**scientist**: I strongly recommend you look more carefully at the help you have been given so far. No, injections and surjections are irrelevant here, unless you want define a bijection (in which case you may assume that

are mutual inverses - don't do that!)

I also suggest you work first on elements rather than sets to get your thinking straight..

As to your example: if

then, one has that

. That is

whose preimage is simply

. Therefore

But it is certainly true that

.

In the above, your function is surjective, but in the general case one must assume that

is "smaller than or equal to"

, otherwise one would have that there exist elements in

which have more than a single image in

. This is not allowed, by the definition of a function.

So what if, say,

? Is it possible that

? Is this a subset of

? If so, how would you describe this function? (Granted, it's an extreme case - but not unusual, think)

PS Please use the Tex capability kindly provided by the team here