# Injective and surjective?

Hello! I hope I've posted this in the correct section...

I'm a 3rd year undergraduate and we're currently studying Vector Spaces (in QM) and I just don't understand what injective (one-to-one) and surjective (onto) mean? As a result I have no idea what an isomorphism is!

I realise this is probably a very simple question, but I'm just struggling so much with the course!

Cheers, Samantha

arildno
Homework Helper
Gold Member
Dearly Missed
Ok, the first thing you have two focus on, is that you have two SETS, call them A and B.
Both A and B has elements, determinable by some criterion.

Now, a MAPPING from A to B takes each element in A and "relates" it to some unique element in B.

To say that a mapping is injective means that there are no two elements in A that are related to the same element in B. Thus, knowing the mapping procedure along with the element in B, we can DEDUCE from this what is the element in A which is related to the known element in A.
If we denote the element in B related to element x in A with f(x), this means that if f(x)=f(y), then x=y (only ONE unique element in A is related to the value of f(x))

To say that a map is SURJECTIVE means that whatever element Y in B you pick out, there exist an x in A so that Y=f(x).
The map covers B, so to speak.

Is this clear?

Yeah! Thanks a lot :-)