<p>Could you explain the following question to me:</p>
<p>(From the Official Blue Math Subject Tests Book)</p>
<p>
[quote]
Q: Which of the following has an element that is less than any other in that set?</p>
<p>I. The set of positive rational numbers.
II. The set of positive rational numbers r such that r^2 (greater-than-or-equal-to sign) 2.
III. The set of positive rational numbers r such that r^2>4.
[/quote]
</p>
<p>The answer is None of the three. Can you please explain this to me. I don't get this question at all.</p>
<p>Thanks in advance. :)</p>
<p>I think you looked at the wrong answers, because in my book the answer is A) None.
So none of the options are correct. I’m not sure why, so anyone mind explaining the answer why?</p>
<p>^ Oh, sorry. Yeah, that was my answer lol. </p>
<p>The answer is A. Can you explain that to me pleaseee? :)</p>
<p>It’s kinda weird. Basically, the idea is that if you pick a value that seems to be THE smallest, you will be able to find an even smaller one that fits the requirements. Like for set I, the smallest number would be 0.0000000000000000000(infinite number of zeros) and then a 1 at the end. If you say 0.01 is smallest, 0.001 is smaller, and so is 0.0001 and 0.00001 and so on.</p>
<p>Point is, you can’t state a definitive answer for any of the 3 sets. So the answer is none of those three.</p>
The sets above are all open in the metric induced by the real numbers. Ergo, no smallest element exists, only infinite sequences of rationals converging to inf(the set). For example, inf(II) = \sqrt(2), which is not rational, so we may take a sequence of rationals converging to sqrt(2) by the density of Q in R.