# Thread: Define functions using domain and codomain

1. ## Define functions using domain and codomain

Let $\displaystyle A=\{ a,l,L \}, B=\{ a,l,o,n,g \}, C=\{ t,h,e \}, D=\{ w,a,t,c,h,t,o,w,e,r \}$. Using these sets as domain and co-domain, define functions having the following properties and explain why the properties hold:

a) A function that is not injective and not surjective.

b) A function that is not injective but is surjective.

c) A function that is not surjective but is injective.

d) A function that is bijective.

2. Where exactly are you stuck?

3. Originally Posted by Nyrox
Where exactly are you stuck?
That's not really the issue.

Any help that can be provided with these questions would be appreciated. I plan to try them myself, but I doubt I'll get them right on my own.

In all honesty, this question wasn't very well described.

4. Well, I'll give you an example for question c): you want it to be injective, but not surjective, that is, you want to distinct elements to have distinct images, but not all elements in the codomain are images from elements of the domain. So, define for instance

$\displaystyle f:C\to B$

as

$\displaystyle f(t)=a$
$\displaystyle f(h)=l$
$\displaystyle f(e)=n$

$\displaystyle f$ satisfies the required conditions. Try working out the other questions: it's really just a matter of knowing the definitions of injective and surjectiveness.