%$%$ (!+-()+ #& *#1%- %/#0 △ABC !"#$% #&'#$()*#1%-*%), # p⇒ q ≡ p′∨q $%&'()*)&) '+((,&,-,'. (p⇒ q)∨p ≡ 1 /($+*+&+ 0123%-)&)4#234 56(p⇒ q) ∨ p ≡ (p′ ∨ q) ∨ p (p⇒ q ≡ p′ ∨ q)

≡ (q ∨ p′) ∨ p ∨ &)& 5%*)67% 84%()*)9≡ q ∨ (p′ ∨ p) (∨ &)& :)-(%67% 84%()*)9≡ q ∨ 1 (p′ ∨ p ≡ 1)≡ 1 (q ∨ 1 ≡ 1) ! !" #$%&'()('(*"# p ⇒ q '/6+((+ 1&%-7%2) ;,&(<6 )'%& q ⇒ p 1&%-7%2)&)& $/*-+ =% ;,&(<6/($+*+ $+-+7(,-, 1-&%'(%- =%-)&)4# # p′ ⇒ q′ '/6+((+ 1&%-7%2)&% p⇒ q '/6+((+ 1&%-7%2)&)& .#)/% $%&)()-# :+&+&$/*-+(+' 3,>(/2+&+ $?4%&(%;)&)4#

p : x = 3 )(% q : x2 = 91&%-7%(%-)&$%& /(+63+-+(,&p⇒ q : (x = 3⇒ x2 = 9)'/6+((+ 1&%-7%2)&)& 3%-2)&) ;,4<&<4#!# (p⇒ q)′ ∧ q >)(%6)' 1&%-7%2)&)& >)- @%()6') /($+*+&+ 0123%-)&)4# ! !+ ,- (/ 01 ,- (/ 23'1)4&'"()* p %-# q 7%)#) 3$#)5# %/#0 p⇔ q 7%-#&%' 3$#)5#/%0

p⇔ q ≡ (p⇒ q) ∧ (q ⇒ p)7(1+$.+/+8-( .($+5-($+) 9# p ,& ,' =% ,& ,' q *%8# :';$;), :+ 3,&<7.(1⇔ 1) ≡ 1, (1⇔ 0) ≡ 0, (0⇔ 1) ≡ 0, (0⇔ 0) ≡ 1'+-,((,-<;(, $, =%-)(%>)()-# B%$%&C

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! !"#$ %& !'()$("() *( +)+p "#$ q %&$'($#$'") p⇔ q *"#$+", %&$'($-"&"&"& !"#$#%"#&!./"'0p ⇔ q %&$'($-"&$ !'! ()%"* '+$,"", )%#&-#. /!01 2#&#'1!&-# 1/#1'2 /1 3$'"#"'3$ 4p /!01 2#&#'1!&!& q4 /"5$ /$ 6,7&7'0p⇔ q !'! ()%"* '+$,"", )%#&-#3!%!% 4+5&, +"-637 !/!%. p !"# q !"#$#%"#&!%!%6(%7 4#5#&"! +"-637 2#&#'"! 8# (#1#&"!4!&9p⇔ q *"#$+", %&$'($-"&"& /68'7#7, 91*#6-7) 1+182/1 :%-9$'"#("+9"'0; < p⇒ q q ⇒ p (p⇒ q) ∧ (q ⇒ p) p⇔ q= = = = = == > > = > >> = = > > >> > = = = =

↑ ↑

p⇔ q ≡ (p⇒ q) ∧ (q ⇒ p) !"#$%#!=0 d, e "," /68'7 3$ ‖ -"(:$-" ;1'1#$##"8" :%-9$'-"&0p : d ‖ e q : e ‖ d%&$'($#$'"&/$& 3$ ;1'1#$##", %?$#"8"&/$&)p⇒ q : d ‖ e ⇒ e ‖ d

q ⇒ p : e ‖ d ⇒ d ‖ e

p⇔ q : d ‖ e ⇔ e ‖ d@2,1'0 A5#$5-$) *7 %'&$, "@"&) (p⇔ q) "," 5%&#B ,6+7##7 %&$'($-" /68'7/7'C51&") *"' @"D9 :$'$,9"'($/"'0 E7 -6&7 7)(p⇔ q) ≡ 1*"@"("&/$ :%-9$'$ $8"?0 0

p : ∠A = ∠B q : ∠A "#$ ∠B 5%&/$+9"'0%&$'($#$'"&/$&)p⇒ q : (∠A = ∠B) !3#. ∠A !"# ∠B ()%4#$1!&9q ⇒ p : :∠A !"# ∠B ()%4#$ !3#. (∠A = ∠B)dir.51?1*"#"'"?0 G H1#/$)p⇔ q : (∠A = ∠B) 6% 6' 8# 6% 6' ∠A !"# ∠B ()%4#$1!&96#7'0G5-1) *B9B& $+ 1@2#1' 5%&/$+ 6#(1, ?6'7&/1 /$8"#/"'C 5%&/$+ 6#(1/282H1#/$ $+ 6#1& 1@2#1' /1 31'/2'0 G H1#/$) (p ⇒ q) %&$'($-" 51&#2+92'C 51&")

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! ! "#$%&$'$% ($)*%* ![(p ⇒ q) ≡ 0] "#$% &'( )*+"*+, -.'.+ )/+"(0 *1#2*$ (0'3$% 452*)#6#)2*,(q ⇒ p) 7508228 /+($9(63 "5:$8"8$; )*+3, [(q ⇒ p) ≡ 1] "#$%&)2()6(, -8 /$+(7 313+, (p ⇔ q) 373 )/+2. 7508228 <($(7'3$9(63 )*+2#0'#$;)*+3, -3$ 13=' <($(7'3$9( "(:32"3$% >8 65+8 8,

(p⇔ q) ≡ 0, )* "* (p⇒ q) 6≡ (q ⇒ p)

-31392($3+"(+ -3$363)2( </6'($(-323$3@%

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Page 18: SMtuba1€¦ · .*/(0+01.'/'(+'(2&2 #0+0& 034 "5$ 60 7&'()'+'(.'& #2(#2(+'(2&'.'&1 *+,&+,(-s≡t 82)9'82:+' # '+2(;2&234".$ 60 7&'()'+'(.'& #2(#2(2&'.'&1 *+),:,&+,(-s ≡ t 82)9'82:+

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Page 19: SMtuba1€¦ · .*/(0+01.'/'(+'(2&2 #0+0& 034 "5$ 60 7&'()'+'(.'& #2(#2(+'(2&'.'&1 *+,&+,(-s≡t 82)9'82:+' # '+2(;2&234".$ 60 7&'()'+'(.'& #2(#2(2&'.'&1 *+),:,&+,(-s ≡ t 82)9'82:+

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Page 20: SMtuba1€¦ · .*/(0+01.'/'(+'(2&2 #0+0& 034 "5$ 60 7&'()'+'(.'& #2(#2(+'(2&'.'&1 *+,&+,(-s≡t 82)9'82:+' # '+2(;2&234".$ 60 7&'()'+'(.'& #2(#2(2&'.'&1 *+),:,&+,(-s ≡ t 82)9'82:+

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Page 21: SMtuba1€¦ · .*/(0+01.'/'(+'(2&2 #0+0& 034 "5$ 60 7&'()'+'(.'& #2(#2(+'(2&'.'&1 *+,&+,(-s≡t 82)9'82:+' # '+2(;2&234".$ 60 7&'()'+'(.'& #2(#2(2&'.'&1 *+),:,&+,(-s ≡ t 82)9'82:+

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