推 PttFund:Yes140.112.218.142 07/23
※ 引述《PttFund (批踢踢基金只進不出)》之銘言：
: 請證明存在一函數 f 使得 f: (0,1)╳(0,1) → (0,1) 為 1-1 且 onto.
其實就是在證R跟R^2有一樣的勢
a=0.a1a2a3....
b=0.b1b2b3....
(捨棄從某位開始都是九的表示法)
f(a,b)=0.a1b1a2b2xxxxxxxxx 即為所求
then f is 1-1, but not onto since 0.919191919191xx isn't in the image
however, R-Q lies in the image of f,
and that im f contains infinitely many rational numbers.
Therefore, there exists g:(0,1) -> (0,1) such that
g leaves all irr fixed, and g maps Q in im(f) to Q 1-1 and onto.
then g(f) satisfies the condition.
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 220.135.132.108