Set Theory, Functions. Injective/Surjective

[ktheo]

Homework Statement

Give f:A→A and g:A→A where f is surjective, g is injective, but f*g is neither surjective nor injecive

The Attempt at a Solution

I don't know why I can't really think of two... I assume it's easiest to do one in ℝ, but when it comes to producing surjective-non-injective functions in general I tend to do them in Z since I find it easiest. I was thinking to do something involving e^x but I'm not sure. How should I approach this? Should I just think of functions I know are neither onto or one-to-one and work with products to find something?

Homework Statement

Assume that f:A→B and g:C→D are bijections. Prove that f^-1 x g^-1 is the two sided inverse of f x g (and in particular, that f x g is a bijection as well).

The Attempt at a Solution

I was wondering if someone could direct me to a similar proof or point me in the direction of some definitions that can help me here. I don't even know how to structure this into a proof.

 

[UltrafastPED]

Just think about what each condition means - and then doodle a solution. For the doodling, see:
http://www.mathsisfun.com/sets/injective-surjective-bijective.html