Results 1 to 6 of 6

Math Help - Surjective, Injective, Bijective

  1. #1
    Newbie
    Joined
    Oct 2009
    Posts
    10

    Surjective, Injective, Bijective

    Hello,

    I would like to pose a question concerning the definitions for the functions mentioned in the title.

    In my book I found the following definitions:

    Function- A relation R from A to B is a function iff:

    1) Each element in the domain is paired with just one element in the range.

    2) The domain of R is equal to A.

    Surjective onto function-

    If every element in the range is paired with at least one element in the domain.

    Injective one to one function-

    If every element in the range is paired with exactly one element in the domain.

    Bijective function-

    Both Surjective and Injective.

    Now here is the question:

    Is the example for an injective function that is not Surjective and therefore not Bijective achieved when there are more elements in the domain than in the range? Are there any other possibilities for having an injective function?

    Than for example we have A= { a, b, c} B = { 1, 2}

    And a and b are paired with 1 whereas c with 2.

    That's a function because every element of A is paired with one element of B. That surjective because every element of B is paired with at least one element of A.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1
    Quote Originally Posted by feliks0 View Post
    Now here is the question:

    Is the example for an injective function that is not Surjective and therefore not Bijective achieved when there are more elements in the domain than in the range? Are there any other possibilities for having an injective function?
    Than for example we have A= { a, b, c} B = { 1, 2}
    And a and b are paired with 1 whereas c with 2.
    That's a function because every element of A is paired with one element of B. That surjective because every element of B is paired with at least one element of A.
    What is the question?
    Is it about injections from A to B? There are none.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2009
    Posts
    10
    If we have the following ordered pairs:

    {<a,1>, <b,2>}

    Is that an Injective function which is not surjective?

    If not, I would like an example of such a function please.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1
    Quote Originally Posted by feliks0 View Post
    If we have the following ordered pairs:
    {<a,1>, <b,2>}
    Is that an Injective function which is not surjective?
    If not, I would like an example of such a function please.
    It is not even a function.
    Any function from A to B must have three pairs because A has three elements.
    Because B has only two terms, there are no injections from A to B.
    Likewise there are no surjections from B to A.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Oct 2009
    Posts
    10
    So could you please give me an example of an injective function which is not also surjective?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1
    Quote Originally Posted by feliks0 View Post
    So could you please give me an example of an injective function which is not also surjective?
    We must change the sets.
    C=\{a,b,c\}~\&~D=\{1,2,3,4\}
    \phi :C \mapsto D,\quad \phi  = \left\{ {(a,4),(b,2),(c,3)} \right\}
    That is an injection which is not a surjection.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. total, injective, surjective, and bijective functions
    Posted in the Discrete Math Forum
    Replies: 6
    Last Post: April 24th 2010, 12:12 AM
  2. Replies: 1
    Last Post: September 21st 2009, 08:01 PM
  3. Is f injective? Surjective? Bijective?
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: May 3rd 2009, 03:27 AM
  4. Injective, Surjective, Bijective
    Posted in the Discrete Math Forum
    Replies: 17
    Last Post: April 2nd 2009, 06:58 AM
  5. injective, surjective or bijective (no2)
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: March 16th 2009, 01:58 AM

Search Tags


/mathhelpforum @mathhelpforum