Results 1 to 7 of 7

Math Help - Injective/Surjective.

  1. #1
    Member
    Joined
    Sep 2008
    Posts
    81

    Injective/Surjective.

    I have a question here that asks to:

    Give an example of a function N --> N that is

    i) onto but not one-to-one
    ii) neither one-to-one nor onto
    iii) both one-to-one and onto.

    If anyone could help me with any of these, it would be greatly appreciate. Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    ux0
    ux0 is offline
    Junior Member
    Joined
    Oct 2009
    Posts
    58
    Quote Originally Posted by GreenDay14 View Post
    I have a question here that asks to:

    Give an example of a function N --> N that is

    i) onto but not one-to-one
    ii) neither one-to-one nor onto
    iii) both one-to-one and onto.

    If anyone could help me with any of these, it would be greatly appreciate. Thanks.
    i) f(x)=1 ... a simple constant function works here, because you can use all values of x in the natural numbers, but you just get one value for f(x) which is in the co-domain.

    ii) not sure if this one is correct but what if we used f(x)=-x, in this case it is mapping the natural numbers to a co-domain that isn't the natural numbers, which is not onto, or one-to-one...


    iii) f(x)=x it is just mapping the natural numbers, to the natural numbers, therefore its onto and one-to-one, but you might want to do a formula proof of each of these using the definitions.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,326
    Thanks
    1298
    Quote Originally Posted by ux0 View Post
    i) f(x)=1 ... a simple constant function works here, because you can use all values of x in the natural numbers, but you just get one value for f(x) which is in the co-domain.
    No, that's not "onto"- if fact, it's very badly not "onto"! Nothing is mapped into any number other than 1! An example of a function that is "onto" but not "one to one" would be f(1)= 1, f(2)= 1, f(n)= n-1 for n> 2.

    i) not sure if this one is correct but what if we used f(x)=-x, in this case it is mapping the natural numbers to a co-domain that isn't the natural numbers, which is not onto, or one-to-one...
    No, that isn't correct either. The problem specifically asked for a function from N to N (N->N) and that is not. Actually, your first example, f(n)= 1 for all n is neither "onto" nor "one to one".

    f(x)=x it is just mapping the natural numbers, to the natural numbers, therefore its onto and one-to-one, but you might want to do a formula proof of each of these using the definitions.
    Yes, this works fine.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    ux0
    ux0 is offline
    Junior Member
    Joined
    Oct 2009
    Posts
    58
    yup i'm a big dummy... even though the question didn't ask, if you had f(x)=x^2... would this be an example of a function that is one-to-one, but not onto, seeing you don't get the entire co-domain, but you do get a unique value for every x...
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,326
    Thanks
    1298
    Quote Originally Posted by ux0 View Post
    yup i'm a big dummy... even though the question didn't ask, if you had f(x)=x^2... would this be an example of a function that is one-to-one, but not onto, seeing you don't get the entire co-domain, but you do get a unique value for every x...
    If you mean from R to R, no, it is not "one to one" because f(-1)= f(1).

    Getting "a unique value for every x" is the definition of function, not "one-to-one". A function if one to one if f(x)= f(y) only when x=y.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    ux0
    ux0 is offline
    Junior Member
    Joined
    Oct 2009
    Posts
    58
    No, I meant it for this example... N \rightarrow N
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member
    Joined
    Aug 2009
    From
    Israel
    Posts
    976
    Quote Originally Posted by ux0 View Post
    No, I meant it for this example... N \rightarrow N
    Yes, if f:\mathbb{N} \to \mathbb{N} is defined by f(x) = x^2 then it is:

    (1) one-one: x^2 = y^2 \Rightarrow x=y

    (2) not onto: What is the source of 3?
    Last edited by Defunkt; October 31st 2009 at 09:12 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Injective, surjective map!
    Posted in the Advanced Algebra Forum
    Replies: 13
    Last Post: December 14th 2010, 12:49 AM
  2. [SOLVED] Injective and Surjective
    Posted in the Calculus Forum
    Replies: 11
    Last Post: September 29th 2010, 02:16 PM
  3. injective and surjective
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: February 12th 2010, 10:18 AM
  4. Injective if and only if Surjective
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: February 3rd 2010, 04:37 PM
  5. Injective/Surjective
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: September 10th 2008, 08:19 AM

Search Tags


/mathhelpforum @mathhelpforum