can someone please help me with this and show all steps plea

can someone please help me with this and show all steps please!!

Define an explicit bijection between N × {1, 2} and N and prove that it is injective and surjective.

Solution

Let, f be defined as follows

f((n,1))=2n-1

f((n,2)=2n

Claim: f is a bijection from Nx{1,2} to N

1. INjective

f((n,1))=f((m,2)) is not possible as they are of opposite parity for any m and n in N

   Case a. f((n,1))=f((m,1))

                  2n-1=2m-1 ie m=n

   Case b. f((n,2))=f((m,2))

                 2n=2m ie m=n

HEnce, f is injective

2. Surjective

Case 1. Let, n be an even natural number, n=2m for some natural number , m

f((m,2))=2m=n

Case 2. Let, n be an odd natural number, n=2m-1 for some natural number,m

f((m,1))=2m-1=n

HEnce, f is surjective

Hence proved