Principle of induction problem

[The1TL]

Homework Statement

Let x ∈ N. Show that there exists for each n ∈ N a natural number denoted by x^n (this is just a notation, but should tell you what we are doing)such that x^1 =x and x^σ(n) =x·x^n.

Homework Equations

σ(n) = n + 1

The Attempt at a Solution

So far my answer is:
1. Let S = {n ∈ N | x^1 = x and x^σ(n) = x * x^n}.
We know that x^1 = x for any x ∈ N and σ(1) = 1+1 = 2. So x^σ(n) = x^2 = x * x^1. Therefore 1 ∈ S. Now assume n ∈ S. We know that x^1 = x regardless of n. Since σ(n) = n + 1, it follows that σ(σ(n)) = σ(n) + 1 = n + 1 + 1. So x^σ(n) = x^(n+1+1) = x^(n+1) * x^1 = x * x^σ(n). Therefore σ(n) ∈ S. By the principle of induction we see that S = N.



My question is: Do I need to go into more detail in showing x^1 = x for this to be a valid proof? and if so how can i do that?

 

[mighty2000]

Yes, it would be helpful to provide more detail in showing that x^1 = x for all x ∈ N. This can be done by using the definition of exponentiation, which states that x^n = x multiplied by itself n times. So, for x^1, we would have x multiplied by itself 1 time, which is simply x. Therefore, x^1 = x for all x ∈ N. This step may seem obvious, but it is important to explicitly state it in your proof.