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Question about injection, surjection, bijection, and mapping

  1. Oct 26, 2008 #1
    f(x) is a bijection if and only if f(x) is both a surjection and a bijection. Now a surjection is when every element of B has at least one mapping, and an injection is when all of the elements have a unique mapping from A, and therefore a bijection is a one-to-one mapping.

    Let's say that f(x) = x3-x+1.

    It's easy to see that it is not injective by showing that f(1)=f(-1)=1
    Since the function is defined for all x, it is surjective (-inf, +inf)

    Then the example finds an interval such that the function is a bijection on a mapping of S to S such that S is a subset of R.

    The example problem says that the function is a bijective on the interval [1,+inf). As I see it, it would also be surjective on the interval (-inf,-1].

    Is that correct and the book just doesn't mention it since it just asks for "a single" interval? or is there a reason that this interval does not work?

    It would seem that the union of [-inf, -1) (1, +inf] would be the interval on which the function is a bijection.

    Thanks a lot for your time.
     
  2. jcsd
  3. Oct 26, 2008 #2

    CRGreathouse

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    [itex](-\infty,-1]\cup[1,+\infty)[/itex] is not an interval, so it doesn't work. But other answers are possible, for example S = {1}.
     
  4. Oct 27, 2008 #3

    Okay cool. The question does state an interval which is more than a single point.

    So [itex](-\infty,-1][/itex] would be an acceptable interval then is the conclusion?
     
  5. Oct 27, 2008 #4

    HallsofIvy

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    Yes, it would. In fact, since the "turning points" are at [itex]x= \pm\sqrt{3}/3[/itex], "maximal" intervals on which f(x)= x3- x+ 1 is an injection are
    [tex]\left[-\infty, -\frac{\sqrt{3}}{3}\right][/tex]
    [tex]\left[-\frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}\right][/tex]
    and
    [tex]\left[\frac{\sqrt{3}}{3}, \infty\right][/tex]
     
  6. Oct 27, 2008 #5

    CRGreathouse

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    {1} = [1, 1] is an interval.
     
  7. Oct 27, 2008 #6

    HallsofIvy

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    Yes, of course. The point was that the question asked to find "an interval which was more than a single point". It wouldn't be necessary to add "which was more than a single point" if there were no intervals containing only a single point.
     
  8. Oct 27, 2008 #7

    CRGreathouse

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    Yes, but that was posted only after my post.
     
  9. Oct 28, 2008 #8
    Sorry about being unclear in the beginning. Thanks for the replies.
     
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