# Thread: one-to-one function

1. ## one-to-one function

I need to prove that there exist a one-to-one function: $\displaystyle f:A\rightarrow A\times B$
where should i start?

thanks in advanced!

2. ## Re: one-to-one function

try picking a function that maps an element a to (a,___) (i'll leave it to you to decide what might go in the blank).

3. ## Re: one-to-one function

do i just need to provide an example of a function that goes from $\displaystyle A\rightarrow A\times B$?
(like $\displaystyle (a, (a, b))$ or $\displaystyle (a, (a, d))$...)

im not sure what exactly do i need to prove here...

4. ## Re: one-to-one function

Originally Posted by Stormey
do i just need to provide an example of a function that goes from $\displaystyle A\rightarrow A\times B$?
(like $\displaystyle (a, (a, b))$ or $\displaystyle (a, (a, d))$...)

im not sure what exactly do i need to prove here...

'Fix' a $\displaystyle b\in B$. Then define $\displaystyle f:A\to (A\times B)$ by $\displaystyle a\mapsto (a,b)$.

Now prove $\displaystyle f$ is one-to-one.

5. ## Re: one-to-one function

And note that the original claim is false when B is empty but A is not.

6. ## Re: one-to-one function

Thanks guys.
appreciate it!