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Math Help - need an integer producing function

  1. #1
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    need an integer producing function

    I need a function such that for any real positive input, the output is an integer--AND is a different integer for each different input.

    Does such a thing exist?
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  2. #2
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    Of course, one-to-one functions have a different y value for all x values.

    y=mx+b would be one function that suits your criteria.
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  3. #3
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    Quote Originally Posted by rainer View Post
    I need a function such that for any real positive input, the output is an integer--AND is a different integer for each different input.

    Does such a thing exist?
    I don't think that a 1-1 map \mathbb{R}^+ \rightarrow \mathbb{Z} is possible since the domain is an uncountable set and the range is countable.
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  4. #4
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    How about: \displaystyle y=m\left\lfloor x\right\rfloor + b where b,m\in\mathbb{Z} \ \mbox{and} \ x\in\mathbb{R}^+\mbox{?}
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    Quote Originally Posted by dwsmith View Post
    How about: \displaystyle y=m\left\lfloor x\right\rfloor + b where b,m\in\mathbb{Z} \ \mbox{and} \ x\in\mathbb{R}^+\mbox{?}
    Correct me if I'm wrong, but this would not always produce an integer y for any real positive x.

    Quote Originally Posted by skeeter View Post
    I don't think that a 1-1 map \mathbb{R}^+ \rightarrow \mathbb{Z} is possible since the domain is an uncountable set and the range is countable.
    Is there a proof or theorem somewhere that says or implies that if "the domain is an uncountable set and the range is countable" then there can be no "one to one map \mathbb{R}^+ \rightarrow \mathbb{Z}" ? I'd be very interested in that.

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  6. #6
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    It is essentially the definition of "uncountable".

    If there were a one-to-one function from an uncountable set to a subset of the positive integers, then f(x)= n would assign a positive integer to each x in the set. We would be "counting" the set.
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