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

Question

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.

Answer 1

You need to give an initial point on the line, and you need to give a rule for recursively generating all other points on the line from the initial one. In other words, your definition will have three parts:

• Such and such a specific point is on the line.
• 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.