wcwcwc asked in 科學數學 · 1 decade ago

about mapping, onto, and set

i know it is easy, but i am having some trouble with the basic theories.

if i am going to construct an example of a continuous function

f:(0,1) -> R

which maps the open set (0,1) onto a closed set.

i understand wt a open set, a closed set is, but once the question come altogether, i am lost.

beside, could anyone give me a simple example to explain "onto" and "mapping"?

thx

3 Answers

Rating
  • prime
    Lv 4
    1 decade ago
    Favorite Answer

    Let

    h(x) = 2* x * pi - pi

    g(x) = tan(x)

    f(x)= g(h(x))

    It is easy to see h(x) is an 1-1, onto and conti mapping from (0,1) to ( pi, - pi ) and

    tan(x) is 1-1 , onto and conti on ( pi, - pi ).

    So f(g(x)) is 1-1, onto and conti on (0,1).

    Since

    lim (x->1-) f = oo

    lim (x->0+) = - oo

    We have f maps (0,1)-> (oo,-oo) = R.

    By def, R is close.(R is both open and close)

    mapping means a function.

    f is onto means the range of f = the codomain of f

    2008-09-19 12:29:12 補充:

    So f(g(x)) is 1-1 and conti on (0,1).

    2008-09-19 12:29:55 補充:

    tan(x) is 1-1 and conti on ( pi, - pi ).

    2008-09-19 12:33:50 補充:

    For briefly, You can say tan(x) is 1-1 , onto and conti from (-pi,pi) to (-oo,oo) imlies

    g(h(x)) is 1-1, onto and conti from (0,1) to (-oo,oo) = R.

    2008-09-19 12:34:16 補充:

    So g(h(x))) is 1-1 and conti on (0,1).

    2008-09-19 17:18:35 補充:

    1. R is both open and close.

    2. f is conti implies f^(-1)(U) is open for all open set U

    so the image of f may be anything.

    2008-09-19 17:23:12 補充:

    1. R is both open and close.

    2. f is conti implies f^(-1)(U) is open for all open set U

    so the image of f may be anything.

  • 1 decade ago

    你是說 f 的映像 (image) 是 closed set in R 吧?

    這很簡單,你不要找單調的函數,而是要找在 (0, 1) 內都有 maximum point 及 minimum point 的函數。

    所以,就定義 f: (0, 1) → R 為  f(x) = sin (2πx)

    明顯地,f 的 image 就是 [-1, 1], a closed subset of R. 因為 -1 及 1 為 f 在 (0, 1) 內的極小與極大值!! 再由 f 為連續,從中值定理得知 f maps (0, 1) onto [-1, 1].

    2008-09-19 17:43:29 補充:

    其實,我只是想舉一個 non-trivial 的例子, f 是常函數,則 f 的 image 為單點集,是一個 R 的閉子集。這問題有什麼困難的?

  • 1 decade ago

    thx, it explains sth.

    so i understand one to one means the function, one x only have one solution.

    so like f(x)=2x is one to one.

    f(x)=x^2 is not, because both x=2, and -2 gives the same solution.

    2008-09-19 13:13:45 補充:

    and on to means the solution appear and cover the space.

    like f(x)=2x is on to, because all value of x cover the space.

    f(x)=x^2 is not onto, because it only cover the >0 side. doesnt cover <0.

    2008-09-19 13:13:54 補充:

    so with my original question.

    with f:(0,1) ->R

    what is an example to map the open set (0,1) onto a closed set?

    the examples u gave me are useful, i have a better picture to understand onto and one to one. but it is an open set. how could that be mapped to a closed set?

    thx again

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