# Gof is injection, does this imply that f is an injection

Let A,B, and C be nonempty sets and lef f map A to B and g map B to C.

If g of f maps A to C is an injection, then f is an injection.

Not sure how to do this proof

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Dick
Homework Helper

It seems like you always start out your posts with something like "Not sure how to do this proof" without attempting it. Why don't you assume f is NOT an injection and see if it leads to a contradiction with g of f being an injection?

Of course, because if I knew how to start it or do it, I wouldn't ask.

Dick
Homework Helper

Of course, because if I knew how to start it or do it, I wouldn't ask.
Sure, if you knew how to do it you wouldn't need to ask. But could you thrash around with attempts a little? Other people do.

You are making an assumption that I haven't. I don't post a futile attempt at it that doesn't go anywhere. With computations for classes like DE and Calc, posting the work allows others to identify errors. I can post work associated with a proof that leads no where so whomever helps would have to start from scratch anyways.

Dick
Homework Helper

You are making an assumption that I haven't. I don't post a futile attempt at it that doesn't go anywhere. With computations for classes like DE and Calc, posting the work allows others to identify errors. I can post work associated with a proof that leads no where so whomever helps would have to start from scratch anyways.
I'm not making the assumption you haven't done anything. I'm just saying you don't tell us what you've tried. That is not valueless even if it didn't work. Check out post 2. Try a proof by contradiction.

Of course, because if I knew how to start it or do it, I wouldn't ask.
More often then not, my students do not know how to start because they are forgetting definitions. One tip I always give my students is to simply write down the definitions of the key words in the problem if you do not know how to start. The is almost always the first thing to try regardless of the problem, and regardless of whether you know how to start.

Since you are asked to show $$f$$ is injective, you need to know what "injective" means. Write down exactly what it means for a function to be injective.

Some math books define injective as: A map $$f:A\to B$$ is injective if $$f(a_1) = f(a_2)$$ implies $$a_1 = a_2$$. This is what you have to prove: that $$f(a_1) = f(a_2) \implies a_1=a_2$$

What are you given? You are given that $$g \circ f : A\to C$$ is injective. What does this mean according to the definition of "injective"? (see the definition above, but apply it to $$g\circ f$$ as opposed to just $$f$$). It may help to switch from $$g\circ f$$ notation to $$g(f(x))$$ notation.

Can you then use the given information to deduce what you need to show?