Results 1 to 4 of 4

Thread: well-defined function (cartesian product)

  1. #1
    Member
    Joined
    Sep 2008
    Posts
    166

    well-defined function (cartesian product)

    Define $\displaystyle \phi: A$ x $\displaystyle B$ ---> $\displaystyle A/C$ x $\displaystyle B/D$ by $\displaystyle \phi((a,b))=(aC,bD)$

    I have to show this map is well-defined, here's what I did:

    Let $\displaystyle (a_1, b_1), (a_2, b_2) \in A$ x $\displaystyle B$, and let $\displaystyle \phi(a_1,b_1)=\phi(a_2,b_2)$. Then $\displaystyle (a_1C,b_1D)=(a_2C,b_2D) \implies (a_1, b_1)=(a_2, b_2)$. Is this implication clear?
    Can I write $\displaystyle (a_1C, b_1D) = (a_1, b_1)(C$ x $\displaystyle D)$?

    Please hlep, thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    Quote Originally Posted by dori1123 View Post
    Define $\displaystyle \phi: A$ x $\displaystyle B$ ---> $\displaystyle A/C$ x $\displaystyle B/D$ by $\displaystyle \phi((a,b))=(aC,bD)$

    I have to show this map is well-defined, here's what I did:

    Let $\displaystyle (a_1, b_1), (a_2, b_2) \in A$ x $\displaystyle B$, and let $\displaystyle \phi(a_1,b_1)=\phi(a_2,b_2)$. Then $\displaystyle (a_1C,b_1D)=(a_2C,b_2D) \implies (a_1, b_1)=(a_2, b_2)$.
    You need to give us more information. For a quotient map of the form $\displaystyle \phi:A\times B\to A/C\times B/D$ to be well-defined, it is necessary that $\displaystyle \phi$ should be constant on the cosets of $\displaystyle C\times D$. You haven't said anything about the nature of $\displaystyle C\times D$ or $\displaystyle \phi$, and you haven't actually proved anything.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Sep 2008
    Posts
    166
    Quote Originally Posted by Opalg View Post
    You need to give us more information. For a quotient map of the form $\displaystyle \phi:A\times B\to A/C\times B/D$ to be well-defined, it is necessary that $\displaystyle \phi$ should be constant on the cosets of $\displaystyle C\times D$. You haven't said anything about the nature of $\displaystyle C\times D$ or $\displaystyle \phi$, and you haven't actually proved anything.
    I am given that C is a normal subgroup of A and D is a normal subgroup of B. What else is needed to show $\displaystyle \phi$ is well-defined?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    In that case, there isn't much to prove. You want to show that $\displaystyle \phi(a,b)$ depends only on the cosets aC and bD, and that is immediately obvious from the way that $\displaystyle \phi$ is defined.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Cartesian product
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Mar 9th 2010, 02:13 AM
  2. Cartesian product
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: Feb 26th 2010, 11:30 AM
  3. Cartesian Product
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: Feb 9th 2010, 11:31 AM
  4. Cartesian Product
    Posted in the Discrete Math Forum
    Replies: 6
    Last Post: Nov 15th 2009, 08:23 AM
  5. Replies: 2
    Last Post: Aug 5th 2009, 10:20 AM

Search Tags


/mathhelpforum @mathhelpforum