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Surjective, injective, bijective how to tell apart

  1. Jun 4, 2006 #1
    Hi,

    I have no problems with recognising a bijective function -> one-to-one mapping e.g. x^3 is bijective wheras x^2 is not.

    But how do you tell weather a function is injective or surjective? I was reading various "math" stuff on this but it has left me only puzzled. Can somebody explain injective and surjective in plain english. How do you tell weather a function is injective or surjective or none?
     
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  3. Jun 4, 2006 #2
    [itex]f:A\rightarrow B[/itex] is surjective if f maps on to the whole of B. For example, [itex]x^2:\mathbb{R}\rightarrow\mathbb{R}[/itex] is not surjective because [itex]x^2[/itex] only maps R onto the positive reals. So [itex]f:A\rightarrow B[/itex] is surjective iff [itex]f(x)=y[/itex], [itex]x\in A[/itex], [itex]\forall y\in B[/itex]. [itex]x^2:\mathbb{R}\rightarrow\mathbb{R}^+[/itex] is however surjective. It's all about the domains and ranges.

    Injective is if f maps each member of A onto one and only one unique element of B, injective is just another word for one-to-one. Again, [itex]x^2:\mathbb{R}\rightarrow\mathbb{R}[/itex] isn't injective because both the -x and +x map onto [itex]x^2[/itex], so it is many to one. [itex]x^2:\mathbb{R}^+\rightarrow\mathbb{R}[/itex] is injective though. [itex]x^3:\mathbb{R}\rightarrow\mathbb{R}[/itex] is injective and surjective, and thus bijective (bijective being both injective and surjective). [itex]x^2:\mathbb{R}^+\rightarrow\mathbb{R}^+[/itex] is then bijective. The important property of injective maps is that they are invertable, surjective maps aren't necessarilly so.
     
    Last edited: Jun 4, 2006
  4. Jun 4, 2006 #3

    matt grime

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    you look at the definition, look at the function and try to figure out if the function satisfies the definition.

    More on the definition in the next post.
     
  5. Jun 4, 2006 #4

    matt grime

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    No, this is not correct. f is injective if f(x)=f(y) implies x=y. This is not what you have written down here.

    With that correction, is the OP now ok with the definition?
     
  6. Jun 4, 2006 #5
    Ok, I did some more reading and thinking.

    If I understand correctly, the point of injection is that it is focused on the mapping that is only one value from domain must be mapped by another value at the codomain. Soo there is one-to-one mapping - injective.

    The point or surjectivity is a bit trickier. The trick lies in the domain and codomain definitions. Here I dont get it. e.g. x^2 is a surjective function if you define the codomain to consist of only positive real numbers R+True?. Meaning - in a surjective function the codomain must not include values that are not mapped to corresponding values of the domain.

    For example if a function is injective(one-to-one mapping) but if its codomain is defined to include values that the function never mapps then the function is not surjective? But if we modify the definition of the codomain we can get a surjective function aswell? Meaning a bijective function should the function be injective in the first place?
     
  7. Jun 5, 2006 #6
  8. Jun 5, 2006 #7

    matt grime

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    One thing to remember is that formally part of the definition of a funtion is the specification of domain and codomain. If you change those you change the function so of course you're going to potentially change its properties. For instance x^2 isn't actually a function until you tell me what is domain and codomain are.


    Now I'm confused because I thought you said you knew what a bijection was. A bijection is a map that is injective and surjective (one-to-one and onto). So what definition of bijection were you using?
     
  9. Jun 5, 2006 #8
    Ah, thank you for that correction. I wasn't sure how to write it mathematically, though now that you've written that it does ring a bell.
     
  10. Jun 5, 2006 #9

    matt grime

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    What you wrote essentially specified what it means to be a function: given some element in the domain we associate a unique element in the codomain, and we write something like f(x) for the element that we associate to x.
     
  11. Jun 5, 2006 #10

    mathwonk

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    another useful, tip: up, down, sideways, how to tell them apart? nose, toes and elbows.
     
  12. Jun 6, 2006 #11
    now I am a bit more confused.

    a bijective function is for me a function that is mapped one-to-one. That means that if you make an inverse function that is for a given value of y you get back only one x. E.g. the inverse of x^2, would be sqrt(x) but that gives to answers +/- sqrt(x). Soo that is not one-to-one and not bijective.

    injective - that is the property of bijective functions -> one-to-one mapping

    surjective - a bit tricky, means the codomain is of same "power" as the domain.

    is any of this correct?
     
  13. Jun 6, 2006 #12
    A bijective function is a function which is BOTH injective and surjective.
     
  14. Jun 6, 2006 #13

    matt grime

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    You cannot define an inverse function if the function is not surjective: how do you define the inverse function on elements of the codomain that are not mapped to?
     
  15. Jun 7, 2006 #14

    mathwonk

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    here is another point of view: given a map f:X-->Y, another map g:Y-->X is a left inverse of f iff gf = id(Y), a right inverse iff fg = id(X), and a 2 sided inverse if both hold.

    then f is injective iff it has a left inverse, surjective iff it has a right inverse (assuming AxCh), and bijective iff it has a 2 sided inverse.
     
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