Define an explicit bijection between N times 1 2 and N and p

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

Define

f((n,1))=2n

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

Proof :

1. f is injective

Note: f((m,1))=f((n,2)) is not possible

Because they are of different parity

So, let, f((m,1))=f((n,1))

So, 2m=2n ie m=n

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

2m-1=2n-1

So, m=n

Hence, f is injective

2. f is surjective

Let, n be in N

Case 1. n is even

So, n =2m for some natural number, m

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

Case 2. n is odd.

SO, n=2m-1 for some natural number, m

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

Hence, f is surjective