Use Strong Induction on the number of times the recursive step is applied to show that $a \le 2b$ each time $(a,b)$ is an element of the set $S$

Question

I need to do the following problem once using strong induction and a second time using structural induction.

I was given the base case of :

$(0,0)$ is an element of the set $S$: $(0,0)\in S$

and a recursive step of :

If $(a,b)\in S$ then also $(a, b+1), (a+1,b+1), (a+2,b+1) \in S$

I've written the base case of $(0,0)$ down, and shown that it gives $(0,1),(1,1),(2,1)$

From my understanding I now need to show $P(j)$, but I have no idea where to go from here, My book does not seem to show any similar examples using set theory.

I've read Rosen's (Rosen Discrete mathematics and its applications 7th edition) notes on strong and well induction, but I can't seem to translate the definition into actual application.

Any help finding where to go from here, would be great.

Answer 1

Show the property holds for the base case.
Assume the property holds for (a,b) and show it holds for
(a, b+1), (a+1, b+1) and (a+2, b+1).