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Math Help - subrings

  1. #1
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    subrings

    Are the following subrings of the ring of all functions from the closed interval [0,1] to \mathbb{R}:
    1. the set of all functions which have only a finite number of zeros, together with the zero function
    2. the set of all functions f such that lim_{x -> 1^{-}} f(x) = 0
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  2. #2
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    Quote Originally Posted by dori1123 View Post
    Are the following subrings of the ring of all functions from the closed interval [0,1] to \mathbb{R}:
    1. the set of all functions which have only a finite number of zeros, together with the zero function
    2. the set of all functions f such that lim_{x -> 1^{-}} f(x) = 0
    For the first one, think about what kind of functions have finite zeros or are constant zero. The polynomials form a ring, but are there other elements in this subring?

    For the second, simply check the subring properties. If you add two functions with this limit property, does the result also have this limit? It is straightforward.
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  3. #3
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    Quote Originally Posted by robeuler View Post
    The polynomials form a ring, but are there other elements in this subring?
    Yes there are many because there is no condition on continuity.
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  4. #4
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    Quote Originally Posted by robeuler View Post
    For the first one, think about what kind of functions have finite zeros or are constant zero.
    A function with finite zeros, does that mean finite number of roots?
    I am trying to show it's closed under multiplication, so the product of two functions with finite number of zeros will also have finite number of zeros. But how can I show that?
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  5. #5
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    Ok you seem to be heading in the wrong direction because the first one is not a subring.

    Here is a counter example:
    f(x)=1
    and
    g(x)=\left\{\begin{array}{cc}-1,&\mbox{ if }x\leq 0.5\\1,&\mbox{ if }x>0.5\end{array}\right.
    Both f and g are in the subring but
    f(x)+g(x)=\left\{\begin{array}{cc}0,&\mbox{ if }<br />
x\leq 0.5\\2, & \mbox{ if } x>0.5\end{array}\right.
    is not.
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  6. #6
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    Why isn't f(x)+g(x)=\left\{\begin{array}{cc}0,&\mbox{ if }<br />
x\leq 0.5\\2, & \mbox{ if } x>0.5\end{array}\right. in the set?
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  7. #7
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    There is an infinite number of zeroes because there is an infinite number of x values between 0 and 0.5.
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  8. #8
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    Thanks, I got it.
    Last edited by dori1123; December 8th 2008 at 05:32 PM.
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