Let A be the set containing all rational numbers that are less than 5. Is there a rational number q in set A such that all other numbers in set A are less than q? Why or why not?
We will try to prove this by contradiction.
Hypothesis: existence of q=a/b which is the largest rational number less than 5, i.e. q=a/b<5 and b≠0, and that no other rational number exists that is larger than q and less than 5.
We will calculate the r, the average between q and 5
r=(q+5)/2
=(a/b+5)/2
=(a+10b)/2b
so that
q<r<5
which means that
r is greater than q,
r is less than 5, and
r is rational.
Therefore that hypothesis that q exists is false.
To determine if there exists a rational number q in set A such that all other numbers in set A are less than q, we need to analyze the properties of set A.
Set A contains all rational numbers that are less than 5. In notation, A can be represented as A = {q | q is a rational number and q < 5}.
Now, let's consider the scenario where we assume there is a rational number q in set A that satisfies the condition that all other numbers in set A are less than q. This means that for any rational number r in set A (excluding q), r < q.
Since set A consists of all rational numbers that are less than 5, we can choose a number, say r0, from set A that is closest to but less than 5. In this case, 5 itself cannot be a member of set A since the condition for set A is q < 5, not q ≤ 5.
Now, assume that q exists and is a member of set A. Since all other numbers in set A are less than q, r0 < q. However, we can find a rational number r1 such that r0 < r1 < q. This is because between any two distinct rational numbers, we can always find another rational number.
For example, consider the numbers r0 = 4.99999 and q = 5 (assume 5 is a rational number), both of which are greater than 4.9999. Now, we can find a rational number r1 such that 4.99999 < r1 < 5. This violates the condition that all numbers in set A (excluding q) should be less than q.
Therefore, there does not exist a rational number q in set A such that all other numbers in set A are less than q. This is because for any rational number q in set A, we can always find another rational number that is greater than some numbers in set A but less than q itself.