Bijection

Sup dudes, 

I understand how something could be injective (one-to-one), and I understand how something could be surjective (onto). But how can something be both? In other words, how can something be bijective? I'm having trouble grasping this simple idea because for some reason, I see injective and bijective as being almost the same thing. 

I know I'm missing something! Does it mean that a function has both injective and surjective parts? I'm currently piecing together the Schroeder-Bernstein Theorem in my head. So if you also know a fun resource for that, do tell! 

Thanks Gaussians!

[Notations used below: <=> means "implies and is implied by" or "is equivalent to", <= means "less than or equal to", >= means "greater than or equal to", f:X->Y means "for every x in X, there is exactly one element f(x) in Y"]

The clearest way to distinguish the concepts is counting preimages. For each element c in Y, we can define n(c) as the number of distinct elements a of X such that f(a)=c. With this definition let f:X->Y be a function:

f injective <=> n(c) <= 1 for all c in Y

f surjective <=> n(c) >= 1 for all c in Y

f bijective <=> n(c) = 1 for all c in Y

I hope these arithmetic versions of the definitions make it clear exactly how the concepts relate.

Haha awesome! Thanks for the help. I have a lot of little questions like this, and it's really helpful to know I can go this group for some help! =)

Small question: Elroch, just to be perfectly clear, for f injective, n(c) is either 0 or 1, right? Can I presume you can't have fractional, distinct elements? Also, preimages are just elements in the domain, correct? 

So for f: A -> B. 

The elements in A are preimages, while the elements in B are the mapped images, right? 

Yes, Maeda. "<=1" in this context can only be zero or one. The preimage of a point b in the range is the set of points a in the domain such that f(a) is b.