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How do you prove that a function is surjective?

  1. Dec 25, 2006 #1
    how do you prove that a function is surjective ?

    i know that surjective means it is an onto function, and (i think) surjective functions have an equal range and codomain?
     
    Last edited by a moderator: Jan 7, 2014
  2. jcsd
  3. Dec 25, 2006 #2

    Hurkyl

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    There are lots of ways one might go about doing it. The most direct is to prove every element in the codomain has at least one preimage. i.e. for a function [itex]f:X \to Y[/itex], to show

    [tex]\forall y \in Y :\exists x \in X: f(x) = y[/tex]
     
  4. Dec 25, 2006 #3
    how can i prove if f(x)= x^3, where the domain and the codomain are both the set of all integers: Z, is surjective or otherwise...the thing is, when i do the prove it comes out to be surjective but my teacher said that it isn't.

    this is what i did:

    y=x^3

    and i said that that y belongs to Z and x^3 belong to Z so it is surjective

    this is obviously wrong, but i don't know what i'm doing wrong!
     
  5. Dec 25, 2006 #4

    Hurkyl

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    Because, to repeat what I said, you need to show for every y, there exists an x such that f(x) = y!


    You claim f is surjective -- that means (for example) that you can find an x such that f(x) = 2.
     
  6. Dec 25, 2006 #5
    'Because, to repeat what I said, you need to show for every y, there exists an x such that f(x) = y!'
    okay, easy! lol
    i read that ten thousand times already! just give it time to sink in...okay it has sunk in

    i guess it is not surjective then...thanx for opening up my eyes
     
  7. Dec 26, 2006 #6

    HallsofIvy

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    Does there exist x in Z such that, for example, f(x)= x3= 2?
     
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