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! B , : - ' ! B . % '%*" &!": p j q jj :p j p ^ q j p _ q j p ;! q j ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;; j jj j j j j j jj j j j j j jj j j j j j jj j j j j (, &! - '% !% + ^ _ :, ". &! , _ ' # "", .. p _ q , " +*+ + . % "" "% #, .. " 7" ", 7. # # "" &' -.
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F (p1 : : : pn) (F ) 2 B '%*"' ': (1) F = pi , (F ) = i (2) F = (), (F ) = () (3) (A) (B) '- , (:A) = :(A) (A ^ B) = (A) ^ (B) (A _ B) = (A) _ (B) (A ;! B) = (A) ;! (B):
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'# F (p1 : : : pn ) A1 : : : An { ) . ( '#" F $"' A1 : : : An " "+ p1 : : : pn , $"', &" F (A1 : : : An), '%*" ": (1) F (p1 : : : pn ) = pi , F (A1 : : : An) = Ai (2) F (p1 : : : pn ) = (), F (A1 : : : An) = () (3) F (p1 : : : pn) = :G(p1 : : : pn) G(A1 : : : An) , F (A1 : : : An) = :G(A1 : : : An) (4) F (p1 : : : pn) = G(p1 : : : pn) H (p1 : : : pn), # & ^, & _, & ;!, G(A1 : : : An) H (A1 : : : An) '- , F (A1 : : : An) = G(A1 : : : An) H (A1 : : : An). ! 2.2. q1 : : : qn { , A1 : : : An 1 : : : n 2 B . , Ai (1 : : : n) = i 2 B . 2.1.
F (A1 (1 : : : n) : : : An(1 : : : n)) = F (1 : : : n):
# "- &# 2.1 "* '! % $"'. & 1. F (p1 : : : pn) G(p1 : : : pn), A1 : : : An { , F (A1 : : : An) G(A1 : : : An). & 2. A1 : : : An B1 : : : Bn { , Ai Bi i = 1 : : : n, F (A1 : : : An) F (B1 : : : Bn). 2 1 & # " 1 + (2.1) (2.2) (2.2) (2.3), - " 2 + (2.4) (2.5) (2.7) (2.8). 2 2 # # " 2 + (2.5) (2.6).
, " '- 0 $"# # , "- # # "* &! , - , " 2, # !. ", $"# # " 1.6 - & &* '!. "/ 2.3. ,- B, " & ! ^ _, " ! : 0" 2 B , "% " - " 1.6, & 7' &' &. (" &" "- B !", " 1.2, &' &. " 0 '* '% ' &' &, ", & P (A) + "- "- A. 12
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- P (x) { " " x & " P " { 0 " . " , " " " " x "- &# & ", & -". ",5 P (x) { "x { ", P (2) = P (6) = P (224 + 1) = P (22 + 1) = (0 ' . /). , " &# M , " P (x) "- " # $'!% M "- B = f g. ( $'! # . "' # #+ "+. "/ n n-"" " "- M &- P : M ;! B . ,- M " &#%, ", "%* M , { "" "". ,- P = f< x1 : : : xn >2 M n j P (x1 : : : xn ) = g &#% P . "" , %& C M n '* ' n-" P , , P = C , ", P (x1 : : : xn ) = ()< x1 : : : xn >2 C: 2.
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'& " # 0-" , - % " B . 0", 0-" P = , % , &# , P = , &# f g. 1 ", %& ' "- M % , M 0 = f g. 2 " ", "- '# . 0" '%* ! 2.5. P (x1 : : : xn) Q(x1 : : : xn) { n- M , (1) P ^Q = P \ Q (2) P _Q = P Q (3) :P = M n n P - %, + + "+. + " ' , * + "+, +*+ + . '# & "+, +*+ Q # x1 : : : xn y1 : : : yk , , + Q , # P (x1 : : : xn). ' P " P 0 (x1 : : : xn y1 : : : yk ), , P 0 (x1 : : : xn y1 : : : yk ) = () P (x1 : : : xn) = . , P 0 ' P & " $ + "+ y1 : : : yk . , P = P M k . '% !% ", +*" Q ", '", Q , *+ + + - "+, & "- # # - 0
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'# P (x) 1 x ^ x 3 Q(y) 2 y ^ y 3 { , R+. 6"" R(x y) = P (x) _ Q(y) S (x y) = P (x) ^ Q(y). P 0 ' P & " $ " y, Q0 ' Q & " $ " x. ( P = P R+ (.1), Q = R+ Q (.2). ( # R = P Q S = P \ Q &- '+ 3 4 . "/ 2.6. '# P (x0 x1 : : : xn) { (n + 1)-" "- M . , n-" Q(x1 : : : xn ) ' P "" '* ( &*) " x0 1' Q(x1 : : : xn) = 9x0 P (x0 x1 : : : xn) (Q(x1 : : : xn ) = 8x0P (x0 x1 : : : xn )) %&+ u1 : : : un 2 M Q(u1 : : : un) = # , ( %&) u 2 M , ( ) P (u u1 : : : un ) = 1.
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Q(x1 : : : xn) = 9x0 P (x0 x1 : : : xn)) &# Q M n Q & P M n+1 P . / ' &" "" " ! ', M = R { ". . 5. ! 2. '# P (x y) y = x2, P R2 # & & XOY . ! # OY # -# '# (.6), .. 9x(y = x2 ) = y 0, "- ' # '+ &- { * y " * x # , !#. ! & # OX #%. % ', 9y(y = x2 ) . / - , #' %& x "- . ! 3. 6"" +" P (x y z) x + z = y R+ . &# # # x + z = y , - -#" OXY Z (.7). ! 0 -# c XOY &- .8. , 9z (x+z = y) = y x, '+ &-: y x # , y ; x 0. ."", # '* #' $, " &# !#+ . - " "# " R, &# & '- % # x + z = y, ! # OZ #% XOY , .. 9z (x + z = y) , - , #' 0" ' z "- # y ; x. &* " " !, "-' " &* '* " #, %*, , # & , '+ "" &*. $ 2.7. P (x) { , x , , - ! . "
:8xP (x) = 9x:P (x) :9xP (x) = 8x:P (x)
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' & " & (1) "- " # & ,. "" , " &# , .. M = fa1 a2 : : : ang, 8xP (x) = P (a1 ) ^ P (a2 ) ^ ^ P (an ) 9xP (x) = P (a1 ) _ P (a2 ) _ _ P (an ) (1) + :(P (a1 ) ^ P (a2 ) ^ ^ P (an )) = :P (a1 ) _ :P (a2 ) _ _ :P (an ) :(P (a1 ) _ P (a2 ) _ _ P (an )) = :P (a1 ) ^ :P (a2 ) ^ ^ :P (an ) % & & ,. ! &# 8y(x2 ; y2 1) ^ 9x(x2 + y2 1): +*+ 3%!% # " x. &# # & :8y(x2 ; y2 1) = 9y:(x2 ; y2 1) = 9y(x2 ; y2 > 1): &# x2 ; y2 > 1 1+ .9. ! # OX # ;1 x 1. / # &# 8y(x2 ; y2 1). &# 9x(x2 + y2 1) # ! # OY ' x2 + y2 1 (.10), .. ;1 y 1. ( #, '- -, " 1, ", &# 1 # ;1 1] ;1 1] (.11). 4.
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(1) QxP (x) = QtP (t), t P (2) :8xP (x) = 9x:P (x) :9xP (x) = 8x:P (x) (3) 9x(P (x) _ Q(x)) = 9xP (x) _ 9xP (x) 8x(P (x) ^ Q(x)) = 8xP (x) ^ 8xQ(x) (4) 9x(P (x) ^ R) = 9xP (x) ^ R 8x(P (x) _ R) = 8xP (x) _ R, R x (5) QxQyP (x y) = QyQxP (x y) . 6 (1) (5) '% . 6 (2) " 2.6. -" (3). 9x(P (x) _ Q(x)) = , '* ' m 2 M , M { " &#, , P (m) _ Q(m) = , . . P (m) = Q(m) = ( 1 , #+ "+, +*+ P Q $ ). " ' 9xP (x) = , # 9xP (x) _ 9xP (x) = . ' " . '#, &, 9xP (x) _ 9xP (x) = . / , & 9xP (x) = , & 9xQ(x) = . " ' '* ' m 2 M , ' ' % P (m) = , # P (m) _ Q(m) = , .. 9x(P (x) _ Q(x)) = . ' " . (3) " " $"# , "', 0 & # " 2.6. ."" " (3) P :P , Q { :Q "" &" " ' !. '"
:9x(:P (x) _ :Q(x)) = :(9x:P (x) _ 9x:P (x)) ' "" (2), { ,. '" 8x:(:P (x) _ :Q(x)) = :9x:P (x) ^ :9x:Q(x) " ' , !, -"' 3%! { 22
(2) !, '" &'" . -" # (4). '# # , '* ' m 2 M , P (m) ^ R = , .. P (m) = R = . ( # , 9xP (x) = , .. # (4). &, # (4), 9xP (x) = R = . 2 #, '* ' m 2 M , P (m) = , P (m) ^ R = , #' R m. % ', # (4). (4) $"# "', 0 & (3) & . R x, QxR = R. ! 6 (1) , " x - QxP (x) '% - # '"" " . ", x "- &# " # "'%, +*'% P '* . ", 9x(y < x) # 9t(y < t), # 9y(y < y). -, < ' 1 '" "- , " , ", + ' + "- +, "%*+  0". - &# R2 # " : f (x t)dx # $'!, .
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#, '* 3%!%, &* { 3%!% " # "# "". (" " " " ! 3.2. #" $ : (1) 9x(P (x) ^ Q(x)) ;! 9xP (x) ^ 9xQ(x) (2) (8xP (x) _ 8xQ(x)) ;! 8x(P (x) _ Q(x)) (3) 9x8yP (x y) ;! 8y9xP (x y) = " ! (1) , '* ' m 2 M , P (m) ^ Q(m) = , .. P (m) = Q(m) = , .. 9xP (x) = 9xQ(x) = . % ', % " ! (1) . " !% (2) "- # , "- $"# " ! (1) ( ' -!). " ! (3) , '* ' m 2 M , 8yP (m y) = . ( # n 2 M P (m n) = , # n 2 M 9xP (x n) = , 0 , 8y9xP (x y) = .
23
& (1) { (3) " ! , , , -. "" , ", , " ! (1) '* m, &*%* ' P , Q. " % '* m, &*%* ' P n, "- m, &* ' Q. '#, ", M = N P (x) x = 2 Q(x) x = 3. ( , 9xP (x) ^ 9xQ(x)) ;! 9x(P (x) ^ Q(x)) = 4& # -# " !, & (2) " M P -, 1, Q(x) x =6 2. ( , (8x(P (x) _ Q(x))) ;! (8xP (x) _ 8xQ(x)) =
, # P "- # %& - , Q { !. , & # " " !, & (3), "", (3) , '* ' 0" x, &* P ' %&" y, % { - y '* ', &* , 0" x, &*%* P '. '# M = N, P (x y) y < x. ( ,
8y9xP (x y) ;! 9x8yP (x y) = #' N - 0" " ,  0". (" " " !, & (3). 4& "# " . "# , %& n-" "- - # &#% , %* & "- M n, M { " &#. , " - % #" "- " M . 0" " A(x) 1' x 2 A #'% *": (9x 2 A) B (8x 2 A) B
9x(x 2 A ^ B) 8x(x 2 A ;! B)
'! # "," "- " A, , , " ". 24
!
3.1 3.1 , - ! . # 0 - "- $"#, #' + , - " 3.1 - 3.2 (' -). ! 3.4. P (x y) { , x y , , - ! . 3.3.
(8x 2 A)(9y 2 B ) P (x y) = (9f 2 B A )(8x 2 A) P (x f (x)) . (&' # # " ! A ;! B B ;! A, A B { " . " ! ; : -" $ " x y ' # f (x). # " ! ;! "", , , -" $ " a 2 A "- Ba = fb 2 B j P (a b) = g 6= . &" - "- Ba 0"' ba " f : A ;! B , - %& a 2 A f (a) = ba . ( , %&" a 2 A P (a f (a)) = , .. (8x 2 A) P (x f (x)) = .
(" &" % " ! ."", # - 3.4 # # . #, "" - 3.4 (, " ! ;!) # " &. )"' ' " & & &'- "- 6].
,
4
6"" # # " ""+ - $"! '- "* + - , "+ '* !. 1 . * "" 1 &', ' "" ' "# , $"' ! %& - '- "# . " 0 # " $'!. - Qx 2 (0 +1) &'" # Qx > 0. , , xlim !a f (x) = b, '%*" &".
8" > 0 9 > 0 8x(jx ; aj < ;! jf (x) ; bj < ") 25
(4.1)
# " +" $"' # - xlim !a f (x) 6= b, .. ! (4.1), , - " (2) " 3.1 -" 3.3, 0 ! 1 9" > 0 8 > 0 9x:(jx ; aj < ;! jf (x) ; bj < ") (4.2) ( # # 1# - " :(p ;! q) p ^ :q ", : (jf (x) ; bj < ") jf (x) ; bj ", '" #'% $"' ' ' - xlim 6 b: !a f (x) =
9" > 0 8 > 0 9x (jx ; aj < ^ jf (x) ; bj ") 2 . # (1) " a { y, " b ; f (y), '" $'! f y. 2 # # $'! f & 1 : 8y8" > 0 9 > 0 8x(jx ; yj < ;! jf (x) ; f (y)j < ") (4.3) "", " # f '%*" &": 8" > 0 9 > 0 8y8x(jx ; aj < ;! jf (x) ; bj < ") (4.4)
# 1# " ! (4) - 3.2, "" (4.4) "" 8y 9 > 0. 0 "- "# "" * " " 8y 8" > 0 (- (5) " 3.1). '# " " (4.3). (" &" " ! (4:4) ;! (4:3), ' -, " $'! ' # - . ( - $"# # & " ! + "', " !, & " ! (4) - 3.2 - (". !% 3). #, " $'!, + - & , " + ( ", $'! y = x2 " R). (" ", $'!, - "' , " 0" '* #' "' . 3 . A { S " S"- , &3" A "- , &" A A, ' x2
A2A
A () 9A 2 A (x 2 A) 26
(4.5)
" " A "- T A T A, A2A ' \ x 2 A () 8A 2 A (x 2 A) ( # - (2) " 3.1 " '% , &+ &3 :
A2A
A=
\
A2A
\
A
A2A
A=
A2A
A
- (4) " 3.1 { &' &+ &3 :
! A2A
A \B =
A2A
(A \ B )
\ ! A2A
A B =
\
(A B ):
A2A
'# A = fAi j i 2 I g. ( S A (T A) % * S Ai i2I T ( Ai ). ( # - (3) " 3.1 % '%* i 2I "- :
i2I
(Ai Bi ) =
! ! \ i2I
Ai
i2I
Bi
i2I
(Ai \ Bi ) =
\ ! \ ! i2I
Ai \
i2I
Bi :
'%* $"# ' -. , "" " $"# * -"- . '# f : X ;! Y A { " "- X . ( " " : ! f
A2A
A =
A2A
f (A)
(4.6)
# (4.6) "", % & "- &- y2f
! A2A
A 9x(x 2
A2A
A ^ y = f (x)
' (4.5), - 0 # -% 9x(9A 2 A(x 2 A) ^ y = f (x)) (4.7) 27
#' y = f (x) - " A 9A 2 A "- 1 & "# "" "" " 9x. #' 9x(x 2 A ^ y = f (x) y 2 f (A) (4.7) # -% 9A 2 A(y 2 f (A)) y 2 f (A) A2A
(4.6). 4 . '# f : X ;! Y . ( ' - ", f { %3 1 8y 2 Y 9x 2 X (y = f (x)). ' - 3.4 - # '%*"': 9g 2 X Y 8x 2 X (y = f (g(y))). - 8x 2 X (y = f (g(y))) , f g = idX , .., g { & f &-. (" "" " ", &- %3 # , " &. 5 . . 1" $"# '% "' ", - ' "- N " "#1 0": 8A(A =6 ;! 9x(x 2 A ^ 8y(y 2 A ;! x y))) (4.8) #' (4.8) %& "- A, " "- "# A :A. " #'" " - " p ;! q :q ;! :p (4.9) . '# (4.8) # '%*"' -%: 8A(:9x(x 2 A ^ 8y(y 2 A ;! x y)) ;! :A =6 ) (4.10) &'" ' " ! (4.10), # 1# '%*" "" - " : :(p ^ q) q ;! :p '", (4.10) # 8A(8x(8y(y 2 A ;! x y) ;! :x 2 A) ;! :A =6 ) (4.11) #'" '%*" " 0 ": :A =6
A = N, :x 2 A x 2 A x y y < x, - "" "!, * " 8y - (4.9). '# '", (4.11) # 8A(8x(8y(y < x ;! y 2 A) ;! x 2 A) ;! A = N) (4.12) 28
, (4.12) # , ! "" '! '%* $".
" # x , , $ x, A , x A, A.* #' " , (4.8) # (4.12), " "" , ' - ", ' "- '# " "#1 0", # ! ' "" '!. ."", 0+ '-+ " # # !$ "- N. (" &" 1 " '% '%* &&*. "", ' "- ' ", - ' "- " "#1 0". , $"# (4.12) (4.8) ", "# N # ' "- . (" ""
#" ' " "- M '%* ' -:
8A (8x(8y(y < x ;! y 2 A) ;! x 2 A) ;! A = ) / ' - ! " $ '!. 2 " 8", - "- "- &# ' . (" &" ! '! "- "# # + ' - # # + "- , # #+ &+ "- . 2' "# ', " 8" 0$$ { # " &. , + ', "- ' # "- R, # + & ' "-". / '* "- " ! $ '! + "- . 7 & &'- " 8" '+ ' - #+ " & "- 6]. M
M
, #)# ,. 1 /
/ 2 x " /1 5
II.
1.
.
" '- , 1 $" -, "* + , "' A
*
,
29
&# + + , '+ -. 2 ' "% -, (-) " , + +. 4& # 0' '!% """ "" &+" # $"# -, '% "* + , - { $"# - + -. 0" , $"#" - - ' # 1# & , 0" &" - &# . '* , 0" $"' '* & & -, " , '% $"'" . -, " +# ", "", "" & * & $'!. 6'", - n;" $'! f (x1 : : : xn) ' n + 1-" P (x1 : : : xn xn+1) f (x1 : : : xn) = xn+1. , 0" " #, & $'!%, 0 '* " '-" $"#+ -. ", $, P (x y) = () y = f (x) $'! f : 8x 9yP (x y) ^ 8y8y0(P (x y) ^ P (x y0 ) ;! y = y0 )] :
- * + &+ &# 8!yP (x y) # "'* ' " y, , P (x y). & # - # + " " $'!#+ %, "% -", $'! % '" '. " &, ", " $'!#+ " + #'% +" " S (x y z ) (x y z ) . ( - 9x(x2 + px + q = 0) - &# '%*" &": 9x8u8v8w8t((x x u) ^ (p x v) ^ S (u v w) ^ S (w q t) ;! t = 0) 2' "# ', $"#" + $'! % f (x y), .. " x + y x y ' # +(x y) (x y) , . " 1 - 1 9x +(+((x x) (x p)) q) = 0] : -, , ", ' '% #, &'" # # &* $" , #', , 0 '"". 30
+" % "- S + " . "/ 5.1. ,- S + " % & '%* &3 (1) C & " x1 : : : xn : : : & "+ ( "+ &'' ' &# ' &' ). (2) "- C " & . (3) ,- $'!#+ " F $ " &-" ' : F ;! N n f0g. 0" '(f ) = n , f { " $'! n "+ (n-" $'!). (4) ,- + B = f g. (5) ,- + " P $ " &-" : P ;! N nf0g. 0" # (p) = n , p { n-" " . (6) " , & : ^ _ : ;! 8 9 = ( ) , ( =< C F P > ' . ' & L . ,*#% ' { jj "*# "- C F P . !# 1. " (0" "- C ) "- " # 0-" $'!# " . ( - "- " # 0-" " . 2. "", $ , "- " , &, "- " "* ""' $ '. #' " - $"' L % & " " $ S , "*# "- + $"' L + maxf@0 jjg. "/ 5.2 ($ 1.
L). ("" L % $ S , '% '%*" " (1) - " c { ". (2) - " " xi { ". (3) f { n-" $'!# " , t1 : : : tn { ", f (t1 : : : tn) { ". (4) " "- &# ' " 1 { 3. (" % & "-+ $'!, '% $'!, "%*+ & "- F , "* .
31
' !, - "+ " "+. "/ 5.3 (3 1.
L ). )"'" L % $ S , '% '%*" " (1) t1 : : : tn { ", p { n-" " , p(t1 : : : tn), ti = tj - { $"' L . (2) A B { $"' L , :A, (A ^ B), (A _ B), (A ;! B), 8xA, 9xA { $"' L (3) $"' L "- &# ' " 1 { 2. )"' (1) % 0"" "". +- " x $"'' A ", +-" $"'' 8xB 9xB. " ' &". )"', "%* &+ +- "+, -". !. 8x(x = y + z ;! 8z(x < z ;! 8y(:x y = z))). .# +- x , +- y z &, # . "/ 5.4. '# =< C F P > { '. & " M ' < M i >, M { "- , " " " M i { &-, * -"' c 2 C 0" i(c) 2 M , -"' n-""' $'!#"' " ' f 2 F { &- i(f ) : M n ;! M , -"' n-""' "' " ' p 2 P { n-" i(p) : M n ;! B . F = , M "#%, P = , M (' #) &. ,*#% " M "*# . !# 1. #1", 0 '"", c i(c), f i(f ), p i(p) - . 2. 6'", '% &'% "' "- # "#, + n-"+ $'! '%*" n + 1-"" " , & 1. ' + &+ " 0" #'% - + 1 -. &*+ - "+ 0 + &' # #. !. 1. ';1 "- " # & " ' < feg f g >. ( " ' 1' 1):8x8y8z (x (y z ) = (x y z )) 2):8x(x e = x ^ e x = x) 3):8x(x x;1 = e ^ x;1 x = e): .
32
'&# ' ;1 , # " 1 8x9y(x y = e ^ y x = e). !, "- '&# ' e. ( " 1 9y8x(x y = x ^ y x = x), # 8x9y8z (z (x y) = z ^ z (y x) = z ^ (x y) z = z ^ (y x) z = z ). 2. "- # "# ' =< fg >. " % - L '%*" &": 1):8x x x 2):8x8y(x y ^ y x ;! x = y) 3):8x8y8z (x y ^ y z ;! x z ) 1 { 0 1 , ' %* * ": 4):8x8y(x y _ y x): ' "- " % - "# ' 0 =< f. " % - L '%*" &" 10):8x :x < x 20):8x8y8z (x < y ^ y < z ;! x < z ) " 0" ' " : 30 ):8x8y(x < y _ y < x _ x = y): " 1)-3) 0 "" 1')-2') '%*" ". x < y "- &# - '%* $"' L : x < y x y ^ :x = y: # " 1')-2') # 0' $"'' " <, , #'# " - " , #, - 1)-3) # -" 1')-2'), - 1)-4) # -" 1')-3'). &, L x y $"' x. 0" "- & # !% {, # ;1 , 0
0
0
33
#' 0 ! %' . "" " : 1):8x8y(x + y = y + x ^ x y = y x) 2):8x8y8z ((x + y) + z = x + (y + z ) ^ (x y) z = x (y z ) 3):8x8y8z (x (y + z ) = x y + x z ) 4):8x(x + 0 = x ^ x 1 = x) 5):8x9y(x + y = 0) 6):8x(:(x = 0) ;! 9y(x y = 1) + p, p { { 0 , ' %* " AChar(p) : 1| + {z + 1} = 0: p
& # + 0 &' & # "- " AChar(0)n :
: 1| + {z + 1} = 0: n
.# n = 2 3 : : : . ( -, , & # & "' , &' & # "- " ACln:
8a08a1 : : : 8an;19x(xn + an;1xn;1 + + a1 x + a0 = 0)
n = 1 2 : : :
!. 7 , " - &# & ", '%* "' "'. 4. { 0 & " &* ' < f0 1g f+ g fg >. " ' { 0 " (" 1 { 6 " 3), " (" 1 { 4 " 2 * ", %* 1 " !":
8x8y8z ((x y ;! (x + z y + z ^ (0 z ;! x z y z ))) : 2 ' + % * "' . * "'" ' , ", - 34
" " + & #, - -# 0" ". 4& # * "' '- "" ' & # " ACln + + n "' 8x(x > 0 ;! 9y(y = x2 )). 5. " K { 0 ' # & ' < f0g f+g ff j 2 K g >, f { " ! '"- . " { 0 " & ' (". " , ! -) & "- " 8x8y(f (f (x)) = f (x) ^ f (x + y) = f (x) + f (y)) "-+ 2 K . ,*# 0 "- " "* K . &+"# '# "- " 1" + ' " "+, &%*+ , #, "-#% "# ". " ' '& , '% &&* ', ', " # "+, - + " " "- . / " '-" $"' , '* ' '# +%. 0"' # &&* " % (2$"' # 0 ' { ' -).
"/
,
6
'# M1 = (M1 i1), M2 = (M2 i2) { & " ' . 7 &- ' : M1 ;! M2 "$"" &+ ", (1) %& " c 2 C '(i1 (c)) = i2(c) (2) %& n-" " f 2 F %&+ m1 : : : mn 2 M1 " " 6.1.
'(i1 (f )(m1 : : : mn)) = i2 (f )('(m1 ) : : : '(mn )) (3) %& n-" " p 2 P %&+ m1 : : : mn 2 M1 "
"
i1 (p)(m1 : : : mn ) = i2(p)('(m1 ) : : : '(mn )):
, "$" ' , !, , - &- & ' + "- % " '" . 35
"/
'# M = (M i) { & " ' . "- M1 M " "'", - 0" i(c) 8c 2 C "' # + ! i(f ) 8f 2 F . ."' "- M1 M &'% "' M1 = (M1 i1), 8m1 : : : mn 2 M1 8f 2 F 8p 2 P 6.2.
i1(f )(m1 : : : mn ) = i(f )(m1 : : : mn ) i1(p)(m1 : : : mn ) = i(p)(m1 : : : mn) : / " " " M. ", +* " t + x1 : : :xn , &'" # t = t(x1 : : :xn ). "/ 6.3 (/ 21 ). '# M = (M i) { & " ' . -"' "' t = t(x1 : : : xn) &- i(t) : M n ;! M '%*" ": (1) t = c, i(t) : M n ;! M { &-: i(t)(m1 : : : mn ) c (2) t = xi , i(t) = i : M n ;! M , i (m1 : : : mn) = mi { i-% " ' (3) i(t1) : : : i(tn) , t = f (t1 : : : tn), i(t) = f (i(t1 ) : : : i(tn)) { ' ! &-. !. "" " - , # # "+ x1 : : : xn, ' $'! #+ "+ $ . "" " ' $'! # " ", "- "+, -* ", # +* ". , " - "+ , "+, ' ' "- , ! " , "+ ' "- x1 : : : xn, ! 0 " $'! x1 : : : xn , - . !. 1. '# K { ! ""' #! C = fc j 2 K g F = f+ g, " i(c) = , ! + '% - '"- #! K . ( !" " % "- " 0" #!. 0" " "' # - ". ", i ((+(x y) +(x y))) = i (+(+((x x) (2 (x y))) (y y))) ( $"' # ?). 36
"" " #! K "- # $'!%, ! " ' '. 2. 2' < f g f: ^ _ ;!g >. (" ' ' { 0 "- ) , + ! { 0 '%* " , .. $'! f gn f g (". 1.1 { 1.3). 3. 6"" '' < fc j 2 Rg f = exp ln sin arcsing > : i(c) = , $'!# " '% '%* $'!, $'!, % !" "-+ " 0 ', % 0"" $'!". 2 , 0 " + !, #' # " % %' $'!. 4& %# ! " %' $'!, '- # & " +" $'!" '# + &# i(t), t = f (t1 : : :tn) & $'! i(f ) " i(t1) : : : i(tn) '- (.!) 1, $"' A " & +- # "+ x1 : : : xn, &'" # A(x1 : : : xn). "/ 6.4 (/ 21 ). '# M = (M i) { & " ' . - $"' A(x1 : : : xn) (n 0) L "- # i(A) : M n ;! f g '%*" ". (1) A " t1 = t2 p(t1 : : : tk ), 8m1 : : : mn 2 M i(t1 = t2 )(m1 : : : mn) = () () i(t1)(m1 : : : mn) = i(t2)(m1 : : : mn ) i(p(t1 : : : tk ))(m1 : : : mn ) = () () i(p)(i(t1 )(m1 : : : mn ) : : : i(tk )(m1 : : : mn )) = : (2) i() = i() = : (3) i(A) i(B) '- ,
i(:A) = :i(A) i(A ^ B) = i(A) ^ i(B) i(A _ B) = i(A) _ i(B) i(A ;! B) = i(A) ;! i(B) i(9xA) = 9x i(A) i(8xA) = 8x i(A): 37
+ m1 : : : mn 2 M i(A)(m1 : : : mn ) = , &'" # M j= A(m1 : : : mn). - - M " +-& "+. "/ 6.5. )"' A(x1 : : : xn) B(x1 : : : xn) # " M, % - , .. 8m1 : : : mn 2 M i(A)(m1 : : : mn ) = i(B)(m1 : : : mn ):
&: A M B. )"' A(x1 : : : xn ) B(x1 : : : xn) # &+ " K, 8M 2 K A M B (A K B). !, $"' A B # (A B), # + &+ " ' . )"' A &*" (` A), A . ! 6.6. A B ()` A ! B ! 6.7. #" ( .. 1 2 Q = 8 9): (1) QxA(x x1 : : : xn ) QyA(y x1 : : : xn ), y A(x x1 : : : xn), A(y x1 : : : xn) ! x A(x x1 : : : xn) y. (2) QxA A, x ! A. (3) :8xA 9x:A :9xA 8x:A: (4) 8x(A ^ B) 8xA ^ 8xB: (5) 8x(A _ B) 8xA _ B, x ! B. (6) 9x(A _ B) 9xA _ 9xB: (7) 9x(A ^ B) 9xA ^ B, x ! B. / - '%*+ %& " (". "' 3.1). "/ 6.8. & " M1 M2 ' % 0" 0 ", %& - A ' '
M1 j= A () M2 j= A 38
!
1. Q R % 0" 0 " #', ", - 8y9x(y = x3 ) R, - Q. 2. R C - % 0" 0 " { - 8y9x(y = x2 ) C , - R. 3. & ' < Z + > < Q + > % 0" 0 " ( " '%* -). 4. ,- #, & ' < R + > < Q + > 0" 0 . .
!
"! !
M1 = (M1 i1), M2 = (M2 i2) % % . - %& ' & < a1 : : : an > & a. m1 : : : mn 2 M1, ' : M1 ;! M2 , & < '(m1 ) : : : '(mn ) > & '(m ). '# ' : M1 ;! M2 { "$". , -" - & &*, " &', '%* ' -. %& $"' F (x) L %& & m 2 M1n M1 j= F (m ) () M2 j= F ('(m )): (6.1) 0 -" ' - " . '# t(x) { ", m 2 M1n. ( 6.9.
.
'(i1 (t)(m )) = i2 (t)('(m )):
(6.2)
/ ' - ' "$" ', " & " , #' 0" ' " %& " # '%* , " & "+. '# t = xj . ( i1 (t)(m ) = mi i2(t)('((m)) = '(mi ) " ", 0" ' ' - - . '# " t1 : : : tk ' - , t = f (t1 : : : tk ), f {k-" $'!# " . ( i1 (t)(m ) = i1 (f )(i1 (t1)(m ) : : : i1(tk )(m )):
% "$" '(i1 (t)(m )) = i2(f )('(i1 (t1 )(m )) : : : '(i1(tk )(m ))):
-% '! - i2(f )(i2 (t1 )('(m )) : : : i2(tk )('(m ))) = i2(t)('(m )) 39
(6.2) -" (6.1) "+ $"'. F (x) " t1(x) = t2(x), m 2 M1n, M1 j= F (m ) , i1(t1 )(m ) = i1(t2 )(m ), M2 j= F ('(m )) , i2(t1 )('(m )) = i2(t2 )('(m )). (6.2) # ', (6.1) 0" ' # ' -% i1(t1 )(m ) = i1(t2 )(m ) () '(i1 (t1 )(m )) = '(i1(t2 )(m ))
3 '. '# # F (x) " p(t(x)), t { & " , p { " . ( M1 j= F (m ) , i1(p)(i1 (t)(m )) = , M2 j= F ('(m )) , i2(p)(i2 (t)('(m ))) = . ' (6.1) # "', i2(p)('(i1 (t)(m ))) = , .. (6.1) 0" ' # ' -% i1(p)(i1 (t)(m )) = () i2(p)('(i1 (t)(m ))) =
, ' { "$". '# $"' F G ' - (6.1) . ( M1 j= F (m ) ^ G(m ) () M1 j= F (m ) M2 j= G(m ) () M2 j= F ('(m )) M2 j= G('(m )) () M2 j= F ('(m )) ^ G('(m )): !: M1 j= :F (m ) () M1 j= F (m ) () M2 j= F ('(m )) () M2 j= :F ('(m )): #+ + # "- # #' -% " . '# $"' F (y x) ' - (6.1) . 6"" $"'' 9yF (y x). -", M1 j= 9yF (y m ). / , '* ' m0 2 M1 , M1 j= F (m0 m ). ( -% '! M2 j= F ('(m0 ) '(m )) , #,
M2 j= 9yF (y '(m )) (6.3) . '#, &, (6.3). ( '* ' n0 2 M2, , M2 j= F (n0 '(m )). #' &- ' %3 '* ' 40
m0 2 M1 , n0 = '(m0 ), .. M2 j= F ('(m0 ) '(m )). ( # ' , (6.1) F "" M1 j= F (m0 m ), .. M1 j= 9yF (y m ). (" "" (6.1) $"' 9yF . &* # "- #, #' - '* !. !#. 1. - &' ' - "2 ' "$ ' %". 2. , + &+ # # - 6.9, & " # +' & " " $"'. "/ 6.10. ,- + -, + & " M " M & Th(M). ,- -, + + "+ K &+ " ' , K & Th(K). - 6.9. ', " M + ", "$+ M.
, x ! 1 1 4 "/
$"' " '% ( $7
2.
.
7.1.
'%) "#'% $"' ( ..$.), "
Q1 x1 : : : Qs xs F (x1 : : : xs y1 : : : yt )
Qi = 8 9, F { & $"' (s 0). $ 7.2. & "! A(x1 : : : xn)
L B . - $"' ..$. B , # $"' A "# $" $"' A. A { 0" $"', & , #, " ..$.. $"' B { ..$. $"' A, QxB { ..$. $"' QxA, Q = 8 9. -, Q = 8, Q0 = 9 &. Q1 x1 : : : Qsxs F { ..$. A, Q01 x1 : : : Q0s xs :F { ..$. :A. '# R1 y1 : : :R1 yt G, - Ri # & 8, & 9, # ..$. B . "", "+ xi yi "', &* , #. , & &-# 0 &" " z1 : : : zt, .
41
+ F , G &" G(z ) $"'', ' " - +- yi G zi . ( ' - 6.7(1) R1 y1 : : : R1 yt G R1 z1 : : : R1 zt G(z ). ( #, #' # - 6.7(5,7), '"
A _ B Q1 x1 : : : Qs xsF _ R1 z1 : : : R1 zt G(z ) R1 z1 : : : R1 zt Q1 x1 : : :Qs xs (F _ G(z )): (" "" ..$. A _ B ! # 0$$ { '* ', " ", %& $"' #'% $"'' ..$. 1 . .
!
.
1:
8xP (x) _ 8xQ(x) 8xP (x) _ 8yQ(y) 8x(P (x) _ 8yQ(y)) 8x8y(P (x) _ Q(y))
9x (P (x y) ;! 8yQ(y z )) 9x (:P (x y) _ 8yQ(y z )) 9x (:P (x y) _ 8uQ(u z )) 9x8u (P (x y) ;! Q(u z ))
2:
"/
7'" #, Th(M) " M =< M i > ' ' % , %& $"' A(x1 : : : xn) L '* ' & $"' B (x1 : : : xn ) L , , 7.3.
A(x1 : : : xn) M B (x1 : : : xn):
(7.1)
(7.1) " " %& " M K, , K ' % . '* ' ", %& $"' A(x1 : : : xn) 1 & '% $"'' B (x1 : : : xn ), ' %*'% (7.1) %& " M K, , K ' 0$$ % . ! &! # # ", #' " $"' ". ' 0$$ % . , " &' $"' , 7.3 #% ". .
42
!
M ( K) "
, 9xB (x x1 : : : xn), B { '" $
$ , M ( K) ! . ' " 7.2 # - # $"' ..$. # " '! ' n ..$. A. n = 0 $"' A { & ' - . -", %& ..$., -* " n , # & $"'. '# 7.4.
.
A = Qn xn : : :Q2 x2 Q1 x1 C (x1 : : : xn y1 : : : ym ): -" , Q1 = 9 "" $"'' 9x1 C (x1 : : : xn y1 : : : ym ). " ) & $"' C # $"'
_k
j=1
D1j ^ ^ Ds j
(7.2)
j
Dij # & & $"', & !. 2 #
0k 1 _ @ 9x1 C 9x1 D1j ^ ^ Ds j A
j
j=1 _k
j=1
9x1 (D1j ^ ^ Ds j ): j
' % - $"' Wkj=1 9x1 (D1j ^ ^ Ds j ) # K & $"' E (x2 : : : ym ). ( # A K Qn xn : : : Q2 x2 E (x2 : : : ym ) * $"' ' % -% '!. Q1 = 8, "" 8x1C :9x1 :C . 0 " :C ' (7.2) -, 1 %" 9x1 . 0 A - # ..$., -* # n ; 1 # "- "# - '!.
" "" % Th(Q) " Q ' < f ! " <. j
43
$
7.5.
Th(Q) % " -
. - " :x < y &'" # y x (". " 2 ! 5). '* - ', "
%# $"' 9y(x1 < y ^ ^ xk < y ^ u1 y ^ ^ ul y ^ y < v1 ^ ^ y < vs^ ^y w1 ^ ^ y wt ) k 0 l 0 0 t 0 (7.3) 2 "+ xi ui vi wi "' &# %*. -", "- # '", & ", +* $"'' (7.3), . - , xi xj , xi < y ^ xj < y xi < y. u v w. xi (ui ) " vj , $"' (7.3) -, #' 0" ' $"'' 9y(v < y ^ y < v) ( - $"'' 9y(v y ^ y < v) ). &+ ' $"' (7.3) # & $"' . " ', xi " wj . '# u " w . ( $"' (7.3) - $"'' u y ^ y u, # $"' y = u. 0" ' +- y "- "# u, (7.3) # & $"'. xi " uj , $"'' y uj "- '&# (7.3) #' 3%! xi < y ^ xi y # $"' xi < y. , vi " wj . , # "# ', &+ "+ $"' (7.3) %*+.
', y (7.3) # (.. k = l = 0, & s = t = 0) $"' (7.3) # & $"' , #' ' " "- Q , "#1 0". - " y &+ , $"' (7.3) # $"'
^k ^s
(x < v ) ^
=1 =1 ^l ^s
^
=1 =1
^k ^t
(x < w )^
=1 =1 ^l ^t
(u < w ) ^
=1 =1
(u w )
(7.4)
"" , # (7.4) (7.3) 1 < . 44
'#, &, $"' (7.4) . / # '%* " maxfx g < minfv g maxfx g < minfw g (7.5) maxfu g < minfv g maxfu g minfw g .# "- '. 1. " (7.5) - . 0" ' , maxfx u g < minfv u g %& y, ' %* ' maxfx u g < y < minfv u g &* ' $"'' (7.3). ( y '* ', #' ' "- Q , .. "-' %&" '" " 0"" "- # . 2. " (7.5) #, "" .
0" ' #, y = maxfu g = minfw g &* $"'' (7.3) '. ! "", # 0$$ , .. 3 ", * %& $"', #'% Q & '% $"''. 2 # Th(Q) ' 0$$ % (". " % 7.3). # " 7.5 " # # 1# ", ' " "- Q '%* -: 1): 8x(9y(x < y) ^ 9y(y < x)) 2): 8x8y(x < y ;! 9z (x < z ^ z < y)): ' "- , " - 1) 2), " " ". &" + + ' + "- DL. (" &" DL # " ' , + - 1') { 3') " 2 ! 5 - 1) { 2) '* &!. (" &" " ! 7.6. DL % " . "- ' ', %& & - 0 ' # $"' , .
45
-* "+. %& $"' # & , & . (" &", K 0 ' ' % , - - # K & , & . / '%* ! 7.7. , K % " , K % %
2 - 7.6 7.7, '" & . "! % % . , % % Q R. ( " " $, '* ' ", %&"' -% ", - 0 . #' '* ' ", %& $"' + + "+ -, '%* ! 7.8. T % " , "! ! " , % , T ( .
' + - 7.7 - & - # , .. ", %* %&"' & "' -% - , " '* '. 2 #, ' + - 7.7 Th(K) ' 0$$ % , 1". & . Th(DL) ( .
, x #
! 3.
8
.
6"" '' =< f0 +1 ;1g f+g f "' Z+ =< Z i > ' , Z, , "- !+ , i { ! +, $'!#+ " . 0", - " i %' ', " - % "" " $'!" ". (, " 1| + {z + 1} (|;1) + {z + (;1)} x| + {z + x} k
k
46
k
&'" # k ;k k x .
" , & % & , "+ & " $"'" ' . #'# ! #% ""' #% - Z, #, - $'!, " " "" ' , " k1 x1 + + kn xn + b ki b 2 Z:
(8.1)
2& , "- -'% "'% $"'' t1 = t2 ' k1 x1 + + kn xn + b = 0
(8.2)
-'% "'% $"'' t1 > t2 { ' k1 x1 + : : : kn xn + b > 0:
(8.3)
&" '% # $"' (8.2) (8.3) t. ( ! $"' (8.2) # $"' t > 0 _ t < 0, ! $"' (8.3) { $"' t 0. / t> 0 t;1 0 t< 0 t+1 0 t= 0 t 0^t 0 %, '% "'% $"'' "- "# 3%! + , ! { 3%! + . # & '% $"'' ), # 0 ", " # &, #'# &' #% 3%! # 3%!, 0 $"' - # Z+ $"' _n m^ (ji) k1 x1 + + kt(ji) xt + b(ji) 0 n
j=1 i=1
(" &" " '%* ! 8.1. )! , ! ! !' , $ ( $ % $ . 47
!
Th(Z+) " . 6"" $"'' 9y(x = 2 y), # , x { , -", # & $"'. "" , & 0 $"' & # & , - 8.1 &, &# & &3 "- , - + # "- !+ 1 " k1 x + b1 0 8.2. .
: : : : : : : : :: : : ks x + bs 0:
, "- , & , & - ! -# (!#) , . , 0" - " & &3 + "- , & "- + " + %& $"' 9y(x = m y), m 2. / $"' x 0( mod m). 61" # '' ' 0 , & "- "+ + " Dm (x) m 2, &'" # 1 " Z+ '%*" &": i(Dm )(x) = () x 0(mod m): '& - " -, ' ' ' " D1 (x), ' - . ( " Z+ ' ! + " Dm $" &' & APr. " " $ 8.3. * ! % " # " " & ' - , " $"''" "". 1. # ax b(mod m) ( , b d = (a m). % % c, ( : x c(mod m) x c + m (mod m) : : : x c + (d ; 1)m (mod m): d
d
48
1
m1 : : : mk . x c1 (mod m1 ) 2 (
).
: : :: : : : : : : : : x ck (mod mk )
( mod m1 : : : mk . % m1 : : : mk % $ d1 : : : dk , "! c1 : : : ck ( x d1 c1 + : : : dk ck (mod m1 : : : mk )
/ ' - & ' " & ". 2'%* ' - -" &&*" " & + ', "' m1 : : : mk % " ". # - , # % . 3. C x c1 (mod m1 ) : : :: : : : : : : : : x ck (mod mk )
( , 8i < j k ci cj (mod (mi mj )). % ( mod m1 : : : mk ]. % m1 : : : mk % $ d1 : : : dk , "! c1 : : : ck ( x d1 c1 + : : : dk ck (mod (m1 : : : mk ])) 8.3. '*+ " ', '# %# $"' :
9x (k1 x + t1 ^ : : : ksx + ts ^ Dm1 (l1 x + u1 ) ^ : : : (8.4) ^ Dm (lt x + ut ) ^ :Dn1 (h1 x + v1) ^ ^ :Dn (hr x + vr ) ki li mi ni hi 2 Z ti ui vi { ", -* " x. 1, D1 (x) "- #, t > 0, .. " $"' (8.5) ' '. t
r
49
." -'% $"'' :Dn (t) #'% $"'' Dn (t ; 1) _ _ Dn (t ; n + 1) (:t 0(n) () t 1(n) _ _ t n ; 1(n)), & 9 3%!%, +" "', '# %# # $"' (8.4) r = 0, .. -*+ $"' :Dn (t). " 0, " ""' 1, "- #, l1 = = lt = 1. 0, " ""' 2 ""' 3, .. " "' , 0 , +" "', '# %# # $"' 9x (k1 x + t1 ^ : : : ksx + ts ^ Dm (x + u) (8.5.) $"' (8.5) ki , 0 $"' , #' %&+ + &+ "+, -* " , ' %  &% -# !# x, + x, ' +*"' 0' $"'' %. " ' $"' (8.5) '
9x(a1 x p ^ ^ af x pf ^ q1 b1 x ^ ^ qq bg x ^ Dm (x + u)):
(8.6)
)"' (8.6) '
_f _g pi pi g ^ qj = minf qj g^ = max f i ai i bj ai bj
i0 =1 j0 =1 ^ 9x(pi0
.#
pi0 ai0
0
0
0
0
ai0 x ^ bj0 x qj0 ^ Dm (xi0 + u))
p g # * $"' = max f a i i i
^f i=1
aipi0 ai0 pi
q g. f $"' qb 00 = min b i , # "# ' $"' j
j
j
j
9x(p ax ^ bx qj0 ^ Dm (xi0 + u))
(8.7)
#, 0 $"' # & $"' bp = aq ^ p 0(a) ^ q 0(b) ^ p + au 0(ma) _ :bp = aq ^ aq bp + abm (8.8) 50
"" , '# $"' (8.7) , bp abx bq. "- ': ) bp = abx = aq &) 6= bp = aq: ' ) bp +p abu 0(qmab), p.. p + au 0(q ma) $"' (8.8) . ' &) a x b , .. p q u u a + u x + u b + u, .. x + u = mt, ma + m t mb + m . ,-' '" * " " "- # ! " # " ' ; 1. " ' 0 # $"' (8.8). &'% ' " '*+ " , - & " - ' 0 # "' - k > 0 k = 0 k 0( mod m), k 2 Z, , '* ' ", %* %&"' & "' ""' -%, " -". ( #, #' - 7.8, " " " 8.3 & . * ! ( "" &' 1929 ' 1"# - '#+ , .. & N+ =< N i > ' 00 =< f1g f+g > ! $'!# " . #, 0 APr '%*" ". "/ 8.4. '# M1 =< M1 i1 > M2 =< M2 i2 > { & " ' 1 2 . 7'" #, Th(M2 ) ' Th(M1 ), %&"' -% A ' 2 "- # - A0 ' 1, M2 j= A () M1 j= A0 : '* ' ", #"' -% A &'" - A0, , Th(M2 ) ' Th(M1 ) 0$$ . ( # '%* ! 8.5. Th(M2) % Th(M1) Th(M1 ) ( , ( Th(M2 ). +!, % Th(M2 ) ( , ( Th(M1). ! 8.6. Th(N+ ) % APr. , & # A0 A '- # "# +- - Qx A +- '%* Qx 0 (". !% 3). .
51
# &, .. APr ' Th(N+ ) " 8.4 (. ). 6'# &' &, - ""', " '#" 1" -# $" $" "" , " "", & " '* " 1" ! "" #& $"! "" (". !% 2). " ' " -, 0 "" '" " &3". , '- $" -" '"-", .. & N ' < f1g f+ g >, 1". #' Dm (x) " Z+ " $"'" ' , + , "+ $"'" ' Z+, " , "+ $"'" ' 0 Z+. / ' '%* . "- L Zn " , '* '% a1 : : : ak b 2 Zn, L = ft1 a1 + + t1 a1 + b j ti 2 N i = 1 : : : kg &3 + "- Z " "- ". " " '%* $ 8.7. M Zn F (x1 : : : xn) c , .. M = f< m1 : : : mn > j Z+ j= F (m1 : : : mn )g , M .
,
9
x $1 $
0" " "" % ' * + . / R ' =< f0 +1 ;1g f+ g f. "", " 4 ! 5 ' ' + % " -1. .# % '' %# &- '& . / ( ' '%* ", -* .("'. $ $ Th(R) % " . ", "- # # 0' "' " &3", " "" '. - "", 4.
.
9.1 (
).
52
- $'! Rn, " "" ' , # " p(x1 : : : xn) !" 0$$!" (". " 1 % 6.3). . " $"' ' "' &# p(x1 : : : xn) = 0 p(x1 : : : xn) > 0. -" 7.4 '# %# $"' 9xB (x), B { 3%! "+ $"' + !. "" :p = 0 p > 0 _ p < 0, :p > 0 p 0 ;p > 0 _ p = 0. ." "* 0+ 0 ! "+ $"', & 9 3%!%, +" "', '# %# $"'
9x(p1 (x) = 0 ^ ^ pn(x) = 0 ^ q1 (x) > 0 ^ ^ qm (x) > 0)
(9.1)
pi qj # " " x, 0$$! + " % "" !" 0$$!" '+ "+, & !" ". #' R p1 (x) = 0 ^ ^ pn(x) = 0 p21 (x) + + p2n(x) = 0
# "" $"' (9.1), -*+ # ' , "-, # . , # "" # $"' 9x(p(x) = 0): (9.2)
', # " p " x + 2, % $"' (9.2) '* '%*" &":
9x(ax2 + bx + c = 0) R a = 0 ^ b = 0 ^ c = 0_ _ a = 0 ^ b 6= 0 _ a 6= 0 ^ b2 ; 4ac 0: .# a b c { " !" 0$$!" '+ "+, & ! .
&*" ' % $"' (9.2) '* "* " 9'". 4& " # 0 , &-# 1" "+ , "" $"''
9x(x4 + bx + c = 0)
(9.3)
, 1, b c { " !" 0$$!" '+ "+, & ! . 0" "- #, b 6= 0, .. " ' $"' (9.3) # $"' c 0. 53
6"" " f (x) = x4 + bx + c. " D(f ) = 256c3 ; 27b4. D(f ) 6= 0, (f f 0 ) = 1. 0" ' " 9'" f " : f0 (x) = f (x) f1 (x) = f 0 (x) = 4x3 + b f2 (x) = ;3bx ; 4c f3 (x) = Db(3f ) :
&" V (1) " # f0 (1) f1 (1) f2 (1) f3 (1), fi (1) { " fi (x) x ! 1, " &" ' 1 0$$! " fi . .# "- 4 ' : a). b > 0 D(f ) > 0 : V (;1) = 2 V (+1) = 2 b). b > 0 D(f ) < 0 : V (;1) = 2 V (+1) = 1 c). b < 0 D(f ) > 0 : V (;1) = 1 V (+1) = 1 d). b < 0 D(f ) < 0 : V (;1) = 2 V (+1) = 1:
' " 9'" * + " f V (;1) ; V (+1), .. '* '% '+ b) d). D(f ) = 0, (f f 0 ) = f2 , .. g = (fff ) { " # , " " * #, " " f ( "", g " - , f , + g #%, !). # $"' (9.3) # $"' 0
b = 0 ^ c 0 _ :b = 0 ^ (D(f ) = 0 _ D(f ) < 0):
(" &" " ( '* ' & &&* " 9'" ' " "#+ . 1 &* "+ $"' Th(R) ', - & " - ' # R & -% n > 0, & -% n 0 n 2 Z. % , '* ' ", %* %&"' & "' ""' -% " -". ( #, " - 7.8, '" & 2. Th(R) ( . " ' - & .(" " & "A decision method for elementary algebra and geometry", 1 ! 40-+ ( 1951 '). # # ( & # -" { & " & !. 0 # & ' * .&", 0" 54
'" ". 2 # # " (, + 0 "- 6]. 6'# ( '* " # $" $" 0" "". '* ' & - " 1 0" &, ", " # " &+ ' . #'# " "" "-  - 0" " $"' # - Th(R). ", - 8x1 y1 x2 y2((x1 =6 x2 _ y1 =6 y2 ) ;! 9A B C (Ax1 + By1 + C = 0^ ^ Ax2 + By2 + C = 0 ^ 8A1 B1 C1 (A1x1 + B1 y1 + C1 = 0^ ^ A1x2 + B1 y2 + C1 = 0 ;! 9t(A1 = tA ^ B1 = tB ^ C1 = tC ))) - "'% "' ", + ". " &"  " 1# " "- &# Th(R). / "", + $''% *# ', &3" 1 . ., #' "- #, & ' * , ' % 1". " "- # 0 ' -, -" , & -. 6"" '' 1 , ' & " " $'!# " . / " " &'" &# sin # R $'!% sin(x). ,- + - ' 1 , + R &'" &# Th1 (R). ! 9.2. Th1(R). " ( . , &'" # # " $", & N ' 2 < f0g f+ g > 1". / &' 1" ' . - 7.8 ', & Th(N) ' 0$$ % , & & 1". . ' 0$$ % . "" "- #, 0 &* ' % . -", Th(N) 0$$ ' Th1 (R) (". 8.4). 0 1 ' - &' # - 8.5. 2 -", ' 1 "- # " Nat(x), , R Nat(x) = () x ; ;'# .
55
", '# # 1 # !# "#1"' -#"' '. " P (t) 8x(sin(x + t) = sin x) ^ t 0: , P (t) = () t { !# $'! sin(x). "#1 -# $"' MP (t) P (t) ^ t > 0 ^ 8s(P (s) ^ s > 0 ;! t s ( # , Nat(x) 9t9s(P (t) ^ MP (s) ^ t = x s: # 1 - '%* ' - (". 8.4 - 8.5). %& $"' F (x1 : : : xn ) ' 2 "- # '% $"'' F 0 (x1 : : : xn) ' 1 , %&+ k1 : : : kn 2 N N j= F (k1 : : : kn ) () R j= F 0 (k1 : : : kn ) / ' - '! % $"' F " $"' F ' 2 F 0 = F . $"' F G &'" $"' F 0 G0 '- , , : (F ^ G)0 = F 0 ^ G0 (F _ G)0 = F 0 _ G0 (:F )0 = :F 0 (8xF )0 = 8x(Nat(x) ;! F 0 ) (9xF )0 = 9x(Nat(x) ^ F 0 ): , # { 1" Th2 (R), ' # " $'!# " ' 1 ', 0 ' ( 0" ' ' &# exp). / &" * 50-+ + # # '. 1 + , " ' '* { , & "+ , "+ $"'" 0 , - 1# + , .. % & &3 "-+ (+, "'+, '+, '+ . .). "", 1" 0 &" & & " e , ", & #, '* ' " 56
p(x y) 2 Zx y] ' %* ' % p( e) = 0. "" , & " '* , & & " Th2 (R), & 1"# , .., "# '- & ( 1" . (" &" # 1" Th2 (R) & # ' & & " e .
,
10
'" & & & , "+ $"'" ' (. " '* ! ', - & $"' ' # 3%! 3%! $"' p(x1 : : : xn) > 0 q(x1 : : : xn ) 0, p q 2 Zx1 : : : xn ]. &# - 3%! # 1 " "#+ . / &- '%*"' %. "/ 10.1. "- M Rn % , "- " 1 " "#+ p1 (x1 : : : xn) > 0 : : : : : : : : :: : : : : : : : : pk (x1 : : : xn) > 0 q1 (x1 : : : xn ) 0 : : : : : : : : :: : : : : : : : : ql (x1 : : : xn) 0
(10.1)
pi qj { " 0$$!". &3 0"+ '&+ "- "- ". (" &" &# Rn, " %& & $"' ' & '& "- . '&+ "- % & &+ "& ("- 1 " &+ ' ). '&+ "- "' # &' + ! . 2'%* ", ' " 9.1, , 0 "' # !. 57
$
$ 0
$ % ! "! ! . . '# A Rn+k { '& "- c1 : : : ck 2 R. #' %& " f (x1 : : : xn y1 : : : yk ) $'! f (x1 : : : xn c1 : : : ck ) & " "+ x1 : : : xn, "- fhx1 : : : xn i jhx1 : : : xn c1 : : : ck i 2 Ag - '&". '# # A { "- 1 " (10.1). -", ! A # x1 '&" "- ". '# c1 : : : ck { & * + 0$$! " , +*+ "' (10.1). &" " 6"" " pi (x1 : : : xn y1 : : : yk ) qj (x1 : : : xn y1 : : : yk ) 2 Zx1 : : : xn y1 : : : yk ] i = 1 : : : k j = 1 : : : l, , pi (x1 : : : xn c1 : : : ck ) = pi (x1 : : : xn) qj (x1 : : : xn c1 : : : ck ) = qj (x1 : : : xn). ' " 8.1 $"' F (x2 : : : xn y1 : : : yk ), 10.2 (
{
).
.
9x1 (p1 (x1 : : : xn y1 : : : yk ) ^ ^ p2 (x1 : : : xn y1 : : : yk )^ ^ q1(x1 : : : xn y1 : : : yk ) ^ ^ q1 (x1 : : : xn y1 : : : yk )) # Th(R) & $"', .. &# # '& "- B . ( # ! "- A # x1 fhx2 : : : xni jhx2 : : : xn c1 : : : ck i 2 B g .. '&" "- " ' ", " &! # .
!
.
& xy = 1 { & "& , # { 0" '& "- . ! # OY & '& "- , fx 2 R j x 6= 0g, &3" '+ 0"+ '&+ "- fx 2 R j x > 0g fx 2 R j x < 0g. 0 . #, 0 "- 0"" '&". 6"" # '' 0 , ' & " "-+ + " c 2 R. 7'" #, - " c ' R . " 10.2 " '%* ", '% " - &'" # " ( { .&. 58
$
A Rn ! ! , ! 0 , A { ! . (" ( { .& " - " & ". %'" "-# " #+ "+. &- "- Rn " , $ # '& "- . ! 10.4. A Rn, B Rk f : A ;! B { ! ! , A = domf rngf { ! ! f { ! . " 10.3 ', %& "- Rm , " $"' ' 0 # '& "- . '# $"' F (x y) $ &- f . ( A = fx 2 n R j 9yF ( x y)g, rngf = fy 2 Rk j 9xF (x y)g, .. & 0 "- " $"'" ' 0 . C . )! ! ! ! ! . "", " x 2 Rn "- A Rn dist(x A) = inf fjx ; yj j y 2 Ag. n { ! , ! 10.5. A R $ f : Rn ;! R , f (x) = dist(x A) ! $ . '# G(x) { $"' ' 0 , %* "- A. 1" $"'' F (x y), %*'% $ $'! f . (" "" - &' . #, 0 $"' " '%* : 10.3.
.
.
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* " ' + 1 (. - # ' # '#. 2" # " &* +. '# " # & " M = hM ii ' = hC F Pi. &" Sn (M ) { " + n-"+ S n M , "+ $"'" ' , S (M ) = S (M ). n2N - + &" "- #, Sn (M ) # " "- M n - n 2 N. #, 0 " &% '%*" ". S1 Sn (M ) { &' & "- M n S2 ij (M ) = fx 2 M n j xi = xj g 2 Sn (M ) 1 i j n S3 A 2 Sn (M ) =) M A A M 2 Sn (M ) S4 A 2 Sn+1 (M ) =) (A) 2 Sn (M ), (x1 : : : xn xn+1) = (x1 : : : xn ) S5 $ $'! i(f ) f 2 F , - & i(P ) P 2 P - S (M ). " M " "- T = S T n , n2N n M T 2 , ' %*+ " S1 { S4. ( , S (M ) { "#1 '' M , &%* " S5. (" ( - .& , '& "- &'% ''' R. / '' # S (R), 0 { ', $''%* " 10.3. ."", #' x y 9z (y = x + z 2), '& "- &'% "#1'% ''', -*'% $ $'! x + y = z x y = z . -, S1 (R) "- , "%*+ " , .. %*+ & &3 + . / " '%*"' %. "/ 10.7. 2'' T R -""# (order1 minimal), - $ $'! x + y = z x y = z T "- , "%*+ " . (" &" '& "- &'% "#1'% ""#'% ''' R. - x = (x1 : : : xn), ex = (ex1 : : : ex ) . ,- fx 2 Rn j P ( x ex ) = 0g, P { " * " 0$$!" 2n "+, " % , & 0 !#+ "- &- : Rn+k ! Rn " % "- ". n
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f(x y) 2 R(n+k) j P (x y "x ey ) = 0 f (x y) = 0g P { % $ 2(n + k) , f { $ , a1 b1] an+k bn+k ] " , ! " " . 0 " ', ' ( & # " $'!# " f # $'!%, '% sin x #" " '%, 0 , "- R, " ' , "% 1# " . / , # 0 $'!# " $'!%, '% sin x %' R, #' 0 " "- N (". # - 9.2). -""#" ''" ", "- , +* 0-""# '' &% "" +1" " " { ", - "- " 1# " . 2 ' , ", + "- 1. %! " ' #" 0$$'% "' $'!+, +*+ ""# ''. 61
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II. " . '# c d { ", -* "+ ( , ). ( '%* - % "" , % "" : (1) c=c (2) c = d ;! (A(c) ! A(d)), - A(d) ' " - A(c) + +- " c d. III. " " ". '# A { - L , " x0 + A. &" A0 { -, ' A " + +- " x x0 . ( A ! A0 { " , " " ". IV. " . '# A(x) { $"' L c & " x, B { - L , -* " x. ( '%* - % "" , % "" . (1) 8xA(x) ;! A(t), t { ", -* "+. (2) :8xA(x) ! 9x:A(x). (3) :9xA(x) ! 8x:A(x). (4) 8x(B ;! A(x)) ! (B ;! 8xA(x)). ! 11.2. , ! . . &*"# " I II - " 1 IV ! $"' &+ "+ ( 6.4). &*"# " III #+ " IV ' + - . "" . '# M =< M i > { .. ' . ( " 4 ', '#"
8x(i(B ) ;! i(A)(x)) ! (i(B ) ;! 8xi(A)(x)) , , *+ 0 , . ." " !% &+ + 3%!% ! # 1# " (4) " 3.1, '" &'" - :
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.
65
3. #, # 0$$ ", .. ', ' 0$$ ( ", ), '* ' ", %& # - ", 0 ## # ". / ' , " 0$$ , .. '* ' ", %* %&"' -
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(6) % ! . F G { , ` F (p1 : : : pn) ! G(p1 : : : pn), A1 : : : An {
L , F (A1 : : : An) G(A1 : : : An) !#. 1. , "" "- # - & . 0" '% " . 2. ## - A1 : : : An, " - Ai & " , & ' 1 '%*+ "' + "#+ , " &&*" # ". , -, +* &&* # % "" . 3. 3%! #'
A ! B A ! B A ;! B B ;! A
. # (1) " '" n = 3. &* ' " . " , A1 ;! A2 A2 ;! A3 { " . " &&* # - A1 ;! A3 . 1:(A1 ;! A2 ) ;! ((A2 ;! A3 ) ;! (A1 ;! A3)) ( ) 2:A1 ;! A2 (" ' %) 3:(A2 ;! A3 ) ;! (A1 ;! A3) (' 1. 2. modus ponens) 4:A2 ;! A3 (" ' %) 5:(A1 ;! A3 ) (' 3. 4. modus ponens) (2) . 6"" 3%!: A ;! B A ;! C A ;! B ^ C '# A ;! B A ;! C { " . ( '%* ## &&*" # " - A ;! B ^ C . 1:(A ;! B ) ;! ((A ;! C ) ;! (A ;! B ^ C )) ( ) 2:A ;! B (" ' %) 3:(A ;! C ) ;! (A ;! B ^ C ) (' 1. 2. modus ponens) 4:A ;! C (" ' %) 5:(A ;! B ^ C ) (' 3. 4. modus ponens) 67
# 3%! 3%! % . -" 0 ". #, '%* ) (A1 ! B1 ) ;! ((A2 ! B2 ) ;! : : : ;! ((An ! Bn ) ;! ;! (F (A1 : : : An) ! F (B1 : : : Bn ) : : : )
(11.2)
"" , " - & ( - " ! 3%!% !, - ,) +" 0 (11.2) $"' (A1 ! B1 ) ^ (A2 ! B2 ) ^ ^ (An ! Bn ) ;! ;! (F (A1 : : : An) ! F (B1 : : : Bn )
(11.3)
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::: ::: ::: :::
An) ! G(A1 : : : An) ( ' %) An) ;! G(A1 : : : An) ( 3a., ". " 3) An) (" ' %) An) (2,3 - modus ponens)
" # " # "" . .# " &'" # # "&' #% " !:
` (p ;! (q ;! r)) ! ((p ;! q) ;! (p ;! r)) 68
(11.4)
1" # # - c = d ;! d = c. 1:c = d ;! (c = c ! d = c) (" 2) 2:(c = c ! d = c) ;! (c = c ;! d = c) ( ) 3:c = d ;! (c = c ;! d = c) (! " !) 4:(c = d ;! c = c) ;! (c = d ;! d = c) (11.4 0 . &.) 5:c = c (" 1) 6:c = c ;! (c = d ;! c = c) ( ) 7:(c = d ;! c = c) (5,6 modus ponens) 8:(c = d ;! d = c) (4,7 modus ponens) -" # # . .# " #'" 0 #% ` (p ;! (q ;! r)) ! (p ^ q ;! r) (11.5) ( # # " : 1:a = b ;! (a = c ! b = c) (" 2) 2:(a = c ! b = c) ;! (b = c ;! a = c) ( ) 3:a = b ;! (b = c ! a = c) (! " !) 4:a = b ^ b = c ;! a = c (11.5 0 . &.) -" # &&* " : %& $"' F (x1 : : : xn) L ` a1 = b1 ^ ^ an = bn ;! (F (a1 : : : an) ! F (b1 : : : bn)) (11.6) # " n = 2 #' + n n + 1 - 1 2. " '% "' , '" ` a = c ;! (F (a b) ! F (c b)) ` b = d ;! (F (c b) ! F (c d)) ( #, #' % ` p ;! q ;! ((r ;! s) ;! (p ^ r ;! q ^ s)) " MP, '" ` a = c ^ b = d ;! (F (a b) ! F (c b)) ^ (F (c b) ! F (c d)): ( #, " % (F (a b) ! F (c b)) ^ (F (c b) ! F (c d)) ;! (F (a b) ! F (c d)) ! ' " !, '" ` a = c ^ b = d ;! (F (a b) ! F (c d)) & #. 69
,
12
6"" # "#+ , %%*+ . ! 12.1. x { A(c) B (c), c { . " " : A(c) (2): A(c) ! B (c) (3): A(c) ! B (c) (1): 8xA (x) 8xA(x) ! 8xB (x) 9xA(x) ! 9xB (x) (1). -", A(c) { ". 1" # 8xA(x). 1:A(c) (" ' %) 2:A(c) ;! (:A(c) ;! ) ( ) 3::A(c) ;! (1,2 { modus ponens) 4:9x:A(x) ;! (3 { &&*) 5:(9x:A(x) ;! ) ;! :9x:A(x) ( ) 6::8xA(x) ! :9x:A(x) (" ) 7:(:8xA(x) ! :9x:A(x)) ;! (:9x:A(x) ;! 8xA(x)) ( (:p ! q) ;! (:q ;! p)) 8::9x:A(x) ;! 8xA(x) (6,7 { modus ponens) 9:(9x:A(x) ;! ) ;! 8xA(x) (5,8 { ! " !) 10:8xA(x) (4,9 { modus ponens) -" (2). 1:8xA(x) ;! A(c) (" ) 2:A(c) ! B (c) (" ' %) 3:8xA(x) ;! B (c) (1,2 { ! " !) 4:8y(8xA(x) ;! B (y)) (3 { (1) 0 -) 5:8y(8xA(x) ;! B (y)) ;! (8xA(x) ;! 8yB (y)) (" ) 6:8xA(x) ;! 8yB (y) (4,5 { modus ponens) 7:8yB (y) ;! 8xB (x) (" " ") 8:8xA(x) ;! 8xB (x) (6,7 { ! " !) 9:8xB (x) ;! 8xA(x) (# ) 10:8xA(x) ! 8xB (x) (8,9 { 3%! { ". . 11.7) .
70
-" (3). 1:A(c) ! B (c) (" ' %) 2::A(c) ! :B (c) ( 0 + & ) 3:8x:A(x) ! 8x:B (x) (1 { (2) 0 -) 4:8x:A(x) ! :9xA(x) (" ) 5:8x:B (x) ! :9xB (x) (" ) 6::9xA(x) ! :9xB (x) (3,4,5 { ! 0 ) 7:9xA(x) ! 9xB (x) ( 0 + & )
-" #, - (" 3.1) % "" .
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D ! x. " " , : 12.2.
(1): ` (9x(A(x) _ B (x)) ! (9xA(x) _ 9xB (x)) (2): ` 8x(A(x) ^ B (x)) ! (8xA(x) ^ 8xB (x)) (3): ` 9x(A(x) ^ D) ! (9xA(x) ^ D) (4): ` 8x(A(x) _ D) ! (8xA(x) _ D) (5): ` QxQyA(x y) = QyQxA(x y)
Q ! 8, ! 9. . # (4):
1:(A(c) _ D) ! (:D ;! A(c)) ( ) 2:8x(A(x) _ D) ! 8x(:D ;! A(x)) (1 { 3 - 12.1) 3:8x(:D ;! A(x)) ! (:D ;! 8xA(x)) (" ) 4:(:D ;! 8xA(x)) ! (8xA(x) _ D) ( ) 5:8x(A(x) _ D) ! (8xA(x) _ D) (2,3,4,5 { ! 0 ) 71
# (3): 1::9x(A(x) ^ D) ! 8x:(A(x) ^ D) (" ) 2::(A(c) ^ D) ! (:A(c) _ :D) ( ) 3:8x:(A(x) ^ D) ! 8x(:A(x) _ :D) (2, 2 - 12.1) 4:8x(:A(x) _ :D) ! (8x:A(x) _ :D) (" (4) * -) 5:8x:A(x) ! :9xA(x) (" ) 6:(8x:A(x) _ :D) ! (:9xA(x) _ :D) ( 0 ") 7:(:9xA(x) _ :D) ! :(9xA(x) ^ D) ( ) 8::9x(A(x) ^ D) ! :(9xA(x) ^ D) (1 { 7 { ! 0 ) 9:9x(A(x) ^ D) ! (9xA(x) ^ D)( 0 + & )
# (2): 1:8x(A(x) ^ B (x)) ;! A(c) ^ B (c) (" ) 2:A(c) ^ B (c) ;! A(c) ( ) 3:8x(A(x) ^ B (x)) ;! A(c) (1,2 { ! " !) 4:8y(8x(A(x) ^ B (x)) ;! A(y)) (1 { 1 - 12.1) 5:8y(8x(A(x) ^ B (x)) ;! A(y)) ! (8x(A(x) ^ B (x)) ;! 8yA(y)) (" ) 6:8yA(y) ! 8xA(x) (" " ") 7:8x(A(x) ^ B (x)) ;! 8xA(x) (1 { 6 { ! " !) 8:8x(A(x) ^ B (x)) ;! 8xB (x) (# ) 9:8x(A(x) ^ B (x)) ;! 8xA(x) ^ 8xB (x) (7,8 { 3%!) 10:8xA(x) ^ 8xB (x) ;! 8xA(x) ( ) 11:8xA(x) ;! A(c) (" ) 12:8xA(x) ^ 8xB (x) ;! A(c) (10,11 { ! " !) 13:8xA(x) ^ 8xB (x) ;! B (c) (# ) 14:8xA(x) ^ 8xB (x) ;! A(c) ^ B (c) (12,13 { 3%!) 72
15:8y(8xA(x) ^ 8xB (x) ;! A(y) ^ B (y)) (14 { 1 - 12.1) 16:8y(8xA(x) ^ 8xB (x) ;! A(y) ^ B (y)) ! (8xA(x) ^ 8xB (x) ;! ;! 8y(A(y) ^ B (y))) (" ) 17:8xA(x) ^ 8xB (x) ;! 8y(A(y) ^ B (y)) (15,16 { modus ponens) 18:8y(A(y) ^ B (y)) ! 8x(A(x) ^ B (x)) (" " ") 19:8xA(x) ^ 8xB (x) ;! 8x(A(x) ^ B (x)) (! " !) 20:8x(A(x) ^ B (x)) ! (8xA(x) ^ 8xB (x)) (9,19 { 3%!)
# " (1) (5) % % ' -. ! 6" 1 "# # " %, " "# - & , #'%* - , - , # ' " . &' # '%* " : .
.
1: ` (8xA(x) _ 8xB (x)) ;! 8x(A(x) _ B (x)) 2: ` 9x(A(x) ^ B (x)) ;! (9xA(x) ^ 9xB (x)) 3: ` 9x8yA(x y) ;! 8y9xA(x)
x 3 "/ '# ; { "- - L . , - A ; 1' ; ` A, '* ' "- - A1 : : : Ak 2 ;, ` A1 ^ ^ Ak ;! A. ! ` A, "! ; ;`A '# B 2 ; 2.
.
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12.4.
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12.5.
; ` A "! B 2 ; ;0 ` B ,
. '# B1 : : : Bk 2 ; ` B1 ^ ^ Bk ;! A. '#, (1) (k) (k) (i) (i) 0 - A(1) 1 : : : As1 : : : A1 : : : As 2 ; , ` A1 : : : As ;! Bi i = 1 : : :k. "" k
i
(k ) (i) (k ) (i) (1) ` A(1) 1 ^ ^ As1 ^ ^ A1 ^ ^ As ;! A1 ^ ^ As i = 1 : : : k #' 0 . 2 # (k) (1) (k) ` A(1) 1 ^ ^ As1 ^ ^ A1 ^ ^ As ;! Bi i = 1 : : : k ' ! " !. ( #, " # 3%!, '" (k) (1) (k) ` A(1) 1 ^ ^ As1 ^ ^ A1 ^ ^ As ;! B1 ^ ^ Bk , !, (k) (1) (k) ` A(1) 1 ^ ^ As1 ^ ^ A1 ^ ^ As ;! A ' ! " !, &'" & ; ` A1 : : : ; ` Ak () ; ` A1 ^ ^ Ak "/ ,- T - L $"# , "' # ", .., 8A 2 L" T ` A =) A 2 T . k
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.
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T { , A1 : : : Ak 2 T ()
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;.
. , #, T (;) { $"# . '# T (;) ` B . #', % T (;) %& - C 2 T (;) " ;, ' - 12.5 ; ` B , .. B 2 T (;) "/ 12.10. ( T , '* ' - A, T ` A T ` :A. " ' T . ,- - ; ", T (;). 74
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T , "! A
L . # ; , "! A
L ;. , T - -, . '#, &, T - A L , T ` A T ` :A, .. % 12.6 A ^ :A 2 T %& - B 2 L ` A ^ :A ;! B , .. 0 . ( # " ', B ` T , .. B 2 T ! 12.12. T , A 2= T :A 62 T , T fAg T f:Ag . 12.11.
.
.
. -" # T fAg . '# T fAg ` B T fAg ` :B . ( '* '% - A1 : : : Ak 2 T , ` A1 ^ ^ Ak ^ A ;! B - B1 : : : Bl 2 T , ` B1 ^ ^ Bl ^ A ;! :B . ' 12.6 C = A1 ^ ^ Ak 2 T , D = B1 ^ ^ Bl 2 T F = C ^ D 2 T . #' ` (C ^ A ;! B ) ;! (C ^ D ^ A ;! B ), F ^ A ;! B 2 T , F ^ A ;! :B 2 T . ` (F ^ A ;! B ) ;! (F ;! (A ;! B ), .. F ;! (A ;! B ) 2 T , .. F 2 T , A ;! B 2 T . A ;! :B 2 T . ` (A ;! B ) ^ (A ;! :B ) ;! :A ( %), .. :A 2 T . . # T f:Ag
"/
1. T ""#, %& - A 2= T "- $"' T fAg . 2. T , %& - A & A 2 T , & :A 2 T . !. ,- - ; , T (;), .. %& - A & ; ` A, & ; ` :A. ! 12.14. T , . . '# T ""# A 2= T . 0" :A 2= T , -% 12.12 "- $"' T fAg , ""# T . '#, &, T . -", ""#. ( '* ' - B 2= T , "- T 0 = T fB g 12.13.
75
. ' T :B 2 T , .. :B 2 T 0 , % B 2 T 0 , .. T 0 . $ 12.15 ( ). T . " " " "" 8. &" T "- + + L . / "- ' 1" % ""# 0" # "- . " "" 8 #, %& ! # S T " +%% #. 0, % #, #, S T { . -", T0 { T0 = T 2S . ( '* '% - B - A1 : : : Ak 2 T0 , , ` A1 ^ ^ Ak ;! B ^ :B , 0 , "- - fA1 : : : Ak g { . #' 0 "- - T0 , '* '% T1 : : : Tk 2 S , , A1 2 T1 : : : Ak 2 Tk . ' S ', T1 : : : Tk "  %%, ", Tk . (, fA1 : : : Ak g Tk , .. Tk { . 0"" S % ! , 1#, T0 { "- $"'. / # "" &'". , " '-" - #, T0 { $"# . / % ' -. !. '# K { &+ " ' . ( K { Th(K) # $"# . "" . Th(K) ` A, '* '% - A1 : : : Ak 2 Th(K), ` A1 ^ ^ Ak ;! A. 6"" #'% .. M 2 K. #' M j= A1 ^ ^ Ak M j= A1 ^ ^ Ak ;! A ( &*" $"'), M j= A, .. A 2 Th(K). , Th(K) Th(M). %& .. M Th(M) , #' %& - A & M j= A, & M j= :A. .
.
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'# T { $"# . , .. M "#% T , 8A 2 T M j= A, .. '" ", T Th(M). C M { "# "- - ;, M { "# T (;). 0 , $"# " "#, 76
( #' %& " ). & ' - & -+ + " "" . $ 13.1 ($ 1 /). .
# 0 " " " -", ' - $'!#+ " #', - +, $'!#+ " ' 1# "' +"' '-% # , &' + + , - +, &3# ! 5, $'!#+ " "- #, & #+ "- $"# , # " & ", , -, " # " . , '# T { ' . - $"' & " A(x) L " '% ' cA , +*'% '' , & " ', T & " - 9xA(x) ;! A(cA ). &" ''% % T . 13.2. T . -" , T 0 , ' & " T # - 9xA(x) ;! A(cA ). -", T 0 . ( '* ' - U 2 T , ` U ^ (9xA(x) ;! A(cA ) ;! B ^ :B - B 1 '. ' 0 + & ( - 11.7(6)) ` U ;! ((9xA(x) ;! A(cA )) ;! B ^ :B ) ( #, #' % ` ((9xA(x) ;! A(cA)) ;! B ^ :B ) ;! :(9xA(x) ;! A(cA )) ! " !, '" ` U ;! :(9xA(x) ;! A(cA )): '# " y, + -. ( -% 12.1(1) ` 8y(U ;! :(9xA(x) ;! A(y))): .
77
" "' -% - & (". ." # - 13.2), '" ` U ;! (9xA(x) ^ :9yA(y)) .. 9xA(x) ^ :9yA(y) 2 T , "-, #' T . ( #, T , '* ' "- - B1 : : : Bn, - Bi " 9xAi (x) ;! Ai(cA ), - U 2 T - B ` U ^ B1 ^ : : : Bn ;! B ^ :B , .. '- "- $"' T fB1 : : : Bn g . 0 "-, #' "' 1 "- T fB1 g , # "- T fB1 g fB2 g , .. " # ## T0 = T Tn+1 = Tn n = 1 S 0 1 : : : -" Te = Tn . ( #, Te { n=0 . "" , 0 , 9A1 : : : An 2 Te, , ` A1 ^ ^ An ;! B ^:B - B , % '* ' m, A1 : : : An 2 Tm , .. '- Tm , "- ' "" 13.2. &" M { "- + ' Te. ."" #1, i
jMj maxf@0 jjg:
(13.1)
"" , i i 2 N { ' Ti , "*# "- - L maxf@0 jjg (". ." 2 % 5.1), .. i + 1-" 1 ' & &, " maxf@0 ji jg . / , 1- 1, jL j = maxf@0 jjg, .. "% '% &'% "*#. . '% - "*# " &3 + 0+ "- , .. "- + + + 0 "*. '# S { , -* T (". ""' &'"). " "- M 1 , - i
i
c c0
c = c0 2 S:
-", { 1 0 . #' c = c { " , c = c 2 S , .. 1 { $ . - c = d ;! d = c { " (". ! ! 11), .. c = d 2 S , S ` d = c, d = c 2 S . (" "" ""# 78
1 . # " " a = b ^ b = c ;! a = c # . '# # M = M= . / "- &' # " " T . ! i " &": c 2 M, i(c) # 0" c 1% . P { n-" " , -" i(P )(i(c1 ) : : : i(cn)) = () P (c1 : : : cn) 2 S: -" - , 0 . '# c1 d1 : : : cn dn i(P )(i(c1 ) : : : i(cn)) = , .. c1 = d1 : : : cn = dn P (c1 : : : cn) 2 S . #' (11.6), # '%*'% "' : ` c1 = d1 ^ ^ cn = dn ;! (P (c1 : : : cn) ;! P (d1 : : : dn)): % " ', (P (c1 : : : cn) ;! P (d1 : : : dn )) 2 S , # P (d1 : : : dn)) 2 S , .. i(P )(i(d1 ) : : : i(dn)) = . i(P )(i(c1 ) : : : i(cn)) = , , %, P (c1 : : : cn) 2= S: ( ' S :P (c1 : : : cn) 2 S: ( #, '- -, '*" &!, '", :P (d1 : : : dn)) 2 S , .. P (d1 : : : dn)) 2= S , ..
i(P )(i(d1 ) : : : i(dn)) = . , & #, .. M = hM ii "#% S , # T , # '%* ' -. %& $"' F (x1 : : : xn ) %&+ c1 : : : cn 2 M
M j= F (i(c1 ) : : : i(cn)) () F (c1 : : : cn) 2 S: (13.2) / ' -, &, &'" # '! % $"' F . ."", "+ $"' ' ! + " . -", , (13.2) F1 F2 , F1 ^ F2 .
"" , M j= F1 ^ F2 0 "', M j= F1 M j= F2 . -%, 0 0 "', F1 2 S F2 2 S . # ' % F1 ^ F2 2 S ( 12.6). -" #, , 13.2 F , - :F . M j= :F , F - M , ' (13.2), F 2= S , .., ' S , :F 2 S . :F - M, M j= F , #' F 0 # (13.2) , F 2 S , .. :F 2= S . '# # F " 9xG(x x1 : : : xn) G ' - (13.2) '- . 6"" # c1 : : : cn 2 M. -", M j= 9xG(x i(c1 ) : : : i(cn)). % $"' 79
", 0 , c 2 M M j= G(i(c) i(c1) : : : i(cn)), .., ' (13.2), G(c c1 : : : cn) 2 S . #, %& $"' A(x) ` A(c) ;! 9xA(x). "" , #' - (". " # - 12.2), '", A(c) ;! 9xA(x) 8x:A(x) ;! :A(c) -, * , # " . (" "" 9xG(x c1 : : : cn) 2 S . '#, &, , 9xG(x c1 : : : cn) 2 S . % n, - - ' Tn , '* ' , c, 9xG(x c1 : : : cn) ;! G(c c1 : : : cn) 2 S , .. G(c c1 : : : cn) 2 S ' ` p ^ (p ;! q) ;! q. (# -% '! M j= G(i(c) i(c1 ) : : : i(cn)), M j= 9xG(x i(c1 ) : : : i(cn)). , " #+ + &* "- '#, #' -% '- ". - (13.2), " " , . C 13.3 ($ { & ). T , , maxf@0 jjg: (13.1). & 13.4. !
, . '# - A &*", " . ( "- f:Ag . "" , ` :A ;! B ^ :B , , #' % ` (:A ;! B ^ :B ) ;! A MP, " '", ` A. # M { "# :A, ' &*" A, " M j= A M j= :A, "- & 13.5 ($ / #.. 2 ). ; , ; . ; " ", ; . 2 # '* '% - A1 : : : An 2 ;, ` A1 ^ ^ An ;! B ^ :B - B . '- "- fA1 : : : Ang , "- #' " "# "- G .
.
80
# " " ..,#! " " " # , &, #' # &+ ". "/ 13.6. '# K &+ c" ' , "' # " (.. -* " - " "). 2 P &+ " K #", , - " " M 2 K &% " P , ', " " M & 0" ". ", K { ' , ""' #", #', - - ' ' G ""' , %&+ a b 2 G ' , - fa bg ""' , .. %&+ a b 2 G a b = b a. 1" "" # ' &# -", .. - - ' %& ' - %. 2 - #", #' '* '% & ' , ' + - ' ( ' % # "). "" ' '- ' + + "- N, % -" N, " (-!). * " # { 0 ' &# !": - - ' ' Q { ! (-!), " ' Q ! . ..,#! & ", "-# " "
# # + &+ ". &, '& 1941 ' + + . ', # "& " &*" " ' #+ " ' ". / & - "" ! , - & , { ".
,
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2'# " ..,#! ", ## "- " " &" "- " $"' !# L "-&'# 1. , %'" 0 " " # # ' " ' . 81
"", ""' ' G ' ", "- 0" "- 1 , 8a b c 2 G(a b ;! a + c b + c): (14.1) , ' G ' ", %& ' H ' ", .. 1 ' (14.1) ' , ' 0"' ' % %& ' . "" ' "+ ' '- ' R ' , , ' Z Q. 1", ' 0 ' 1 R. %& ! ' '- "" ' , '% # ' #. "" , '# Zm = f0 1 : : :m ; 1g { ' mod m. -", Zm "- -&'# 1 4, ' %* (14.1). (, & 0 4 1, & 1 4 0. " ', " m ' (14.1) c = 1 ' , m = 0 Zm, '", 0 4 1 4 2 4 4 m ;1 4 0
.. ' 1 0" Zm , .
' " . , -", ' " ' #". '# G { & ' , - - ' ' ". -"' 0"' g 2 G " " cg (0" 0 2 G '%* "' " - "). 6"" '' = hfcg j g 2 Gg f+g fg ( " # , 0 + "- + " ). '# LOAG { "- " ' + & + ' ' ( # ""!). 6"" "- - L ; = LOAG f:cg1 = cg2 j g1 6= g2 g1 g2 2 Gg fcg1 + cg2 = cg3 j g1 + g2 = g3 g1 g2 g3 2 Gg:
- "- H ; " "#. "" . " 1# "- fcg1 : : : cg g, +*+ - H. 6"" ' ' H ' G, -'% 0"" g1 : : : gn. ' % 0 ' ' ". ' -'% ' cg 0" gi, " '", H "# H. n
n
82
( # " " ,#! "- ; " "# M = hM ii. #' M j= LOAG, M { ' & ' . " &- ' : G ! M , - '(g) = i(cg ). g1 = 6 g2 , M j= :cg1 = cg2 , .. i(cg1 ) 6= i(cg2 ). / , ' 3 . , g1 + g2 = g3, M j= cg1 + cg2 = cg3 , .. i(cg1 ) + i(cg2 ) = i(cg3 ). / , '(g1 + g2 ) = '(g1 ) + '(g2). (" &" ' { 3 ""$", .. '(G) "$ G, '(G) ' , ' ' ' M . 2 # G ' " 0 1. " " # '  "*, " & ". 2. -&'# - + # '  +, " + #.
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T ! ! " , k maxf@0 jjg '# M { & "# T . & " ' "- #+ + " fd j 2 Ag, jAj = k, T { "- - fd =6 d j = 6 2 Ag. ' "- $"' T 0 . "" , %& "- F T 0 - 1# "- + fd1 : : : d g. 6"" " "- - F "# M, "- fd1 : : : d g #" 0"" M. , " &" # ' "# F { - F , +* T , #' M, - fd =6 d i =6 j i j = 1 : : : n , #' 0 " 0"". (" "", - "- T 0 " "#, .. " " ,#! "- T 0 " "#, " " " { 2" '* ' "#, "*# + k, .. "*# 1 ' k. 2 ' "*# %& " T 0  & k, #' "- 0 ", * ! d " "*# k ( ! + ' "- T 0 ). (" &" T 0 , # T " "# "* k .
n
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j
83
, "- # " " { 2" { ( ", & " "- &# " "- " - #% "$" "" , M = hM ii { & " ' , T = Th(M), " " { 2" { ( # "* k > maxfjj jM jg "# M0 = hM 0 i0i T , "*# k. ' , T , Th(M0 ) = Th(M), 0 " "' &# "$", #' "% "*. 2'! ", 1# , ' " + & "+ 1 "# 0" "". 0" # 0 " ' ' " & ". ( " . 4% "-# "# $"' # $"# !, " " " "". ' + 1# , " "- $"' # "' + . 7'" &# " " " &' ". #' " - % "- " , "- # + + - " . ( $"' t = sup X 1 '%*" &":
8x(X (x) ;! x t) ^ 8s(8x(X (x) ;! x s) ;! s t): ( # " + 1 :
8X (9s(8x(X (x) ;! x s)) ;! 9t(t = sup X )): ' "" , %& ' , ' %* " + , "$ % * + . 0 . . # " $, &# X # "- '#+ .
*# ( 1" ' ' # ), "", " " { 2" { ( '!, " , "*# + $ , "$. "/ 14.2. , T "* k, " "* k "$. 84
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) T
).
k maxfjj @0g, T . T , '*c ' - A L , A 2= T :A 2= T . 0" ' -% 12.12 "- ; = T fAg ;0 = T f:Ag . " " { 2" { ( '* ' "# M "- ;, "*# k "# M0 "- ;0 , "*# - k. #' & 0 " % "" T , ' % 1 " "$. (, ' - 6.9, 0 " 0" 0 , "-, .. M j= A, M0 j= :A !. 1 . '%* $ 5 . "!
# 0 " "- , ", 9]. ', Th(DL) + + .'. "- , " ! 7, "* , #, . '* ' 0 $ & " ! 7 "* % , #' $, %& " 0" 0 (". - 7.7 ) ' . 2 . $ 62 . ! . % ', & "'+ %& "* , #, . " 0 (". " 3 ! 5) # " " & " ACln %& n 2 N & " AChar(p) p, & " AChar(0)n, n 2 N. "", '!, 1 0+ '+ "+, &*. " '%* $ .
! , "! . "", & "'+ "* . "" , &+ { & " Q "$ &"' "%, ", Q(e), ' " Q e, #' " 0+ .
85
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1. /.,#. ""'% '. " . { ,.: ', 1971. 2. .. "#. % ' + $'!. { ,.: ,, 1983. 3. .7', 6.-$$. "# . { ,.: ,, 1994. 4. /./. ,"" 0" "". { ,.: ,, 1987. 5. .0. ( "- ''"- . { ,.: ,, 1969. 6. .. , .,. . ,*# &+ "- . & &. { - , - , 1998. 7. .. , .,. . /" "" "- . 4# 2. , &. { - , - , 1988. 8. .. , .,. . /" "" "- . 4# III. , &. { - , - , 1990.
86