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1. m-²Ͳ 1-²Ͳ. m-Ͳ
A m- B, i g , i x N x A g(x) B.
e e i A m B, g: A m B, e , e g m- A B.
e Ae A |m B, ei A m B ei B m A.
ee ie m-eiei: A mB A m B B m A.
m-ii 1-ii.
A 1- B, i i' g , i x N x A g(x) B. e e i A 1 B.
i ie <1 , |1 1-eiei 1.
eeei i m-ii 1-ii.
r1) A m B, A 1B.
r2) ie 1 m eei i i.
r3) A m B m ; 1.
r4) A m B B , A ; e 1.
r5) A m B B , A ; e 1.
r6) A , e A m e m A; e 1.
r7) A m N A=N e 1.
r8) A m A= e 1.
r9) N m A A .
r10) N 1A A i eie .
r11) m A A N.
r12) A e i B B N, A m B.
r13) i B A m A B A m B A.
r14) i B A m A B A m B A.
r15) i A N A 1A N A N m A
1. , {x | = o} m {x | C}.
={x | = o} ={x | C}.
f(x, y)= ()+. s-m-n- s(x) , f(x, y)= s(x)() , y. . : ()=0 s(x)()=. = o s(x) C. , s(x) . s(x) : m .
g(x, y)= () . s-m-n- t(x) , g(x, y)= t(x)() , y. . : ()=C t(x)()=0. C t(x) = o. , B t(x) A. t(x) : B m A.
m B m A. m .
2. , {x | = o} m {x | }
={x | = o} ={x | }.
f(x, y)=0 (). s-m-n- s(x) , f(x, y)= s(x)() , y. . : ()=0 s(x)() . = o s(x) . , s(x) . s(x) : m .
g(x, y)= o( ( s-m-n- t(x) , g(x, y)= t(x)() , y. . : () t(x)()=0. t(x) = o. , B t(x) A. t(x) : B m A.
m B m A. m .
3. , {x | } m {x | D eie}.
={x | } ={x | D eie}.
f(x, y) = . s-m-n- s(x) , f(x, y)= s(x)() , y. . , . (z) z, s(x)()=f(x, y)=0 y. , s(x) = o, Ds(x) = N, s(x) . , . N , () . y f(x, y) , s(x)() . , Ds(x) , s(x) . s(x) , s(x) : m .
g(x, y) :
g(x, 0) = z( (z) );
g(x, y+1) = z( (z) & z g(x, 0) & z g(x, 1) & & z g(x, y)).
g(x, y) , s-m-n- t(x) , g(x, y)= t(x)() , y. . , Dx . g(x, y) N, t(x)() N. , t(x) , t(x) A. , Dx . t(x) , . , . , B t(x) A. t(x) : B m A.
m B m A. m .
4. : {x | D eie} m {x | eie}.
={x | D eie} ={x | eie}.
s-m-n- s(x) t(x) , N s(x)=Dx Dt(x)=x ( 4 5 7.3). s(x) B B t(x) A. , s(x) : m t(x) : B m A. m .
. {x | = o} m {x | C} m {x | } m
m {x | D eie} m {x | eie}.
5. , D m {x | }.
={x | }. g i . f(, )= . s-m-n- s(x) , s(x)()=f(x, y) , y. D s(x)()=f(x, y)=g(y) y, s(x)=g , s(x) . D s(x)() y, s(x)=f , s(x) . , D s(x) , s(x) : D m .
6. , D m {x | D }.
={x | D }. g , Dg . i f(, )= . s-m-n- s(x) , s(x)()=f(x, y) , y. D () , i t , () t i. t () i, f(x, y) t. s(x)() t, s(x) . , Ds(x) . , D s(x) .
D. () , N () i. , N s(x)()=f(x, y)=g(). Ds(x)=Dg , s(x) .
D s(x) , s(x) : D m .
eiei i m : dm(A)={B | A mB}. i eiei m-ee.
ie m i i m-eei ie , e m :
a m b, A mB i A a, B b
e a4x)). i , Ef .
e f(x)>4x i x Df . eie. i, i n N {0, ...,4n} i n eeei Ef , f(n)>4n, i eee Ef i eeei f(0), ..., f(n-1). n N {0, ...,4n} i >3n eeei , i eie.
e B i eie . i B=Eg e g. e k ie i g, g k . e f(x)= x( z( x(z)>4x)) ee, k i eie e. e, f(k) Ek Ef = Eg Ef = B Ef , B Ef
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