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Thread: sujective function

  1. #1
    Junior Member
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    sujective function

    Is there a surjective function from R^m to R^n where m<n ?

    My guess is no but I don't know how to justify it.

    Thansk
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  2. #2
    Senior Member
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    No surjective function from $\displaystyle \mathbb{R}^{m}$ to $\displaystyle \mathbb{R}^{n}$ with $\displaystyle m<n$?

    Actually, for all $\displaystyle m$ and $\displaystyle n$ in $\displaystyle \mathbb{N}-\{0\}$, $\displaystyle \mathbb{R}^{m}$ and $\displaystyle \mathbb{R}^{n}$ are equipotent (i.e. there exists a bijective function between them)!

    Of course, if $\displaystyle m=0$, $\displaystyle \mathbb{R}^{0}$ is finite while $\displaystyle \mathbb{R}^{n}$ is infinite, so there is no surjection.

    Let $\displaystyle A$ and $\displaystyle B$ be two sets, Cantor-Shröder-Bernstein theorem states that if there exists an injection from $\displaystyle A$ to $\displaystyle B$ and an injection from $\displaystyle B$ to $\displaystyle A$, then there exists a bijection between them.

    So, to start, you can find an injective function from $\displaystyle \mathbb{R}$ to $\displaystyle ]0,1[$, to prove they are equipotent, then do the same for $\displaystyle \mathbb{R}^{2}$ and $\displaystyle ]0,1[^{2}$, and finaly find a surjective function from $\displaystyle ]0,1[$ to $\displaystyle ]0,1[^{2}$, so that will prove there is a surjection from $\displaystyle \mathbb{R}$ to $\displaystyle \mathbb{R}^{2}$.

    But there can be a quicker way...
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