How to inductively define a set?

Question

I am trying to define a set inductively.

Suppose the set I want to define is:

S = {(a, b) | a, b ∈ Z,(a − b) mod 3 = 0}.

I know that to define this inductively I need a basis, some rule to make a new element from some existing one in S, and some end/closure.

Answer 1

Hint: Start with $(0,0)$, $(1,1)$, and $(2,2)$. If $(x,y)$ is in, so is $(x\pm 3,y)$ and $(x,y\pm 3)$.