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Thread: sujective function

  1. #1
    Junior Member
    Oct 2008

    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.

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  2. #2
    Senior Member
    Nov 2008
    No surjective function from \mathbb{R}^{m} to \mathbb{R}^{n} with m<n?

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

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

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

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

    But there can be a quicker way...
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