Homework Help: Injective Compositition

1. Apr 10, 2012

dijkarte

Given two functions:
f:A --> B
g:B --> C
How to show that if the (g ° f) is injection, then f is injection?

I tried this:

We need to show that g(f(a)) = g(f(b)) ==> a = b holds true for all a, b in A. But there's nothing said about function g.

2. Apr 10, 2012

dijkarte

I've tried using function mapping diagrams and actually it showed this proposition is wrong.
(g ° f) injective ==> g and f are injective.

3. Apr 10, 2012

micromass

No, you don't need to show that, that's given.

You need to show that f is an injection. That is: f(a)=f(b) ==> a=b. That is what you need to show.

4. Apr 10, 2012

dijkarte

You are absolutely right, my bad expressing the problem...

And yeah my post should have been moved under elementary school math ;)

But it's not a homework either, it's a question my professor did not have time to clarify well!

5. Apr 10, 2012

micromass

Doesn't really matter. It's the style of homework, so it belongs here. It's irrelevant whether it is actually homework.

So, got any ideas??

You have f(a)=f(b) and you need to prove a=b.
Convert it to g(f(a))=g(f(b)) in some way.

6. Apr 10, 2012

dijkarte

But I think in order to show that f(a) = f(b) ==> a = b, g has to be given as injection as well, though I could prove that both functions g and f are injections using function mapping diagram.

7. Apr 10, 2012

micromass

No, you don't need that g is an injection.
And if gf is an injection, then it does NOT imply that g is an injection.

8. Apr 10, 2012

dijkarte

Ok I could prove it by contradiction. Assuming f(x) is not injection, then

Then there's the case where f(a) = f(b) and a != b for some a, b

Then g(f(a)) = g(f(b)) where a != b, which contradicts the given argument.

9. Apr 10, 2012

micromass

That is ok. But there is no need for a contradiction argument.

If f(a)=f(b). Taking g of both sides, we get g(f(a))=g(f(b)). By hypothesis, this implies a=b.

10. Apr 10, 2012

dijkarte

Got it! Any good reference that helps with doing proper proofs?

Thanks.

11. Apr 10, 2012

micromass

The book "How to think like a mathematician" by Kevin Houston is a good book.