Question

Give a recursive definition for the following set of ordered pairs of positive integers (
`a|b` means that a is a factor of b):
`S=`{
`(a,b)|a \in Z^+, b \in Z^+, a|b`}

Answer 1

A recursive definition for the set S of ordered pairs of positive integers (a,b) where a is a factor of b can be given as follows:

1. The ordered pair (1, b) is in S for all positive integers b.

2. If a is a factor of b, then (a, b) is in S if and only if (a, b/a) is in S.

The first rule establishes the base case for the recursion, stating that (1, b) is in S for all b. The second rule provides the recursive step, stating that (a, b) is in S if and only if (a, b/a) is in S, where b/a is an integer division of b by a, and a is a factor of b. This recursive definition ensures that all pairs (a, b) in S have a as a factor of b.