# Give a recursive definition of the set of points [m, n] that lie on the line n = 3m

I need to give a recursive definition of the set of points [m, n] that lie on the line n = 3m in N cartesian (cross) product N; where N = the set of all natural numbers. I need to use s as the operator in the definition.

I understand I need to show set of ordered pairs of points that lie on the given line, I am confused on how to start or go about the recursive definition that produces this set of ordered pairs.

• If $\langle m,n\rangle$ is on the line, so is some-specific-point-derived-from-$\langle m,n\rangle$-using-$s$.
• A point $\langle k,\ell\rangle$ is on the line if and only if it is forced to be there by the first two clauses of this definition.
HINT: If $\langle m,n\rangle$ is on the line, so is $\langle s(m),?\rangle$; what should replace the question mark?
The initial point will depend on whether your $\Bbb N$, like mine, contains $0$, or whether instead it’s just the set of positive integers.
• @taylor.tackett: For your initial point you would use $\langle 1,n\rangle$ for some $n$; what does that $n$ have to be? For the other question, if $\langle m,n\rangle$ is on the line, then you know that $n=3m$. If you replace $m$ by $m+1$, by what must you replace $n$? It’s not $n+1$. Sep 6 '16 at 4:25