Equipotent Sets We know by definition that if a bijection between two sets A and B exists,then A and B are equivalent.The book I was reading took the function f:Z→N 
f(x)= -2x,for x<0
      2x+1,for x=>0
     which is a bijection(it can be easily proven),but I'm a little confused.
   How can Z and N be equivalent sets if N is a subset of Z ?

One immediate (though rather dangerous! .. see paragraph below) way to wrap your mind around this is to consider the fact that both sets are of infinite size. Yes, the one is a strict subset of the other, but they are both infinite ... so maybe it's a little easier to digest that way.
Then again, the really surprising thing about cardinality is that not any two infinite sets have the same cardinality .. that there different 'kinds' of 'degrees' of infinity, if you want. As it so happens, $\mathbb{N}$ and $\mathbb{Z}$ are of the 'same' kind though; in both cases, you can put all elements in a list ... something which is not possible for the real numbers. However, it's exactly this fact that makes the notion of cardinality and equipotence an interesting and useful notion!
