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1 2(beta4++)3WORKEDOUTANDTEXIFIEDBY 4(E-mail:xiaoxinpan@163.com)HONOREDREVIEWER (chouxiaoya@bbs.pku.edu.cn)September1,2004 !# $ % & ’ )(...... *,+.-0/#1.2,354 6 798 : ; = 8 ?@BAThanksatrillion!!!Bow......1CEDGF5H IKJ#L M NPO5Q5R SPT#U V SXW)Y Z.NX[]\ ^ I _a‘]b N2002c6de1‘fN2003c1de2gihEj2kmlonqprtsoumvwyx{zi| } ~ f N fIr q o N i umvf i umv N q u v(J GSE S f ¡o¢) _{£aNE⁄ ¥emailƒ § G ¤ ¤K'3‘ “ «a‹ nbeta› fiHofl ‘ “ –f†·‡vm Gfi•¶H‚NEHofl› fi•„o”( » ffi‰…) +N ¿G ‘m“ ` ´mˆ ‘fN+fiH ˜=´mˆgHG¯„i”m ¿˘ ˙` ¶Gfi¨ m ‘NE˚i¸ f†t‡ ˝ ˛rˇs fi•v ( fiH ˜— sH10) 4 ¤K \‚^iI02CS a ‚ Iq ˘ fi^ ˜ afim f ‹m ¤[f^m BBSÆfichouxiaoyaStedySakaruSyitianxingSxuening¢a tª _ ˙^q˜G • fi‰Ł l Ø S Œiº q zGfi ‹ ¤ ¤ 'Contents1 mæææ32 f X ııı223 527łłł662Chapter1øøøœœœ1:(1)f2g;(2)f1;4;9;16;25;36;49;64;81;100;121;144;169;196g;(3)f1;8;27;64g;(4)f0;1;2;:::g;(5)f2;3g;(6)fa;b;c;d;e;f;g;h;i;j;k;l;m;n;o;p;q;r;s;t;u;v;w;x;y;zg:2:(1)f(x;y)jx;y2R^x2+y21g;(2)fj==4+k^k2Zg;(3)fxjx2N^x8g;(4)f(x;y;z)jx;y;z2N^x2+y2=z2g;(5)fxjx2R^x2+5x+6=0g.3:(1);(4);(5);(6);(8);(9)ßf ] G ß 4:(1) Proof:A2B^BC()A2B^8x(x2B!x2C)( )=)A2B^(A2B!A2C)(x/A)=)A2C( )Q.E.D.(2) A=fag;B=ffagg;C=ffag;fbgg A2B^BC !A*C (3) A=fag;B=fa;bg;C=ffa;bg;fb;cgg AB^B23C !A=2C (4) A=fag;B=fa;bg;C=ffa;bg;fb;cgg AB^B2C !A*C 5: A=fag;B=ffagg;C=fffaggg A2B^B2C !A=2C 6:(1)0 ;1 fag;fbg;fcg2 fa;bg;fa;cg;fb;cg3 fa;b;cg f;;fag;fbg;fcg;fa;bg;fa;cg;fb;cg;fa;b;cgg(2)0 ;1 f1g;ff2;3gg2 ff1g;f2;3gg f;;f1g;ff2;3gg;ff1g;f2;3ggg(3)0 ;1 f;g;ff;gg2 f;;f;gg f;;f;g;ff;gg;f;;f;ggg(4)0 ;1 ff1;2gg f;;ff1;2ggg(5)0 ;1 ff;;1gg;f1g2 ff;;1g;1g f;;ff;;1gg;f1g;ff;;1g;1gg7:# 8:(1)f4g;(2)f1;3;5g;(3)f2;3;4;5g;(4)f2;3;4;5g;(5)f;;f4gg;(6)ff1g;f1;4gg.49:(1)f 7; 6; 5; 4; 3; 2; 1;0;1;2;3;4;5;6;7;8;9;12;15;16;18;21;24;27;30;32;64g;(2);;(3)f 7; 6; 5; 4; 3; 2; 1;4;5g;(4)f 7; 6; 5; 4; 3; 2; 1;0;3;4;5;6;9;12;15;18;21;24;27;30g.10:$ %P(A)=f;;fagg PP(A)=f;;f;g;ffagg;f;;faggg ’&(1);(2);(4);(5) G 11:Proof:( )+* ,A B=A A\B=(A B)\B(A B=A)=(A\B)\B(-+. / 021)=A\(B\B)(3 æ 1)=A\;(4’521)=;(6 1)7 8* ,A\B=; A=A\E(9;: 1)=A\(B[B)(=?1)=(A\B)[(A\B)(8+@1)=;[(A\B)(A\B=;)=A\B(9;: 1)=A B(-+. / 021)A B C D ’E F A B=A,A\B=;Q.E.D.12:G HJI K L’ Lemma1.1M2N OP mæAPB QA B=A,A\B=; Proof:R S11TU Q.E.D.Lemma1.2M2N OP mæAPB QA B=;,AB Proof:A B=;():9x(x2(A B))(; )5()8x:(x2(A B))(V W+X Y[Z \)()8x:(x2A^x=2B)(] M^- )()8x:(x2A^:x2B)(=2 )()8x(:x2A_x2B)(_ T ‘2a bdcfe2g 1)()8x(x2A!x2B)(h i[Y[Z \)()AB( )Q.E.D.(1)j (A B)[(A C)=Ak?l mnkA\B\C=; Proof:(A B)[(A C)=A()A (B\C)=A(bdcfe2g 1)()A\(B\C)=;(Lemma1:1)()A\B\C=;(3 æ 1)Q.E.D.(2)j (A B)[(A C)=;k?l mnkA(B\C) Proof:(A B)[(A C)=;()A (B\C)=;(bdcfe2g 1)()A(B\C)(Lemma1:2)Q.E.D.(3)j (A B)\(A C)=;k?l mnkA(B[C) Proof:(A B)\(A C)=;()A (B[C)=;(bdcfe2g 1)()A(B[C)(Lemma1:2)Q.E.D.(4)j (A B)\(A C)=Ak?l mnkA\(B[C)=; Proof:(A B)\(A C)=A()A (B[C)=A(bdcfe2g 1)()A\(B[C)=;(Lemma1:1)Q.E.D.13:(1)G HJI K L’ Lemma1.3M2N OP mæAPB Q A\BAPA\BBProof:6M o N Ox x2A\B()x2A^x2B( mæU.[ )=)x2A(_ T ‘2a prq;1)& A\BA 9s E[HU A\BB Q.E.D.Lemma1.4M2N OP mæAPB Q AA[BPBA[BProof:M o N Ox x2A=)x2A_x2B(_ T ‘2a t u21)()x2A[B( mæ+v2 )& AA[B 9s E[HU BA[B Q.E.D.wH x2T Proof:(A B) C=(A\B)\C(-+. / 021)A\B(Lemma1:3)(A\B)[(A\C)(Lemma1:4)=A\(B[C)(8+@1)=A\(B\C)(bdcfe2g 1)=A (B C)(-+. / 021)Q.E.D.(2)j k?l mnkA\C=;yP (1)=;Y z Proof:G H7 8* kA\C=;y (A B) C=(A\B)\C(-+. / 021)=(A\C)\B(3 æ 1J{ . 021)=(A C)\B(-+. / 021)=A\B(Lemma1:1)=(A\B)[;(9;: 1)=(A\B)[(A\C)(A\C=;)=A\(B[C)(8+@1)=A\(B\C)(bdcfe2g 1)=A (B C)(-+. / 021)wH( )J* , |] 2} ~x x2A^x2C y) x JX oB 2 x=2(A B) CPx2A (B C) ^ 2 U (A B) C=A (B C)4’5U 7Q.E.D.14:Proof:B=E\B(9;: 1)=(A[A)\B(=?1)=(A\B)[(A\B)(8+@1)=(A\C)[(A\C)( [ )=(A[A)\C(8+@1)=E\C(=?1)=C(9;: 1)Q.E.D.15:A=B=D=G{C=F=H 16:(1)f3;4;f3g;f4gg;(2);;(3)f;;f;gg;17:(1)f;;ff;gg;fff;ggg;ff;g;ff;gggg;(2)f;;f;g;ff;gg;f;;f;ggg;(3)ff;g;ff;ggg;18:(1)f;;1;2;3g;(2);;(3);;(4);.19:(1)A[B;(2)A;(3)B.20:G HJI K L’ Lemma1.5M2N OP mæA;B;C;D QAB^CD)A[CB[DProof:8M o N Ox x2A[C()x2A_x2C( mæ+v2 )()(x2A_x2C)^(x2A!x2B^x2C!x2D)( [ & )=)x2B_x2D( * )()x2B[D( mæ+v2 )Q.E.D.Lemma1.6M2N OP mæA;B;C;D QAB^CD)A\CB\DProof:M o N Ox x2A\C()x2A^x2C( mæU.[ )=)x2B^x2C( [ & )=)x2B^x2D( [ & )()x2B\D( mæU.[ )Q.E.D.wH x2TU Proof:A=A\E(9;: 1)=A\(C[C)(=?1)=(A\C)[(A\C)(8+@1)(B\C)[(B\C)(T &Lemma1:5)=B\(C[C)(8+@1)=B\E(=?1)=B(9;: 1)Q.E.D.21:(1)j A\B=Ak?l mnkAB Proof:A\B=A()8x(x2A\B$x2A)(Y z2 )()8x((x2A^x2B)$x2A)( mæU.[ )()8x(((x2A^x2B)!x2A)^(x2A!(x2A^x2B)))(Y 2 +3 W )()8x((:(x2A^x2B)_x2A)^(:x2A_(x2A^x2B)))(h i[Y[Z \)()8x((:x2A_:x2B_x2A)^(:x2A_(x2A^x2B)))(_ T ‘2a bdcfe2g 1)()8x((:x2A_x2A_:x2B)^(:x2A_(x2A^x2B)))(_ T ‘2a . 021)9()8x((:x2A_x2A_:x2B)^((:x2A_x2A)^(