The set X is defined as
- 12 ∈ X
- 15 ∈ X
- if x, y ∈ X, then x + y ∈ X
- if x, y ∈ X, then x − y ∈ X
Claim: for every natural number n, 3n ∈ X
I know I should induct on natural numbers that means my base step is n=0 and my induction step will be n=k+1.
So how do I go about doing the induction step?
My first line is: Assume 3n ∈ X for some n ∈ N.
Usually I would use the definition of X to make a claim about what 3n equals but I'm not sure what I can conclude from 3n ∈ X since the definitions are all split up. Am I supposed to use proof by cases or something? I know I should be proving 3(n+1)∈ X but I have no idea how I could get there.
Thank you.
I figured it out with the help of the replies. Here is how I proved it:
To prove 3(n+1) = 3n + 3 ∈ X, I had to prove that 3 and 3n are members of X.
So I proved 3n is a member of X because of the induction hypothesis.
Then proved 3 is a member of X because 15,12 ∈ X so 15-12 = 3 ∈ X
Prove by induction on $n$.
$3=15-12\in X,\;6=3+3\in X$. So it's true for n=1,2. Suppose it's true for $n\geq2$ (that's $3n\in X$), then $3(n+1)=3n+3\in X$. By induction, prove it.