## Examples 2.1.2:

Find the bijections to prove the following statements:

In each of the three cases we have to find a bijection between the two pairs of sets. - Let
**E**be the set of all even integers,**O**be the set of odd integers. Then*card(***E**) = card(**O**) - Let
**E**be the set of even integers,**Z**be the set of all integers. Then*card(***E**) = card(**Z**) - Let
**N**be the set of natural numbers,**Z**be the set of all integers. Then*card(***N**) = card(**Z**)

- Define the function
*f(n) = n + 1*with domain**E**and range**O**. Then the function*f*is clearly one-to-one and onto, hence it is a bijection. Now*f*is a bijection between**E**and**O**, so that*card(*.**E**) = card(**O**) - Define the function
*f(n) = 2n*with domain**Z and**range**E**. Then it is straight-forward to show that this function is one-to-one and onto, giving the required bijection. Hence,*card(*.**Z**) = card(**E**) - Define the following function:
*f(n) = n / 2*if*n*is even and*f(n) =- (n-1) / 2*if*n*is odd, with domain**N**and range**Z**. Again, it is not hard to show that this function is one-to-one and onto, and therefore*card(*.**N**) = card(**Z**)