Please confirm whether I am right or not?

Question

5) Let S = { n belongs to natural numbers | n Ξ 1 (mod 3) }. Give an example of a bijective function f: S --> N.

Choose 3x + 1, where x is natural number /and the range belongs or a subset of natural numbers.

Now to prove the function is injective, we must show that if f(x1) = f(x2), then x1 = x2.

Suppose f(x1) = f(x2), then 3x1 + 1 = 3x2 + 1.

substract 1 from both sides.

3x1 = 3x2

divide both sides by 3.

x1 = x2

Therefore, the function is injective.

Let y belongs to the set of natural numbers, then we must show that there is an x belonging to S such that f(x) = y.

Choose x = (y - 1) / 3, then f(x) = f[(y - 1) / 3].

f(x) = 3 * [(y - 1) / 3] + 1

f(x) = y

Therefore f is sujective.

Since f is surjective and injective, f is bijective.

Answer 1

your function is from N to S. Do the inverse function ((x-1)/3) and you're right. (modifying the proof of course-but your methods are right and will still apply.)

Answer 2

As Mitch said:

Your ideas and understanding of the problem are absolutely correct. You just need to clean up the wording a little bit.