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Principle of induction problem

  1. Oct 30, 2011 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations
    σ(n) = n + 1


    3. 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?
     
  2. jcsd
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