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.


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.

  • $\begingroup$ Should s(n) replace the question mark in this case? Mine does not contain 0, so I would use 1 as the initial point correct? $\endgroup$ Sep 6 '16 at 4:14
  • $\begingroup$ @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 $n+1$. $\endgroup$ Sep 6 '16 at 4:25

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