# Injection, surjection, and bijection

1. Oct 13, 2009

### Koshi

I'm having trouble understanding just what is the difference between the three types of maps: injective, surjective, and bijective maps. I understand it has something to do with the values, for example if we have T(x): X -> Y, that the values in X are all in Y or that some of them are in Y...
Honestly I'm just incredibly confused about the terms. If someone could give me a straightforward way of explaining each of them I would very much appreciate it.

2. Oct 13, 2009

### Ben Niehoff

3. Oct 13, 2009

### Moo Of Doom

Remember the definition of a function f : X --> Y. It must satisfy two essential conditions:

1. Every element of X gets mapped to something in Y.
2. That something in Y is unique for each element of X.

Injections and surjections are special kinds of functions that also have one of these properties going in the other direction:

(Surj.) Every element of Y is mapped to by some element of X.
(Inj.) The element of X that maps to a particular value in Y is unique.

A function which is both surjective and injective is called bijective.

4. Oct 13, 2009

### Koshi

Wow, thank you so much! That was exactly the explanation I was looking for.
This will make my linear class so much easier to follow

5. Oct 13, 2009

### Moo Of Doom

Glad to have been of help. :)