I find these a little tricky too. I normally just look at a lot of them and try and understand why they work (or just memorise them).

The easiest way of doing it after you have the "suspected" bijection is just to show it has an inverse. I find it's faster than using the definitions of injectivity and surjectivity.Also, how can I verify that the functions above are one-to-one and onto by using the definitions? Do I just do it the usual way, for injective, let f(x)=f(y) then show x=y, and for onto, show there exists x in A such that f(x)=y for every y in B (here I consider f from A to B)? Any suggestion and help is appreciated.