Which of the following has an element that is less than any another element in that set?
I. The set of positive rational numbers
II. The set of positive rational numbers $r$ such that $r^2$ is greater than or equal to $2$
III. The set of positive rational numbers such that $r^2 > 4$.
The answer is None.
I am not very clear why. Can anyone please explain intuitively? I know that there is no such thing as the smallest positive rational number.
On
HINT:
An important point for these types of questions are understanding the properties of rational numbers. For example, by contradiction, question I is not true. Let us suppose there is a smallest positive rational number $\frac{p}{q}$. We can multiply by $\frac{1}{2}$ to get $\frac{p}{2q}$. However, this is smaller than $\frac{p}{q}$, so there is no such thing as "the smallest positive rational number" (that is larger or smaller than a given value).
Can you use a similar type of argument for questions II and III with the appropriate bounds, which are $r>\sqrt2$ for question II and $r>2$ for question III?
Note: The negative root is not required because the question specifies the numbers have to be positive.
" 'there is no such thing as the smallest positive rational number' which is why None of the choices has a smallest element." Be a little careful! The is no smallest rational number but a set of rational numbers could have a smallest element in the set.
Example: The set of rational numbers $r$ so that $r \ge 5$ has a smallest element. $r = 5$ is in the set and it is the smallest element in the set.
Of the set of all rational numbers so that $r^2 \le 0$. That set has only one element at all. $0$. $O$ is the smallest element in that set.
Okay, but what about the questions in the exercise:
I) The set of positive rational numbers.
There is no smallest positive rational number so no, the set doesn't have a smallest element.
If asked why, you should be able to justify it. If $r$ is in the set then $\frac r2 < r$ is also in the set, so none can be smallest.
II) The set of all positive rational numbers so that $r^2 \ge 2$.
$\sqrt{2}$ is irrational so for any rational number, $r$ either $r^2 < 2$ or $r^2 > 2$. So for any $r^2 > 2$ there is another $\sqrt 2< s < r$ and $s^2 > 0$ so there is no smallest element.
But NOTE. If the question was all rational numbers so that $r^2 \ge 4$ that does have a smallest element. If $r = 2$ then $r^2 \ge 4$ and for any number $q < 2$ then $q^2 < 4$ so $2$ is the smallest element of the set.
III. The set of all positive rationals where $r^2 > 4$. This is just the set where $r > 2$. It has no smallest as there is always some $2< s < r$ for any rational $r > 2$.