ch8-math dis

1
8.3 ??????????
?????????????????.??????, ???????.
??8.8 ???????A???B?????, ??A?B ????, ??
AB, ??, ?A?B???, ???AB.
?????????????.
?8.9 (1) ZN ?f Z?N,
?f?Z?N?????, ??, ? ZN.
2
8.3 ??????????
(2) N ? NN
???N ? N?N?????, ???N ? N??????????????,
?????.N ? N????????????????(??????)?????????.
??????????????, ??????????N ?
N?N????????.????????????, ?lt0, 0gt????, ?????????
lt0, 0gt, lt0, 1gt, lt1, 0gt, lt0, 2gt, lt1, 1gt, lt2, 0gt,
??? ??? ??? ??? ??? ? 0??? 1??
2??? 3??? 4 ? ? 5
3
8.3 ??????????
?ltm, ngt???????, ?????????k. ??m, n?k?????.
1). ??ltm, ngt???????????????12(mn)
(mn1)(mn)/2 2). ???ltm,
ngt????????????????ltm, ngt???????m. ??, ltm,
ngt???(mn1)(mn)/2 m 1??. ??, ?? k
(mn1)(mn)/2 m ??????, ???N ?
N?N?????f N ? N?N, f(ltm, ngt) (mn1)(mn)/2
m.
4
(3) NQ (??????)???????p/q(p,q????qgt0)???????.??,
???????????(???).
8.3 ??????????
?????????????????????
???0/1?????, ???????????????????.???????????N?Q?
??, ???????????????.?? 1/1, 2/2, 3/3,
?????1. ???? ??????, ????????????????.?
1/1???, ??, 2/2, 3/3, ?????. ???????????f N?Q,
??f(n)?n??????. ???? NQ.
5
8.3 ??????????
(4) (0, 1)R (?? (0, 1) x x?R, ?0ltxlt1
) ?f (0, 1)?R, f(x) tan(2x-1)?/2. ??,
f??????, ?ranf R. ??, f?????, ? (0, 1)R.
6
8.3 ??????????
(5) 0,1(0,1) (?? (0, 1) x x?R, 0ltxlt1
, 0,1 x x?R, 0 ? x ? 1
) ??0,1?(0,1)??????????0?1?????.??, ????????
1/2, 1/22, 1/23, , 1/2n,
??????????
0, 1, 1/2, 1/22, , 1/2n, ?
?? ?? ??? ? 1/2, 1/22,?1/23,
1/24, , 1/2n2,
??0,1????????????, ??, ?????????.
7
8.3 ??????????
?????f 0,1?(0,1)??????????????. ???? 0,
1 (0, 1).
(6) 0,1a,b (?a, b?R, a lt b)
??????lt0, agt?lt1, bgt?????. ??????????? f 0,
1?a, b, f(x) (b-a)x a ???? 0, 1a,
b. ????? (0, 1)(a, b).(?a, b?R, a lt b)
8
8.3 ??????????
?8.10 ?A?????, ?P(A) 0, 1 A.
?
????f P(A)?0,1A, f(A) ?A,
?A?P(A). ???A???A?????. ??, f????. ?????g?0,1
A, ??, ?g A? 0, 1 . ?, B x x?A, g(x) 1
?B?A, ?B g, ?, ?B?P(A), f(B) g. ????
f????. ??????? P(A)0,1A.
9
8.3 ??????????
??8.6 ?A, B?C?????,
(1) AA (2) ?AB, ?BA (3) ?AB, BC, ?AC
????.???????? ???????, ??????. ??????????8.6??
NZQN ? N, R0, 1(0, 1) ??????????
??????(?????, ????????????)??????R??. ??
N?R????? ?NR, ????????????????? ?NR,
??N???????????R??.
10
8.3 ??????????
??8.7(????) (1) NR (2) ?????A, ??
AP(A).
?
(1) ???? N0,1, ???? NR. ????? ????f
N?0,1??????. ???0, 1??????. ??x?0, 1, ?? x
0.x1x2, (0 ? xi ? 9) ???????
0.24999?0.25000. ????????x???. ???????????,
????????, ??x???0.25000.???????, ????f N?0,
1?????????????.
Georg Cantor is the father of set theory.
11
8.3 ??????????
?f N?0, 1??N?0, 1???????.????f?????? f(0)
0.a1(1)a2(1) f(1) 0.a1(2)a2(2) f(n-1)
0.a1(n)a2(n) ?y?0, 1???????, y?????0.b1b2,
??? bi ? ai(i), i1, 2, . ??, y????????,
?y?????????????????. ???? y ? ranf, ? f?????.
12
8.3 ??????????
(2) ???(1)?????, ?? ????g A?P(A)??????. ?g
A?P(A)??A?P(A)???. ????B B x x?A, x?g(x)
?B?P(A), ???x?A, ?? x?B ? x?g(x). ??, ?? x?A,
?? B ? g(x), ? B?rang. ??????????
NP(N). ??????????? N0, 1N. ???, P(N), 0,
1N?R???N?????. ???????????????????.
13
8.3 ??????????
??8.9 (1) ??????A???B?????, ??B???A,
??AB?B?????A, ???AB.
(2) ?A?B???, ?AB?AB, ??B????A,
??AltB ??B??????A, ???AltB.
?? NN, NR, AP(A), RNNltR, AltP(A),
NltN.
14
8.3 ??????????
??8.8 ?A, B?C???, ?(1) AA(2) ?AB, BA,
?AB(3) ?AB, BC, ?AC.
????????. ?????????????????????,
???????????????????. ??????,
???????A???B???????????.????, ????????f A?B?g
B?A, ????????
15
8.3 ??????????
???? 0, 1N0, 1. ?x?0,1????,
0.x1x2?x??????. ?????????, ???????????????1???,
?? x 0.1010111.?????x??0.1011000. ??x?0,1),
x 0.x1x2?x??????. ??f 0,1?0,1N, ?? f(x)
tx, tx N?0,1, tx(n) xn1, n 0, 1, 2,
?? x 0.10110100, ????x???tx? n ? 0 1
2 3 4 5 6 7 tx(n) 1 0 1 1 0 1 0
0 ??, tx?0,1N, ???x, y?0,1, x ? y, ? tx ?
ty, f(x) ? f(y).?????f 0,1?0,1N????.
16
8.3 ??????????
??????f???, ?????? 0,1N0,1.???????.??f
?????. ??t?0,1N, ??t(0) 0, t(n) 1, n 1,
2, . ??f?????, ??x 0.011????f(x)
t.???x????, ???x????0.100. ??,
?????x?0,1, ?? f(x) t. ??????, ????????g
0,1N?0,1. g????????f??, ??t?0, 1N, t N?0,
1, g(t) 0.x1x2, ??xn1 t(n). ?????,
?0.x1x2???x??????.??t1, t2?0, 1N, g(t1)
0.0111, g(t2) 0.1000. ??g(t1)?g(t2)????????,
?g(t1) g(t2) ?????????, ?g(t1) ?
g(t2). ???????????????, ?????g????.
17
8.3 ??????????
???8.8?? 0,1 N 0,1. ????????? 0,1
N R. ???????, ???? N Z Q N ? N R
a,b (c,d) 0,1 N P(N) 0,1 A P(A)
N lt R A lt P(A) ??a, b, (c,
d)??????????????.
18
8.3 ??????????
??, ????????????????.??????????????.?????????
?. ????????, ???????????,
???????????????????, ??????????????. ??????,
??????????????.
??8.10 ???? ???n??????????????, ?
0 ? n n? n
?a? a ?a???, ??a, ? a a? a
19
8.3 ??????????
????????????????????.????????? 1.
??????n, ? n n 2. ??????n?m, ?m?n, ?m n
3. ??????n?m, ?m?n, ?m?n 4. ??????n?m,
???? m?n, m n, n?m, ????????,
????????????. ?????????, ??????????????, ?
??????m?n, m n ? m n m lt n ? m ? n
20
8.3 ??????????
??????????????????????.
??8.11 ????????, ????????????? ???????????,
??????.
?? a, b, c????, ??3 0,1,2, a, b, c
0, 1, 2 3 ?N?R?????, ????????N?R??. ??????
???? ???????????????.
21
8.3 ??????????
??8.12 (1) ??????A, ??A????????????A???,
??cardA(????A), ? cardA n ? A n
(2) ?????N??????0(??????) , ?cardN 0 (3)
???R??????, ? cardR ?
??8.13 ?A?B???, ?
(1) cardA cardB ? AB (2) cardA ? cardB ?
AB (3) Card lt cardB ? cardA ? cardB?cardA ?
cardB
22
8.3 ??????????
???????? cardZ cardQ cardN ? N ?0
cardP(N) card2N carda, b card(c, d) (2N
0, 1N) ?0 lt ? ????? ??????????????.????,
????. ???????A, ??? A lt P(A), ??, ? cardA lt
cardP(A) ???????????.
23
8.3 ??????????
???????????????, ??? 0, 1, 2, , n, , ?, ?0,

• 0, 1, , n, ??????, ????????, ??????
• ?0, ?, ????????, ???????
• ?0????????, ??????????? cardP(R)?.

24
8.3 ??????????
??8.14 ?cardA ? ?0, ????A????????.
?? a, b, c , 5, ???Z, ????Q, ??N ?
N?????? ???????, ???????????????.??????????????
?????. ?????????, ????????, ???????????. ???R?????
?R???????????.
25
8.3 ??????????
???????????

• ?????????????
• ???????????
• ??????????????
• ????????????????
• ???A???P(A)?????

26
8.3 ??????????
?8.11 ????????. (1) T x x???BASEBALL????
(2) B x x?R, x2 9, 2x 8 (3) C
P(A), A 1, 3, 7, 11 ? (1) ?T B, A, S,
E, L?? cardT 5 (2) ?B ??? cardB 0
(3) ?A 4?? cardP(A) P(A) 24 16
27
?8.12 ?A?B???, cardA ?0, cardB n, n????, n ?
0, ?cardA ? B. ?
1) ?cardA ?0, cardB n?? A?B?????. ? A
a0, a1, a2, , B b0, b1, b2, , bn-1
??ltai, bjgt, ltak, blgt?A ? B, ? ltai, bjgt
ltak, blgt ? ik?jl ????f A ? B?N f(ltai, bjgt)
inj, (i0,1,, j0,1,,n-1) ??, f?A ?
B?N?????. ??, cardA ? B cardN ?0. 2)
?????????? ?cardA ?0, cardB n??
A?B?????. ????3?? A ? B?????, ??, cardA ?
B??0. ?B ? ??, cardA ? cardA ? B. ???? cardA ? B
?0.
28
8.3 ??????????? ----?????????,?