# Thread: well-defined function (cartesian product)

1. ## 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 $\displaystyle D)$?

2. Originally Posted by dori1123
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.

3. Originally Posted by Opalg
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?

4. 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.