I.
0 i i =- 3
0~ ~8 ~3 p.0~ 9 4 0~ ~5 5~6 6~3 7~8 p 35 q ~p p jjk q p HjK q p.~3 3~33 9 34~38 0~ 39~43 ~3 44~45 4 p.36~37 3 46~57 5~9 i a+bi a+bi p.48~49 3 58~59 0 60~6 6~63 64 65
9 (Cantor, G.) (Zermelo, E.) 908 (Fraenkel,.) (Neumann, J. L. von)(paul (Isaak) Bernays)(Godel, K.) (ristoteles) (Leibniz, G. W.) 9 (Boole, G.) (De Morgan,.) (Jevons, W. S.) (Schroder, F.) (Frege, G.) (Peano, G.) (Russell, B.) ( ratio) 500 (Cardano, G.) (Descartes, R.) (imaginary number) (Euler, L.) i 9 (Gauss, K. F.) (988) (987) (000) (998)
~3 00 B C B B I. ;B B; ;B={}=B; 'B B' 'B={} =B' u. ('B)'C='(B'C) a+b=b+a 'B=B' a_b=b_a ;B=B; (a+b)+c =a+(b+c) ('B)'C='(B'C) (a_b)_c (;B);C=;(B;C) =a_(b_c) a_(b+c) ;(B'C)=(;B)'(;C) =a_b+a_c '(B;C)=('B);('C)
;B==B; 'B==B' B C (;B);C =;(B;C) (;B);C B ;B=B; 'B=B' B ; C B C B C ;B C (;B);C ;(B;C) ={ 3 5} B={3 5 6} ;B=B; 'B=B' ; B C B C B C B B ;B=u B B B;C ;(B;C) (;B);C=;(B;C) ={ 3} B={5 6} ;B=u C={ 3 5} D={ 3 5} C;D={35} B C ('B)'C='(B'C) B ={x x 0 } B={x x 0} B C (;B);C=;(B;C) ={x <x<3} B={x 4 x 5} ('B)'C='(B'C) ={ } (;B);C=;(B;C)=;B;C ('B)'C='(B'C)='B'C 3 4 I. ;B={3 5} B;={3 5} ('B)'C ;B=B; 'B={ 3 5 6} B'={ 3 5 6} 'B=B' ={ 3 5 7} B={ 5 0} ;B={ 5} B ={x <x<3} B={x 4 x 5} ;B=u B ={ } { } u {} {} { } {} {} { } {} ' B C B C B C 'B C ('B)'C '(B'C) ' B C B C B C B'C '(B'C) ('B)'C='(B'C) ={ 3 5} B={ 3 5 7} C={3 6 7} ;B={3 5} (;B);C={3} B;C={3 7} ;(B;C)={3} (;B);C=;(B;C)
={ 3 5} B={ 3 5 7} C={3 6 7} ={ 3 5} B={ 3 4} C={3 4 5 6} (;B);C=;(B;C) ('B)'C='(B'C) ;(B'C)=(;B)'(;C) '(B;C)=('B);('C) B C ;(B'C)=(;B)'(;C) ;(B'C) B C ;B=B; 'B=B' ; B C B C B C B'C ;(B'C) (;B)'(;C) (;B);C=;(B;C) ('B)'C='(B'C) ;(B'C)=(;B)'(;C) '(B;C)=('B);('C) B C ' B C B C B={ } C={3} ('B)'C={ 3 4 5} ;B ;C (;B)'(;C) ;(B'C)=(;B)'(;C) B C '(B;C)=('B);('C) Ç ={x x< x } ={ 3 4 5 6} ={ 3 5} B={ 3 5} ;(Ç 'B) B C ;(B'C)=(;B)'(;C) '(B;C)=('B);('C) B C B C 5 6 I. 'B={ 3 5 7} ('B)'C={ 3 5 6 7} B'C={ 3 5 6 7} '(B'C)={ 3 5 6 7} ('B)'C='(B'C) '(B;C) ' B C B ('B);('C) ; B C B 'B C B C B;C '(B;C) C B C 'C ('B);('C) '(B;C)=('B);('C) ={ 3 5} B={ 3 4} C={3 4 5 6} B'C={ 3 4 5 6} ;(B'C)={ 3 5} (;B)'(;C)={ 3 5} ;(B'C)=(;B)'(;C) B;C={3 4} '(B;C)={ 3 4 5} 'B={ 3 4 5} 'C={ 3 4 5 6} ('B);('C)={ 3 4 5} '(B;C)=('B);('C) B={ } C={3} B'C={ 3} ('B)'C='(B'C)={ 3 4 5} {4 5},,{ 3 4 5} {4 5} { 4 5} { 4 5} {3 4 5} { 4 5} { 3 4 5} { 3 4 5} { 3 4 5}
wma MB(,048 KB) wma MB B ;B Ç BÇ (;B)Ç Ç 'BÇ ;B (;B)Ç wma Ç MB BÇ Ç 'BÇ (;B)Ç =Ç 'BÇ B (;B)Ç =Ç 'BÇ (;B)Ç B B ('B)Ç ;B Ç 'BÇ (;B)Ç B B B ' B B Ç BÇ Ç 'BÇ (;B)Ç =Ç 'BÇ 7 'B ('B) C Ç ;BÇ ={ 3 4 5 6} ={ 3 5} B={ 3 5} ;Ç =u ;(Ç 'B)=(;Ç )'(;B)=;B={3 5} B B B ; C B C C ;B C ('B)Ç =Ç ;BÇ 'B C CÇ ('B);CÇ ;B Ç BÇ ={ 3 4 5 6 7 8 9} ={ 4 6 8} B={ 3 6} ;B={ 6} (;B)Ç ={ 3 4 5 7 8 9} Ç ={ 3 5 7 9} BÇ ={4 5 7 8 9} Ç 'BÇ ={ 3 4 5 7 8 9} (;B)Ç =Ç 'BÇ 'B={,, 3, 4, 6, 8} ('B)Ç ={5, 7, 9}, Ç ;BÇ ={5, 7, 9} ('B)Ç =Ç ;BÇ
-B B -B ;BÇ (De Morgan,. ; 806~87) B ('B)Ç =Ç ;BÇ B (;B)Ç=Ç 'BÇ ('B)Ç=Ç ;BÇ (;B)Ç=Ç 'BÇ ('B)Ç=Ç ;BÇ B ; B B ={ 3 4 5 6 7 8 9} ={x x } B={x x 6 } (;B)Ç =Ç 'BÇ ('B)Ç =Ç ;BÇ B C ;B C -B=;BÇ (-B)Ç B B B -B=;BÇ (-B)Ç =Ç 'B B C B C B 'B={ 3} B -B (-B) C Ç 'B B B B ' 8 I. ={ } B={3} ={ 3} B={} ={ 3} B={} ={ 3} B=u C B C 'B (-B)Ç =Ç 'B B C B C ;(B'C)Ç (;B)-C 'B={ 3} B =u B={ 3} ={} B={ 3} ={} B={ 3} ={3} B={ } ={x x } B={x -3 x<3} C={x 0 x 5} ('C);(B'C) {x -3 x 5} 40 a b a b 30 5 7 B C (-B)'(B-C)=u (-C)'(;B)
35 B B 5 B B B n(;b) n(ç ;BÇ )= n(;bç )=5 B 5 n(;b) n() n(ç ;BÇ )= n('b)=4 8 n() n('b) n()=n(;bç )+n(;b) 3 n(;b) 9 n(;b) 9 3 9 40 B B B n(;b) n(ç ;BÇ )= n(;bç )=5 n(;b) n() n()=40 n(ç ;BÇ )=n(('b)ç )=n()-n('b)= n('b)=9 n(),('b) n() 9 n() 9 n(;bç )=5 n()=n(;bç )+n(;b) n() 9 n(;bç )+n(;b) 9 n(;bç )=5 6 n(;b) 4 n(;b) 4 6 Ç ;BÇ =('B)Ç n()=35 n(ç ;BÇ )= n(ç ;BÇ )=n(('b)ç )=n()-n('b) =35-n('B)= n('b)=4 8 n(),('b) n() 4 8 n() 4 n(;bç )=5 =(;BÇ )'(;B) (;BÇ );(;B)=u n()=n(;bç )+n(;b) 8 n() 4 8 n(;bç )+n(;b) 4 n(;bç )=5 3 n(;b) 9 n(;b) 9 3 856
4 B ;B=u B B ;B=B; 'B=B' B C (;B);C=;(B;C) ('B)'C='(B'C) B C ;(B'C)=(;B)'(;C) '(B;C)=('B);('C) B (;B)Ç =Ç 'BÇ ('B)Ç =Ç ;BÇ ;B=u B (Z) B -B=;BÇ (Y),('B) ;('B)=(Z) ={ 4 6 8 0} B'C={ 3 4 6} ;(B'C)={ 4 6}(Y) Ç ;BÇ =('B)Ç (Y) 0 I. ={ 3 5} B={ 4} B -B='BÇ B ;('B)= ={ 4 6 8 0} B={ 4 6} C={ 3 6} ;(B'C)={ } B Ç ;BÇ (;B)Ç ={ 3 5} B={ 4} u {3} {5} {3 5} ={ 3 4 5 6 7 8 9 0} ={ 4 6 8 0} B={ 3 5 7} C={ 3 6} B;C={ 3} '(B;C)={ 3 4 6 8 0} 'B={ 3 4 5 6 7 8 0} ('B);C={ 3 6} BÇ 'CÇ =(B;C)Ç B;C={ 3} BÇ 'CÇ ={ 4 5 6 7 8 9 0} B-={3 5 7} (B-)Ç ={ 4 6 8 9 0} ' B C -B B - C B -C (-B)'(-C) B C C B B B;C -(B;C) (-B)'(-C)=-(B;C) C C
link ={x x 0 } ={x x} B={x x } C={x x 6} '(B;C) ('B);C BÇ 'CÇ (B-)Ç ={ 3 5} B={ 4} n(;b)=n()+n(b)-n('b) =5+0-9 =6 -B n(-b)=n()-n(;b) =5-6 =9 9 B C (-B)'(-C)=-(B;C) ={ 3 5 7 9} B={ 3 5 7} ('B);(Ç 'BÇ ) 'B={ 3 5 7 9} ;B={3 5 7} ('B);(Ç 'BÇ )=('B);(;B)Ç =('B)-(;B) ={ 9} { 9} 35 5 0 6 35 B n()=35 n()=5 n(b)=0 n(ç ;BÇ )=6 n('b)=n()-n(ç ;BÇ ) =35-6 =9 806 (Boole, G.) (duality) E E E E E ' E ' E E a(bc)=(ab)(ac) 55 55 55 55 55 a(bc)=(ab)(ac) E ' E ' ' ; ; '
5~6 940 TV Twenty Questions 006 (I) 8 3 3 p p p ~pp ~p not p I. 006 (I) 006 006.. ~p p ~p p ~p ~(~p) (<) (>) (æ) ++3+4 0
~p p ~(~p)=p p ~p p ~p ++3+4>0 p ~p p q p x q x x= x= p q p x p q x x p q x p x p p ~p p P ~p PÇ px 6 { 3 6} qx-4>0 {x x>} 3 x x x x p q P Q P={x x p} Q={x x q} p q {x x p x q} ={x x p}'{x x q} =P'Q p q {x x p x q} ={x x p};{x x q} =P;Q (P'Q)Ç =PÇ ;QÇ (P;Q)Ç =PÇ 'QÇ ~p q=~p ~q ~p q=~p ~q x= p q x= p q p x x {}. x x 5 6 4 3_5=5 4 3 9
x 3 x 6 {6 8 y}={x x 6} x x= ' {-' '} px 3 qx = ={ 3 4 5 6} px 3 qx = { 3 4 5 6} ~px 3 {4 5 6} ~qx { 3 5} px> q0<x< 3 B C D B C D B C D ~px {x x } ~qx 0 xæ {x x 0 xæ} B C D p x p x p 4 I. (Russel, B.. W.) 90 H H={x x H} x<h H x H x H H x<h
p P x p yy x p yy P= P+ P+u P=u ={ 3} x x 6 x x 5 ={ 3 4 5 6} x <36 { 3 4 5} x x <36 x-=5 {3} x x-=5 x 6 P P={ 3} P= x 5 Q Q=u ={ 3 4 5 6} x x <36 x x-=5 px {} 5 B C D B C D x p x ~p x p x ~p x x+> x x+ x x +x=0 x x +x+0.0 ={x xæ} ={x x } 5 >5 5 5 x -4+0 0<x< x -4=0 {- } x 0 xæ {x x 0 xæ} x x > x x +x-3=0 x x x x +x-3+0
7~8 p 55 q p q p 55 q p 55 q p 55 q x= x =x= px = q p q p 555 q p q 6 I. p q p q p 35 q p q p q.. p q p q p 55 q p q p p 55 q p q p 55 q p q p 55 q p jjjk q q jjjk r p jjjk r
x= x = Q px=qx = P - P Q P={} Q={- } P,Q p q P Q p 555 q P,Q P,Q p q p 555 q D B C D 000 00 400 600 p 555 q C B p q P Q p q P Q p 555 q P,Q P,Q p 555 q B C C B B C 4 x =4 x= p q P Q p q p q q p p 555 q q 555 p ~p 555 ~q ~q 555 ~p px 4 qx P={4 8 y } Q={ 4 6 8 0 y } P,Q p 555 q ~p ~q ~q ~p x=- x =4 x+ x=0 x -x=0 x< x-<5 0 x -x=0 x=0 x= x -x=0 x+0 a+b=0 ab=0 a>0 b>0 a+b>0 7 8 I. x+0 x -x+0 x= x+0 x -x=0 x -x+0 x+0 x -x+0 x+0 x+ x< P P={x x<} x-<5 x<3 x-<5 Q Q={x x<3} Q P 3 x P,Q 3 0 0 { 5 0} { 3 4 6 } { 5 0} { 3 4 6 } a= b=- a+b=0 ab+0 {00 600} {00 400 600} {00 400 600} {00 600} {000 400} {000} {000} {000 400}
a =b a=b x=0 y=0 xy=0 p 555 q ~q 555 ~p p q P Q Q Q C p 555 q P,Q P P,Q QÇ,PÇ ~q 555 ~p P C (Cu) (Li) (Sr) p 555 q ~q 555 ~p P C Q C p 555 q ~q 555 ~p p 555 q ~q 555 ~p p 555 q p jjk q n n n n n n n n=k k n =(k) =4k =(k ) k n n n n a b ab a b 9 p q P Q p jjk q P,Q p HjK q P=Q 30 I. p q q p a=0 ab=0 qp p q pq a=0 ab=0 ab=0 a=0 p jjk q q jjk p p q p q p HjK q q p a=b a =b a +b a+b a+b a +b xy=0 x=0 y=0 x+0 y+0 xy+0 xy+0 x+0 y+0 a b a b ab a=m- b=n- m n ab=(m-)(n-)=(mn-m-n+)- mn-m-n+ ab p jjjk q p HjjK q
p q px= qx = px= qx+=5 x= x = p 5555 q p jjk q x=- x = x+ q 5555 p p q p 5555 q q 5555 p p HjK q p q p q px 3 qx pa =b qa=b px=y=0 qx+y=0 px+0 y+0 qxy+0 p p q p<x< q q jjjk p q 3 <x< 7 4 r p jjjk r r0 x p<x< p q p r q r p 5555 q q 5555 r p q r P Q R p 3555 q q 3555 r P,Q Q,R P,R p q r P Q R p 3555 q q 3555 r P,Q Q,R P,R 3 p 3555 r p 3555 r ~r 3555 ~p p q 3 3 6 9 p 55 q 3 q 55 p q jjjk p p q a=-b a =b a+b p 55 q a=b a =b q 55 p q jjjk p p q x=y=0 x+y=0 p 55 q p jjjk q x=- y= x+y=0 x+0 y+0 q 55 p p q x+0 y+0 xy+0 p 55 q p jjjk q xy+0 x+0 y+0 q 55 p q jjjk p p q x +x x+ x+ x +x x =x x= x= x =x p q pa b qa=b px +y =0 qx=y=0 x y 0<x< x-a> x+a<5 a a -
~p p jjjk q p HjjK q 9 p p p p p p 555 q ~p 555 ~q ~q 555 ~p p 555 q p q q p p 555 q q 555 p p q 6 3 6 3 (Y) p q P Q Q,P q 55 p (Y) (Z) x = x=x + x+ (Z) p 55 q p q (Y) px=- q x = p 55 q q 55 p p jjjk q (Z) 6 3 6 3 p q P Q Q,P p 555 q x = x=x + x+ p 555 q p q p q px=- q x = p jjk q 3 I. p 55 q a=' b=-' a+b=0 a+b 360 ={x x} x >0 P P={x x x+0} P+ ={x x} x -3x+=0 Q Q={ } Q+u am=bm a=b a+b am+bm am+bm a+b x > x> x x x x
link x x >0 x x -3x+=0 a b a+b 360 a=b am=bm x> x > a b a+b> a> b> p q p-<x< qx<3 pbc qbc B=C pxy=0 qx +y =0 x y pç 'B= q,b 33 BC =70 B=70 C=40 BC B+C p 55 q B=C BC q 55 p q jjjk p p q x=0 y= xy=0 x +y +0 p 55 q x +y =0 x=y=0 xy=0 p 55 q q jjjk p p q Ç 'B= (Ç 'B)Ç =Ç =u ;BÇ =u -B=u,B p 55 q p jjjk q,b Ç 'B= q 55 p q jjjk p p q a b a b a+b a b a+b += p q P Q P={x -<x<} Q={x x<3} Q P - 3 x P,Q p 55 q q 55 p p q
0~ 4 3 4 % 3 7 % 4 % 7 %.4 0.7 3 +3-3 {x x } _3 {x x } 3 +3=5 _3=6 N a b a+b<n ab<n 34 I. +3-3 {x x } 3 3= 3 ' -' '+(-')=0 '-'=0 '_'= ' '= Z Y _3 {x x } 3
a b a Á b< Á + Y -3=- 3= 3 a + b a Y b a+b ay b 0 + Y 3 + 3 Y 3 + 0 3 Y R 0 R a b c identity element R a a+0=0+a=a a_=_a=a 0 0 a+b=b+a ab=ba (a+b)+c=a+(b+c) (ab)c=a(bc) a(b+c)=ab+ac (a+b)c=ac+bc 3(+'3 )+(+'3) =3+6'3++'3 =3++6'3+'3 =3++(6+)'3 =5+7'3 35 0 inverse element 36 I. R a a+(-a)=(-a)+a=0 a_ = _a= (a+0) a a -a (a+0) a -a a a a 3(+'3 )+(+'3) =3+6'3++'3 =3++6'3+'3 =3++(6+)'3 =5+7'3 0 3 + + Y 0 3 Y 3 + -3 0 3 Y - 3 ={x x=n- n} a- b- a b (a-)+(b-)=(a+b-) (a-)-(b-)=(a-b) (a-)_(b-)=(ab-a-b+)-< (a-) (b-)= a- b-
a -a (a+0) a -3 -(-3)=3-3 - 3 5 3 5 - ' -' ' ' = ' -' -(-')=-+' +' -' = -' 5 3 3 3 5 R a 0 a+0=0+a=a a=a=a a -a a+(-a)=(-a)+a=0 a a = a= (a+0) a a a -'3 -'3 -(-'3 )=-+'3 -'3-3 ' +'3 = =+'3 -'3 (-'3 )(+'3) 3 5 -' 3 B4 4 4 B4 a B4 4 b a b -+'3 +'3 37 B4 4 3 : 3 '6 æ : '= : 4 B4 '6 a= B4 4 '6 b= = '6 3 a b '6 '6 a_b= _ = 3 a b ac=bc c(c+0) c (ac)_ =(bc)_ c c a_{c_ }=b_{c_ } c c a_=b_ a=b a b c(c+0) ab=bc a=b
a b c a+c=b+c a=b a+c=b+c -c (a+c)+(-c)=(b+c)+(-c) a+{c+(-c)}=b+{c+(-c)} a+0=b+0 a=b {- } -_(-)=< -_ =-< _=< 38 I. a b c(c+0) ac=bc a=b a b ab=0 a=0 b=0 ={0 } R a b a b=ab+a+b a a e=e a=a e e x=x =e x a=0 a+0 a a ab=0 a _(ab)= _0 a a { _a}_b=0 a _b=0 b=0 ab=0 a=0 b=0 e a a e=e a=a a e=a ae+a+e=a (a+)e=0 a e=0 0 x x=x =e x=e x++x=0 e=0) x+=0 x=- - {- 0 } 0 '+ --' -+' ={x x=3n n} 3a 3b (a b 3a+3b=3(a+b) 3a-3b=3(a-b) 3a_3b=3 3ab a b a+b a-b 3ab
~3 B C 70 cm B 65 cm C 74 cm B 5 cm C 4 cm B -5 cm C +4 cm B C 0 -.5-0. -9.5-4.3 3.5 8.0 ( C) 5 C a a>0 a=0 a<0-9.5-4.3 -.5-0. 3.5 8.0 3æ -3 - - 0 3 39 a a>0 a=0 a<0 a>0 -a<0 a>0 b>0 a+b>0 ab>0 R R ' =R ; =u R ( ) ( ) (least upper bound)
a>0 b>0 a+b>0 ab>0 a a>0 a=0 a<0 a>0 HjK -a<0 a b c a>b b>c a>c a>b c>0 ac>bc a>0 b>0 a+b>0 ab>0 a>b b>c a-b>0 b-c>0 a a æ0 a-c=(a-b)+(b-c)>0 a>c a>b a-b>0 c>0 a a<0 a=0 a>0 a=0 a =a a=0 a>0 a =a a>0 ac-bc=(a-b)c>0 ac>bc a<0 -a>0 a<0 a =(-a) (-a)>0 a a æ0 a b c a>b a+c>b+c a b a<0 b<0 a+b<0 ab>0 a>b c<0 ac<bc a +b =0 a=0 b=0 40 I. a b a-b a-b>0 a-b=0 a-b<0 a-b>0 HjK a>b a-b=0 HjK a=b a-b<0 HjK a<b a b a>b a=b a<b a b c a>b a=b a<b a>b b>c a>c a>b a+c>b+c a>b c>0 ac>bc a>b c<0 ac<bc 4 a<0 b<0 -a>0 -b>0 (-a)+(-b)>0 (-a)+(-b)=-(a+b)>0 a+b<0 a<0 b<0 -a>0 -b>0 (-a) (-b)>0 (-a) (-b)=ab>0 ab>0 a<0 b<0 a+b<0 ab>0 a æ0 b æ0 a +b =0 a =0 b =0 a=0 b=0 a>b a-b>0 (a+c)-(b+c)=a-b>0 a+c>b+c a>b a-b>0 c<0 -c>0 (a-b) (-c)>0 -ac+bc>0 bc>ac a>b c<0 ac<bc x y xy xy=x+y xy [ x xy=x +y y x y xy=x=x+y x>y [ xy=y=x +y x=y=0 x= y=- x y x= y=- xy=y=x+y x<y [ x=y=0 xy=x=x +y x y x= y=-
a a a 3 =3-3 =3 a>0 a 0 a a<0 a a 0 a aæ0 a =a [ a<0 a =-a a = -a -4<0-4 =-(-4)=4 >0 = -4 + =4+=6-4 + (-4) 5 - -5 -'3 + -'3 (-4)=-8<0 (-4) = -8 =-(-8)=8 5>0 5 =5 a= -a + +a a- + a- + a-3-5<0-5 =-(-5)=5 5 - -5 =5-5=0 -'3<0 -'3 =-(-'3)=-+'3 -'3>0 -'3 =-'3 -'3 + -'3 =-+'3+(-'3)= 4 I. 0<a<3 a-3 + a+ -3<a-3<0 a-3 =-(a-3)=3-a <a+<4 a+ =a+ a-3 + a+ =(3-a)+(a+) =4 4 a= -a<0 -a =-(-a)=-+a +a>0 +a =+a -a + +a =-+a+(+a) =+3a a= +3a=+3 =7 a->0 a- =a- a-=0 a- =a- a-3<0 a-3 =-(a-3)=-a+3 a- + a- + a-3 =a-+a--a+3 =a a= a= a- + a- + a-3 = - + - + -3 = + 0 + - =+= 0 0 0 0 0 0 (006)
-<x x+ + x- x+ + 4-x x- - x-5 x <x 5 x>5 a a =a a b aæ0 bæ0 ab =ab= a b a<0 bæ0 ab =-ab=(-a) b= a b aæ0 b<0 ab =-ab=a (-b)= a b a<0 b<0 ab =ab=(-a) (-b)= a b ~ a b ab = a b aæ0 a =a a<0 a =(-a) =a a a =a a b ab = a b a a -a a a a -a a a>0 a =a>-a a<0 a =-a>a a=0 a =a=-a a b c a b a c a b b a a b c 43 x+>0 x- 0 x+ + x- =(x+)-(x-) =x+-x+=3 x+>0 4-xæ0 x+ + 4-x =(x+)+(4-x) =x+4+4-x=8 b a b>a a c c a c>a b c a c a>c b>a>c b a c x x- 0 x-5<0 x- - x-5 =-(x-)-{-(x-5)} =-x++x-5=-4 <x 5 x->0 x-5 0 x- - x-5 =(x-)-{-(x-5)} =x-+x-5=x-6 x>5 x->0 x-5>0 x- - x-5 =(x-)-(x-5) =x--x+5=4 '+'3 +'5 '+'3<+'5 a= a-'3 + a-'5 '5-'3 a b M(a b)= {(a+b)+ a-b } M(a b)-m(a-b b-a)+m(-a -b) 0
4 a b a Á b< Á R a b c a(b+c)=ab+ac (a+b)c=ac+bc R a a+0=0+a=a a_=_a=a 0 3 3= 3 (Y) 0 (Z) '- '+ = ='+ '- ('-)('+) (Z) ('3-)-(-)='3->0 '3->- (Y) -'<0 -'>0 -' + -' =-(-')+(-')= (Z) ={n+ n } m+ n+ m n (m+)(n+)=4mn+m+n+ =(mn+m+n)+ m n mn+m+n (m+)(n+)< 44 I. R a a+(-a)=(-a)+a=0 a_ = _a= (a+0) a a -a (a+0) a a '- '+ '3- - '3-<- -' + -' a -a a 3-' -(3-')=-3+' 3-' 3-' 3+' = =3+' (3-')(3+') (a-) + (a+b) =0 a-=0 a+b=0 a= b=-
={n+ n} 3-' (a-) +(a+b) =0 a b a>b>0 < a b -<a 3 a+ + a-3 link cm cm cm cm 3 cm 4 cm 5 cm a= b=3 c=5 a'c+b='5+3 b'c=3'5 (a'c+b)-b'c=('5+3)-3'5 =3-'5>0 a'c+b>b'c 7 4cm cm a cm b cm c cm a'c+b b'c a>b>0 ab>0 a-b>0 b-a - = <0 a b ab < a b a>- a+>0 a 3 a-3 0 a+ + a-3 =a+-(a-3) =5 45 (real number) (irrational number) (rational number) a a b(b+0) b (integer) 0 (natural number) 3 (complex number) a b i a+bi
5~9 (Heron) a b c h (a-b) h =c - a=8 b=4 c=5 h =-63 c b a h x =- i 3x-4=0 x -x-3=0 x -=0 x -x-=0 3x-4=0 x -=0 x +=0 x -x-3=0 x -x-=0 x = x= ' 46 I. (Cardano, G.) (Bombelli, R.) (Descartes, R.) (imaginary number) (Eular, L.) i (Cauchy,. L.) a+b' - (Gauss, K. F.) a+bi x +=0
i imaginary number 0 x =- - i i=' - i =- (i=' -) i =- x =- x=' -=i x=-' -=-i a b i a+bi a+bi a b a+bi a b c+di c d a=c b=d a+bi=c+di a b c d a+bi=c+di HjK a=c b=d a+bi=0 HjK a=0 b=0 a b -5i=a+bi a= b=-5 3+i 5-i 3 5i x y (x+)+(3x+y)i=0 (x+y)+(x-y)i=3-i a a+0i a+bi (b+0) a=0 bi (b+0) x y x+ 3x+y x+=0 3x+y=0 x=- y=3 x y x+y x-y a+bi [ a (b=0) bi (a=0 b+0) [ a+bi (a+0 b+0) x+y=3 x-y=- x= y= x=- y=3 x= y= x y (x+)+(y+3)i=0 i -4+i 0 +'3-3i - 5 47 48 I. (x-y)+i=x+yi x y 3+i 3 5-i 5 - x y x+ y+3 x+=0 y+3=0 = +0i 0 3 3 3 3 x=- y=- 5i=0+5i 0 5 x y x-y a+bi a b x-y=x =y x=- y= a+bi a=0 bi (b+0) 0 +'3 - i -4+i -3i i -3i 5 5 'å5= 5 " 3 =3 " (-3) =3
a+bi a b a-bi a+bi a + b i a + b i =a-bi 4-3i i - 3+i a b c d (a+bi)+(c+di)=(a+c)+(b+d)i (a+bi)(c+di)=(ac-bd)+(ad+bc)i a+bi a b a + bi =a+bi (4+3i)+(+i) (+i)(3+i) (4+3i)+(+i)=(4+)+(3+)i=6+4i (+i)(3+i)=6+4i+3i+i =6+(4+3)i+_(-) =(6-)+7i=4+7i 6+4i 4+7i a+b' a b (4-3' )+(-') (4-3' )(-') ' i (5+i)+(+3i) (3+4i)(3-i) (-3+4i)+(-3i) (-+3i) z=-5i z+z zz a b c d (a+bi)+(c+di)=(a+c)+(b+d)i i =- a b c d (a+bi)(c+di)=ac+adi+bci+bdi =ac+(ad+bc)i+bd_(-) =(ac-bd)+(ad+bc)i 49 V X 50 I. impedance I E Z E=I Z +3i 5-i X 4+3i -i - 3-i a+bi a b a+bi =a-bi a-bi=a+bi -b=b b=0 i 6-4' 4-0' i i i i - (5+i)+(+3i)=(5+)+(+3)i=6+4i (-3+4i)+(-3i)=(-3+)+(4-3)i=-+i (3+4i)(3-i)=9-3i+i-4i =9+(-3+)i-4_(-) =3+9i
B a b a=6+4i b=8+7i 4 ml 6 ml 7 ml 8 ml B B a b B a b a+b=(6+4i)+(8+7i) =(6+8)+(4+7)i =4+i b+a=(8+7i)+(6+4i) C C z z z =(8+6)+(7+4)i =4+i a b a+b=b+a z +z =z +z z z =z z (z +z )+z =z +(z +z ) (z z )z =z (z z ) z (z +z )=z z +z z (z +z )z =z z +z z 5 (-+3i) =(-) +_(-)_3i+(3i) =4-i+9i =4-i+9_(-) =-5-i z=-5i z =+5i z+z =(-5i)+(+5i)=(+)+(-5+5)i=4 zz =(-5i)(+5i)=4+0i-0i-5i =4-5_(-) =9 a b c d z =a+bi z =c+di z +z =(a+bi)+(c+di)=(a+c)+(b+d)i z +z =(c+di)+(a+bi)=(c+a)+(d+b)i =(a+c)+(b+d)i z +z =z +z I=+3i() Z=5-i(X) E=(+3i)(5-i) =5-i+5i-6i =5+(-+5)i-6_(-) =+3i (V) z z =(a+bi)(c+di)=ac+adi+bci+bdi =(ac-bd)+(ad+bc)i z z =(c+di)(a+bi)=ca+cbi+dai+dbi =(ca-db)+(cb+da)i =(ac-bd)+(ad+bc)i z z =z z
z =+i z =-i z +z =z +z z z =z z (a+bi)(x+yi)=(x+yi)(a+bi)= (ax-by)+(ay+bx)i= z +z =(+i)+(-i)=3+i z +z =(-i)+(+i)=3+i z +z =z +z z z =(+i)(-i)=4+3i z z =(-i)(+i)=4+3i z z =z z ax-by= ay+bx=0 a b x= y=a +b a +b a+bi a b - i a +b a +b z =-i z =-3i z =3+i (z +z )+z =z +(z +z ) z (z +z )=z z +z z z z+(-z)=(-z)+z =0 z_ = _z z z z+ =(z+0) z=a+bi a b -z (z+0) z -z=-(a+bi)=-a-bi a b = = - i z a+bi a +b a +b z z+0=0+z=z z_=_z=z a+bi a b (a+bi)+0=0+(a+bi)=a+bi (a+bi)_=_(a+bi)=a+bi 0 a+bi x+yi x y (a+bi)+(x+yi)=(x+yi)+(a+bi)=0 (a+x)+(b+y)i=0 a+x=0 b+y=0 x=-a y=-b C 0 a+bi -a-bi a b a+bi - i a +b a +b a+bi -i 5 I. -a-bi a+bi (a+bi+0) x+yi x y --(-)i=-+i -i - - i= + i +(-) +(-) 53 z =-i z =-3i z =3+i (z +z )+z ={(-i)+(-3i)}+(3+i) =(3-4i)+(3+i) =6-3i z +(z +z )=(-i)+{(-3i)+(3+i)} =(-i)+(5-i) =6-3i (z +z )+z =z +(z +z ) z (z +z )=(-i){(-3i)+(3+i)} =(-i)(5-i) =3-7i z z +z z =(-i)(-3i)+(-i)(3+i) =(--5i)+(4-i) =3-7i z (z +z )=z z +z z a+bi a b (a+bi)+0=0+(a+bi)=a+bi (a+bi)_=_(a+bi)=a+bi 0 0 a+bi a b a+bi a b x+yix y (a+bi)+(x+yi)=(x+yi)+(a+bi)=0 yy (a+x)+(b+y)i=0 a+x=0 b+y=0 x=-a y=-b x+yi=-a-bi a+bi -a-bi
54 I. z -z (z+0) z 3-4i +i a b c d (a+bi)-(c+di)=(a-c)+(b-d)i a+bi ac+bd bc-ad = + i c+di+0 c+di c +d c +d a+b'3 a b 4-3'3 (4-3'3 )-(-'3) -'3 '3 i a b c d (a+bi)-(c+di)=(a+bi)+(-c-di) =(a-c)+(b-d)i a+bi =(a+bi)_ c+di c+di c d =(a+bi)_{ - i} c +d c +d ac ad bc bd = - i+ i- i c +d c +d c +d c +d ac+bd bc-ad = + i c+di+0 c +d c +d z=a+bi a b -z (z+0) z -z=-(a+bi)=-a-bi a b = = - i z a+bi a +b a +b 3-4i -3-(-4)i=-3+4i 3-4i 3-4 3 4 - i= + i 3 +4 3 +4 5 5 +i --i +i - i= - i + + 5 5 a+bi a b x+yi x y (a+bi)(x+yi)=(x+yi)(a+bi)= (ax-by)+(ay+bx)i= ax-by= yy ay+bx=0 yy +> a x-aby=a yy _a +> b x+aby=0 yy _b +> (a +b )x=a x= y=- a b x+yi= - i a +b a +b a+bi a a +b a a +b b - i a +b b a +b yy i -'3 --'3 z z = z zz z (z+0)
a+bi (a+bi)(c-di) (ac+bd)+(bc-ad)i = = c+di (c+di)(c-di) c +d ac+bd bc-ad = + i c+di+0 c +d c +d - (-i)-(-3i) -3i +i 'i -'i ('i) =- (-'i) =- (-i)-(-3i)=(-i)+(-+3i) =(-)+(-+3)i =-+i -3i (-3i)(-i) -i-3i+3i = = +i (+i)(-i) -i --5i 5 = =- - i 5 -+i - - i 'i -'i x =- x - 'i -'i a -a 'ai ' -a='ai (+6i)-(3+i) (7+4i)-(3-3i) -i -4i +i a>0 -a 'ai ' -a='ai a+bi a b -i +i - { +i } +i -i -i -5 '5i -9 '9i= 3i ' -3='3i -' -49=-' 49 i=-7i z -5i+(-i)z=6+5i 55 56 I. -3-5 - - 9 5 (+6i)-(3+i)=(+6i)+(-3-i) =(-3)+(6-)i =-+4i (7+4i)-(3-3i)=(7+4i)+(-3+3i) =(7-3)+(4+3)i=4+7i +4i +4i = = -4i (-4i)(+4i) -6i +4i 4 = = + i 7 7 7 -i (-i) -i+i -i = = = =-i +i (+i)(-i) -i -i +i - +i -i (-i)(-i) (+i)(+i) = - (+i)(-i) (-i)(+i) -5i 5i = - =-i 5 5 +i (+i) +i = =i { } =i =- -i (-i)(+i) -i -5i+(-i)z=6+5i (-i)z=5+0i 5+0i (5+0i)(+i) z= = =5i -i (-i)(+i) ' - - ' - 'i
i ' -7 ' -36 -Æ - 4 -' -6 'ƒ-8 'å 3'i '3 + = + '3 ' - '3 'i ='3 'i+ ='6i-'6i=0 ' '3 i a>0 b>0 'a'b='aåb 'a =Æ a 'b b ' -3_' ' -3_' -5 ' -6 ' ' ' -8 ' -3_'='3i_'='3_'i='6i ' -3_' -5='3i_'5i='3_'5i =-'å5 ' -6 '6i = ='3i ' ' ' ' 'i = = =- i ' -8 '8i 'i '6i -'å5 '3i - i 'ƒ-48-'ƒ- (+' - )(+' -8 ) -' -3 'ƒ-8 + 'ß 'ƒ- '3 ' - i i i i ifi y n i«i n i«i =- i =-i i = ifi =i y k n=4k-3 i«=i n=4k- i«=- n=4k- i«=-i n=4k i«= '6=" (-)_(-3)=' -_' -3='i_'3i='6i =-'6 57 a -a 'ai ' -a='ai a>0 b>0 ' -a_' -b='ai_'bi ='ƒa_b i =-'aåb +" (-a)(-b) " (-)_(-3)=' -_' -3 '5 '3i 5i i i 3 5 i '7i 6i - i -4i 'ƒ-48-'ƒ-=4'3 i-'3 i='3i (+' -)(+' -8)=(+' i)(+'i) =+4'i+'i+4i =-+5'i -' -3 -'3i (-'3 i)_'3i = = 'ƒ- '3i '3i_'3i 3+'3i '3 = =- - i -6 3 x y (x+yi)-3(y-xi)=9+7i x=3 y=- '6-' - a+bi a b '6+' - '3 - i +i { } «-i +{ } «=- -i +i n
- a+bi 'ai 'ai 0 - i i = a b a+bi a b a b a+bi b=0 a=0 b+0 a b a+bi a-bi a+bi a -a ' -a= 3i=0+3i 3i 0 3 (Y) +' (+')i 0 (Z) 5-3i 5-3 i 5+3i(Z) (Y) - 'i(y) z=(+i)a-a+3-i =a+ai-a+3-i =(-a+3)+(a-)i z 0 a-=0 a= z 0 -a+3=0 a=3 58 I. 3i 3 0 (+')i 5-3i 5+3i - -'i (x+5)+(3y-6)i=0 x+5=03y-6=0 x=-5y= (x+3i)(-i)=(x+6)+(3-x)i =8-yi x+6=83-x=-y x=y= -i =-i -+'3i - = i -+'3i -'3i = --'3i
link z=(+i)a-a+3-i a z z -i (-i)(-i) -i = = =-i +i (+i)(-i) -i { }00=(-i) = +i x y (x+5)+(3y-6)i=0 (x+3i)(-i)=8-yi i -+'3i z=-4i z z z+z zz + z z z i+i +i +y+i { -i } +i ' -8='i ' -3=3i ' -='3i 'å ' ' -8+' -3 ' -- ' -3 '3 ='_'i+'3i_'3i- '3i =4i-6+i =-6+6i a+bi a b '' -8+' -3'ƒ-- 'å ' -3 59 z=-4i z =+4i z+z =(-4i)+(+4i)=4 zz =(-4i)(+4i)=4-6i =0 = = + i z -4i 0 5 + = z+z 4 = = z z zz 0 5 i i =-i =-ii = k i 4k-3 =(i ) i= i=i i 4k- =(i ) i = (-)=- i 4k- =(i ) i = (-i)=-i i =(i ) = = i+i +i +y+i =(i--i+)+y+(i--i+)=0 z=a+bi a b P(a b) a x (a 0) bi y (0 b) x y O z=a+bi P OP z z z = a+bi =" a +b z z=a+bi P OP r OP x h z=r(cos h+i sin h) h z
B ;B=u B ;B=B; 'B=B' (;B);C=;(B;C) ('B)'C='(B'C) ;(B'C)=(;B)'(;C) '(B;C)=('B);('C) (;B)Ç =Ç 'BÇ, ('B)Ç =Ç ;BÇ a b a Á b< Á 0 a -a a a+0 a 3 a b a>b HjjK a-b>0 a=b HjjK a-b=0 a<b HjjK a-b<0 p p p ~p p 555 q p q P Q p 555 q HjK P,Q p 555 q HjK P Q 3 p q q p ~p ~q ~q ~p a b c d a+bi=c+di HjK a=c b=d a b a+bi a +b i =a-bi 3 a b c d (a+bi)+(c+di)=(a+c)+(b+d)i (a+bi)-(c+di)=(a-c)+(b-d)i 4 (a+bi)(c+di)=(ac-bd)+(ad+bc)i p jjk q p q q p a+bi ac+bd bc-ad p HjK q p q = + i c+di+0 c+di c +d c +d 60 I. 6 4 -a (aæ0) a =[ a = -a a =a -a (a<0) a=0 b= c= abc=0 -(B-C)Ç =-(B;CÇ )Ç a=b=c=0 =;(B;CÇ ) a=b=c=0 abc=0 =(;B);CÇ abc=0 a=b=c=0 =(;(BÇ )Ç );CÇ a +b +c =0 a=b=c=0 =(-BÇ )-C a=b=c=0 a +b +c =0 a +b +c =0 a=b=c=0 p 555 q x< x-a<3 PQ a+3 P={x x<} Q=[x x< ] P,Q x< x-a<3 a+3 æ aæ- a'b' ab a'+b'=(a+b)' a'-b'=(a-b)'
0 B C -(B-C)Ç =(-BÇ )-C 07 +i { } +'3i { } ' 0 x x-a3 a 08 z=-i z- + z - z z z z 03 a b c abc=0 a=b=c=0 09 B B=('B);(Ç 'BÇ ) a +b +c =0 a=b=c=0 B B 04 ={a' a} 0 ={ 3 4} B={ 3 6 8} B 05 R ab=a+b+ ()(-3) 3 0 979 005 06 6 I. (+i)z+(4-z i)=0 z z z 63 a+ba-b ' '_'= ' '= 3 x 3x=x3=e 3+x+=- x=-7 ()(-3) =(++)(-3) =5(-3) =5+(-3)+ =4 e a ae=ea=a a+e+=a e=- z=a+bi ab zæ=a-bi (+i)z+(4-zæi) =(+i)(a+bi)+{4-(a-bi)i} =(a-b+ai+bi)+(4-b-ai) =(a-3b+4)+(a+b)i=0 a-3b+4=0a+b=0 a=-b= z=-+i
+i { }=i ' +i { } +i =[{ } ] fi =ifi ' ' =(i ) i =- +'3i -+'3i { }= +'3i +'3i +'3i { }3={ }{ } -+'3i +'3i = _ =- +'3i { } +'3i =[{ }3]33_ = --'3i +'3i ={ 3 4} B={ 3 6 8} ;B={4} B={468} p 555 q z=-i z =+i z+z =zz = z- z - (zz -z )+(zz -z) + = z z zz zz = -(z+z ) zz - = = B=('B);(Ç 'BÇ ) =('B);(;B)Ç =('B)-(;B) B B (Hilbert, D.) 4 y
3 0 9 n( n 0) n (strike) (ball) 37 3 3 3 3 4 5 6 7 8 9 0 3 4 5 6 7 8 9 0 9 6 B 30 4560 96 5700 6 6 7 0 5 64 I. 6 5 6 5 5 65 6 5 65 8 (Euler, L.) 970 (number place) 984 9 8 9 3 9 3 8 6 4 7 9 5 4 9 5 8 6 7 3 5 7 3 9 8 4 6 5 3 9 7 4 6 8 6 7 4 8 5 3 9 9 8 6 4 3 7 5 7 9 3 5 6 8 4 8 6 9 4 5 3 7 4 3 5 7 6 8 9
68 I.
5 3 3 3 9 6 8_ 6 7_ 5_ 3_ 4 0 6 _ 9_ 4_ 4 8 8 8 3 0 9 7 5 6 4 0 6 0 5 6 3 7 9 3 4 6 4 8 8 8 9 4-5+5i 5 % 5 % 69 5 % 9 3 6 8 7 5 3 9 4-8 -3-4 6i i 8i - -5-9 7i 5i 3i -6-7 - i 9i 4i 3 0 % -8+6i -3+i -4+8i -+7i -5+5i -9+3i -6+i -7+9i -+4i 6+ + 8+ 8 3 0 7+ 5+ 3+ 9 7 5 + 9+ 4+ 4 6 6i i 8i -5+6i -5+i -5+8i 7i 5i 3i -5+7i -5+5i -5+3i i9i 4i -5+i -5+9i -5+4i 35 %
B ('B);(Ç 'B) (-i )z +iz=3-i z z z 5+i 3-5i 7-i B Ç -+3i -3-i BÇ -B ={x x 0 } B -B={3 6 7} Ç ;BÇ ={8 9} B 4 6 8 0 p q r p 555 q ~r 555 ~q r 555 q p q q p q r x +x x+ -i z= +i +z+z +z +y+zfi a b ={a a=«n } a b ab=ab-(a+b)+6
('B);(Ç 'B)=(;Ç )'B =u'b=b a-3b=3 a-b=- a=-3 b=- z -3-i ={ 3 4 5 6 7 8 9} Ç ;BÇ =('B)Ç ={8 9} 'B=-('B)Ç ={ 3 4 5 6 7} -B={3 6 7} B=('B)-(-B)={ 4 5} B x+ x +x x=0 x =x x= x=0 x= x =x ={ y} + =4+8= - =4-8=-4 μ «m n μ _«= m+n m+n m+n < 0 z z=a+bi a b z =a-bi (-i)z +iz=(-i)(a-bi)+i(a+bi) =(a-3b)+(a-b)i (a-3b)+(a-b)i=3-i p q r P Q R p 3 q P,Q ~r 3 ~q RÇ,QÇ Q,R r 3 q R,Q P,Q=R p q q p q r -i z= =-i +i +z+z +z +y+zfi =+(-i)+(-i) +(-i) +y+(-i)fi =+(-i)+(-)+i+y++(-i)+(-) =-i -i e ae=ea=ae-(a+e)+6=a e(a-)-3(a-)=0 (e-3)(a-)=0 e=3 x x=x=x-(+x)+6=3 x= 50 % 50 %