Prove that |Z| = |N| by constructing an explicit bijection f : N → Z

Can some one help?!!! Thank you so much!!!

The key to this is finding the function that maps N to Z. Basically you want to map all the even numbers to the negative integers, and the odd numbers to 0 and the positive integers. So:

for all n = even, f(n) = -(n/2)
for all n = odd, f(n) = (n-1)/2

for all n E N.

Then just prove that your function f(n) is injective and surjective for the cases n = even, and n = odd!