Let 2Z 2n n elementof Z Find a function f N rightarrow 2Z t

Let 2Z = {2n: n elementof Z}. Find a function f: N rightarrow 2Z that is 1-1 and onto. Prove that your function is onto (no proof for 1-1 is required).

Solution

The function from Z to 2Z that is one one and onto is

f(z)=2z

So we look for a function fron N to Z that is one one and onto

g(2m)=m-1,m=1,2,3,....

g(2m-1)=-m,m=1,2,3,....

THis is a function from N to Z

Now we compose f and g to get a function from N to 2Z

f(g(2m))=2g(2m)=2(m-1),m=1,2,...

f(g(2m-1))=2g(2m-1)=-2m,m=1,2,....

Let, 2k>=0

So,2k=2(m-1) gives m=k+1

Let, 2k<0

-2m=2k

m=-k

HEnce fog is the required fucntion from N to 2Z