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Prove N and ∅ is inductive

  1. May 29, 2013 #1
    1. The problem statement, all variables and given/known data

    Use principle of Mathematical Induction,

    Prove N (set of natural numbers) is inductive.
    Prove ∅ is inductive

    2. Relevant equations
    Principle of Mathematical Induction

    3. The attempt at a solution

    For N
    Let S be a subset of N
    1) 1 is element of S.
    2) Suppose S is inductive for some natural numbers. If x is an element of S, then x+1 is an element of S.
    3) By PMI, N is inductive for every natural number n.

    Is that correct?

    For ∅
    Let S be a subset of ∅?
    I don't know how to start. Would anyone give me a hint?

    Thanks!
     
  2. jcsd
  3. May 30, 2013 #2

    CompuChip

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    Science Advisor
    Homework Helper

    Which definition of inductive set are you using? I checked MathWorld, and it suggested:
    Clearly, ∅ does not satisfy this.
     
  4. May 31, 2013 #3
    Thanks. I know how to prove the empty set one now. An empty set's successor is {∅} and that one's successor is {∅,{∅}}, so on. I looked that one up on WolframAlpha.
     
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