Is the Empty Set an Inductive Set?

[xlu2]

Homework Statement

Use principle of Mathematical Induction,

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

Homework Equations

Principle of Mathematical Induction

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!

 

[CompuChip]

Which definition of inductive set are you using? I checked MathWorld, and it suggested:

nonempty partially ordered set in which every element has a successor

Clearly, ∅ does not satisfy this.

 

[xlu2]

Which definition of inductive set are you using? I checked MathWorld, and it suggested:


Clearly, ∅ does not satisfy this.

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.