Question

The set S contains some real numbers, according to the following three rules.

(i) 1/1 is in S

(ii) If a/b is in S, where a/b is written in lowest terms (that is, a and b have highest common factor 1), then b/2a is in S.

(iii) If a/b and c/d are in S, where they are written in lowest terms, then a+b/c+d is in S.

These rules are exhaustive: if these rules do not imply that a number is in S, then that number is not in S. Can you describe which numbers are in S?

Answer 1

The set S exists in the rational interval $Q$ ∩ $[\frac{1}{2} , 1]$ ; also, the rational numbers $\frac{a}{b}$, in which 0 < a ≤ b ≤ 2a.

This is because $\frac{1}{1}$ has this form and the changes preserve the property of being in the given rational interval. If a ≤ b ≤ 2a, then  $\frac{b}{2a}$  abides by the given criterion,

∵ b ≤ 2a ≤ 2b

Also, if $\frac{a}{b}$ and $\frac{c}{d}$ follow the rule stated above, then so will $\frac{a+c}{b+d}$,

∵ a + c ≤ b + d ≤ 2a + 2c = 2(a + c)

Ans) S = Q ∩ $[\frac{1}{2}, 1]$