1、 Bq Hf G g 2011 S 8 kotaromath.titech.ac.jp1 Bqfw 1.1 B B : $ 0 w B B set qMO *1 :w B wwx BpK *2 : .w B N T : .w B Zg : .w B Q : .w B R.A B RoMqmqmw 0 fw Bw A i element, member|KMx tlox : point qqUK B A t 0o x U A wA pK x U A wA psMqMOqfg x A“, x 6 A“ q XhqQy 2 N, 0 =N,*3 R, N =R pK BwG . $t BG btx|
2、Qy(1.1) x|x x :Tm x x 2 p srwOt A E b | fUhbV E q bqU MqXt x U : “ pKqU s pKqVx (1.1) x N|x x 2 p q bqK x tb E P(x) ;MoA = x|P(x) qbqV x A“ pKqx P(x) U Rqm “ qq pK1.2 A B B B wbowA U B A wA pKqV B x A w B subset pKqMMB A“ q b *4bsj *5B A“ x B x A“:qXt B = x|P(x), A = x|Q(x) q oMqV| B A“ pKqx w x t 0
3、oP(x) U Rqmsy Q(x) U Rqm “ q *6q pK2011 4 D 12 *1 jwOsstx Bw )QoMsM Bgtlo)Q (20 HD s txalhg $B ) x B “ tqr bU|w c QwppxsM*2 : .w B N T Z, Q RbqUpVUUKyr bqKQcOMO Bw O oSO :w Rxr s Hw $pOxc*3 0 :qbvQK*4 sw XwJ px|wq B j Aq oMOpK*5 “ x pKq *6 8x : P(x) ) Q(x)“ srq X1 B A qmhsM B B empty set qMM q X Bx
4、w Bw BpK Bw s mw B A, B U A B Tm B A hoMsy x A pKqq x BpKqx pKbsj mw Bx bA = B“ A B Tm B A“: B t B A pKlo A = B psMqV| B x A w B proper subset qMloB ( A“ hx B $ A“ q X1.3 B 4 B B mw B A, B t 0oAB = x|x A hx x B=x|(x A)(x B);AB = x|x A Tm x B =x|(x A)(x B)fg A, B w B union, intersecion qMO *7 B A, B
5、t 0oU Rqm(1.2) AB = B A A AB AB = B A AB AA = A A = AA = A AA = AU Rqmt| B A, B, C t 0oAO S| O (1.3) (AB)C = A(B C) A(B C) = (AB)(AC)(AB)C = A(B C) A(B C) = (AB)(AC)U Rqm4 B K B X w BwQOspx X wq BKMx . BsrqMOqtb X w B A t 0oX A = Ac = x X|x 6 A = x X|(x A) A w (X tSZ ) 4 B complement qMO *8qXt(1.4)
6、(Ac)c = A; Ac A = X; Ac A = ; Xc = U Rqm*7 P Tm Q“ (P and Q) wq P Q“, P hx Q“ (P or Q) wq P _Q“ q X*8 X nA“ wq X A“ q XqKh P psM “ q :P“ q X2wO . B X w B A, B t 0oU Rqm ( de Morgan wO )(1.5) (AB)c = Ac Bc; (AB)c = Ac Bc:tx;M *94 J 1.1. E P, Q t 0o(P Q) (P)(Q); (P Q) (P)(Q)U Rqm1.4 B B B A w B .w B A
7、 w B power set q| P(A) hx 2A q X J1-1 B A, B Uqt X w Bsy| AB x X w BpK B Y U B A, B Mw Bsy| Y x AB w BpK1-2 wO (1.5) sM1-3 B A = 1;2;3 yw B P(A) ywA boKsM1-4 t n xwA m B A w B P(A) wA wx :x 2n pK1-5 R2 = (x;y)|x;y R q|X = (x;y) R2 |xy = 0; O = (0;0)A = (x;y) R2 |x = 0; B = (x;y) R2 |y = 0qbqV| A, B
8、p X, O sM1-6 ( ) V w X, Y t 0oX + Y = x+y|x X;y Y X q Y w ( qow ) qMO X X + Y pKqsM X Y X + Y pKqsM X Y = X + Y qshw A G Ex?T*9 wO qzy32 B X wA x X t 0o| B Y wA f(x) Y 0 d 0 wF f X TY w | X f w , Y f w qMO *10 f w U X| U Y pKqf: X Yq Xw f U x X t 0o f(x) Y 0 d|qMOqbqVx| “w Et 7“ ;Mof: X 3 x 7 f(x) Y
9、q XqXt| U R C pKOs f : X R (C) X w : ( : : : : ) qMOqUK Mv f : X Y q A X t 0of|A: A 3 x 7 f(x) Yp)Q f|A: A Y f w A w Mv qMO qo f: X Y U)QoMqV| A X, U Y t 0of(A) = f(x)|x A Y; f1(U) = x X|f(x) Ufg f t A w , U w o q *11 f(A) xf(A) = y Y |f(x) = y qs x A U Ob q bqpV f : X Y q A;B X, U;V Y t 0oU Rqm(2.1
10、)f(AB) = f(A)f(B) f(AB) f(A)f(B)f1(U V ) = f1(U)f1(V ) f1(U V ) = f1(U)f1(V )f(A)f(B) f(AB) f1(U)f1(V ) = f1(U V )A f1(f(A) Uf(f1(U) u B X, Y t 0o|X Y = (x;y)|x X;y Y X q Y w u qMO 2.1. R2 = (x;y)|x;y R x RR qsbqUpV u X Y t 0o|(2.2) X : X Y 3 (x;y) 7 x X; Y : X Y 3 (x;y) 7 y Yfg H R | H R w qMO2011
11、4 D 19 (2011 4 D 26 Y )*10 qMOt? f t X w f(X) wpOqK*11 MGiU|A x 2 X t 0o f(x) 2 Y pKU| A X t 0o f(A) Y pKh| f 1 xo :wGq apKU| f U opsM p 4 f : X Y t 0ograph(f) =(x;f(x) X Y |x X= (x;y) X Y |x X;y = f(x) X Y f w qMO o f : X Y U pKqx f(X) = Y U RqmqpKh| o pKqx|x1 6= x2 f(x1) 6= f(x2)U w x1, x2 X t 0o
12、RqbqpK f U Tm opKqV f x o pKqMO 2.2. B X T X w idX : X 3 x 7 x X s pKqMO s x opK B X w B A t 0oiA: A 3 x 7 x Xp iA: A X A qMOA x opKtA iA U pKhw A G Ex A = X qsqpK R f: X Y q g: Y Z t 0og f : X 3 x 7 g f(x) = g(f(x) Zp)Q g f : X Z f q g w R qMOo f : X Y t 0o| g: Y X pg f = idX; f g = idYqswU ObqV g
13、f w o qMM| g = f1 q Xg 2.3. f wo U Obhw A G Ex f U oqsqpK J2-1 B X = 1;:;m| Y = 1;:;n t 0o X T Y w .w Bx nm xwA Ts2-2 (2.1) | spsMwx sU RqhsM .KsM2-3 f : X Y U ( o ) pKhw A G Ex| Y|graph(f) U ( o ) qsqpKhi Y : X Y Y x H R wpK2-4 f : X Y , g: Y z t 0o g f U osy f x opK| g f U s g x pK2-5 f : X Y q g:
14、 Y X p| g f = idX Tm f x psMOs .KsM2-6 X, Y | f : X Y qb5 f U opKhw A Ex f1(0Y) = 0X qsqpKhi 0X, 0Y xfg X, Y wpK X q Y wiUqtvp bqV| f U pKhw A G Ex f U opKqpK2-7 R3 w BX = S2 (0;0;1) S2 =(x;y;z) R3 |x2 + y2 + z2 = 1t 0o|f : X 3 (x;y;z) 7 f(x;y;z) = ( ; ) R2; hi = x1 + z; = y1 + zt f: X R2 w x opKq|o
15、 sM63 Bw S 0 s 2 mw B X, Y wt o f : X Y U ObqV| X q Y x 0 s pKqMO4 J 3.1. B X, Y , Z t 0o X q X x 0 spK X q Y U 0 ssy Y q X x 0 spK X q Y U 0 s|Tm Y q Z U 0 ss X q Z x 0 spK 3.2. : . B N xwwq 0 spK : m t 0o m w : .w B Nm = k N|k = mM f(k) = k + m1 qbq f : N Nm x opK T : .w B ZM| f : N Z f(x) =x2 (x
16、x : )x12 (x x- : )q yM u N N g : .w B Q 3.3. : .w B R xwwq 0 spK (0;1)M| f(x) = 12(1 + x=(1 +|x|) x o f : R (0;1) )Q (0;1M| (0;1 qww o f: (0;1) (0;1 f(x) =12n 1(x = 12n;n = 1;2;:)x (otherwise)q q|x opK 3.4. mw B X, Y U 0 spKqV| X q Y w Sx sM qMM| |X| = |Y| q Xv Bqv B4 J 3.5. : m, n t 0o 1;:;m q 1;:;
17、n U 0 spKsy m = n pK m tb : $ T m1 = n1 pK2011 4 D 26 (2011 6 D 14 Y )*12 w xhMoM wt “ tyoM| tb bq7 3.6. BpsM B X U v B pK|qx|K : m U Oo m Os oSXwqbwqV g(x;y)= 0:x1y1x2y2 : qbq g x o Bw Sg 3.13. B X t 0o |X| | yj = f(j) q|fw G : y1 = 0:a11a12a13 :y2 = 0:a21a22a23 :.qoSXhi| : 2x $tsOs oSX j = 1;2;: t
18、 0o bj 0;1;:;9 bj 6= ajj qsOtq : 0:b1b2 : x y1, y2,.wMcqs104 u RgD B B 2;:;n tlo 4 Zh B qx n xw B A1, A2, ., Anw BwqpK B = N = 1;2;: t 4 Zh B qx| Bw (v ) A1;A2;: wqpK = R qo| : R t 0o U = x Q|x bq|ba| = |ban + an a|5|ban|+|an a| = |an b|+|an a|:p| w Yw : “ t 0o n = N s |an b| 0. w x, y V t 0o hx;yi
19、= hy;xi. w x, y, z V q , R t 0o h x+ y;zi = hx;zi+ hy;zi.M| : :qb n 0 A U Y pKqx| w x Rn 0 t 0otxAx 0 qsqpKp| x xqs| 11 q oMwqV|sM n 0 A U Y pKhw A G Ex A w Ubo Yw :pK Rn w w ux (x;y) 7 txAy wtTZhi A x Y 0 pK7-8 g 7.13 sM s b tx| uww s;M .)7-9 g 7.15 sM ( g 7.10 ;M .)238 mw qm 8.1. psM B X t 0o| d:
20、X X R U X w m (m : ) pKqx|w EhbqpK w x, y X t 0o d(x;y) = 0 sx x = y wqVp|fwqVtv ( Y Q ) w x, y X t 0o d(x;y) = d(y;x) ( 0 Q ) w x, y, z X t 0o d(x;z) 5 d(x;y) + d(y;z) ( s ).wqV| B X qm : d w (X;d) m qMO 8.2. Rm wm dE xm :pK *17 (Rm;dE) m i qMO wpsM B X tm bqUpVM| x, y X t 0oddisc(x;y) =1 (x 6= y)0
21、 (x = y)q q ddisc x X wm :pK mm q| (X;ddisc) mmqqm 8.3. R w V w qx , | |: V 3x |x| R p w x V t 0o |x|= 0 sx x UwqVt Rqj|fwtv w x V q R t 0o | x|=| |x| w x, y V t 0o |x+y|5|x|+|y|.hbwpK | | U h (V;| |) qMOg 8.4. (V;| |) t 0o d(x;y) = |yx| q q d x V wmqs 8.5. R 3 x t 0o 0 |x| 0 d | | x R w)QwT mx R wm
22、pK Rm 3x = (x1;:;xm) t 0o|x|1 := |x1|+|x2|+|xm|qSXq| | |0 x Rm w)QwT Rm wm : d1 q X Rm 3x = (x1;:;xm) t 0o|x| := max|x1|;|x2|;:;|xm|2011 5 D 28 (2011 6 D 21 Y )*17 sx ot d q MhU|sxMMsmq zbht dE q Xqtb24qSXq| | | x Rm w)QwT Rm wm : d q Xg 8.6. R w V w u h ; i ( J 7-7 ) t 0o|x| :=hx;xiq q|x V w)Q t| u
23、tow s() |hx;yi|5|x|y|U RqmM| x = 0 s sU RqmU| x6= 0 wqVx|v = y hx;yi|x|2 x t 0o hv;vi= 0qMOT () Uhijt w H 1| H 2 w Qx uw QTbYt wp H 3 w Q ( s ) fO|x+y| = (hx+y;x+yi)1=2 = |x|2 + 2hx;yi+|y|21=25|x|2 + 2|x|y|+|y|21=2 = |x|+|y|: 8.7. Rm w u ( H 7 ) T xp|fU Rm wmxmpK :w ) qmw Q 8.8. m (X;d) w : xn U x X
24、 t ) b qxlimnd(xn;x) = 0U RqmqpKwqlimnxn = xq Xw q 7tbqUpV4 J 8.9. m (X;d) w : xn U x t ) |Tm y t ) bsy x = y pK 8.10. B X w 2 mwm : d1, d2 U pKqx| Yw : A, B pAd1(x;y) 5 d2(x;y) 5 Bd1(x;y) (x;y X)U RqmwU ObqpK4 J 8.11. mU pK|qMOx| B X wm : .w Bw )Q J 8.12. B X w 2 mwm d1, d2 U pKqbwqV| X w : xn U d1
25、 to xt ) bqq d2 to x t ) bqx pK25 Yw : A, B t 0o Ad1 5d2 5Bd1 U RqloMqb xn U d1 to x t ) bsy|0 5d2(xn;x) 5Bd1(xn;x) 0 (n )iT xn x d2 to x t ) bot xn U d2 to x t ) bsy d1(xn;x) 5A 1d2(xn;x) swp xn x d1 to x t ) b 8.13. Rm wm dE, S| 8.5 w d1, d xMt pKM1md1(x;y) 5 dE(x;y) 5 d1(x;y); d(x;y) 5 d1(x;y) 5
26、md(x;y)U Rqm J8-1 g 8.4 sM8-2 8.5 w | |0, | | xqt Rm wpKqsM8-3 t| p 1 qs : p qm | x = (x1;:;xm) t 0o|x|p :=(|x1|p +|x2|p +|xm|p)1=pq q| | |p x Rm w)Q ( x WXM ) w xRm t 0olimp+|x|p = |x|pKqTsM8-4 m (X;d) w B U X t 0o d0 = d|UU qbq (U;d0) xmpK8-5 m (X;dX),(Y;dY ) t 0od: (X Y )(X Y ) 3(x1;y1);(x2;y2)7 dX(x1;x2) + dY (y1;y2) Rx X Y wm)Q dX q dY w umqMO8-6 B X wm : d t 0o|p)Q dj xm :pKqsM d1(x;y) = log1 + d(x;y). o Cs C2- : : 0;) R p (0) = 0, 00(x) 0 R3Q x = (x0;x1;x2), y = (y0;y1;y2) t 0o d(x
