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+c b+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

Answer / Step-by-step explanation:

First of all, recalling the statement in the question, it is observed that

S is contained within the rational interval:

Q ∩ [ 1/2 , 1 ] , where the rational number is a /b where 0 < a ≤ b ≤ 2a.

It should also be noted that the reason for the statement above is that:

1 / 1 when transformed preserve the property of being in this interval.

Also, If a ≤ b ≤ 2a, then b / 2a obeys the requirement, since b ≤ 2a ≤ 2b. And if a / b and c/d obey the requirement, then so does (a+c )/ (b+d), since a + c ≤ b + d ≤ 2a + 2c = 2 ( a +  c).

We could also write it such that:

Where S contains some real numbers, according to the following three rules in the question,

(i) 1/1 is in S (ii) 1/2 is in S, and (iii) If a/b and c/d are in S, where they are written in the lowest terms then (a+c)/(b+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?

The answer will still be the same: S=Q∩[1/2,1]