<li>Which of the following has an element that is less than any other element in that set?
I. The set of positive rational numbers
II. The set of positive rational numbers r such that r^2 >= 2
III. The set of positive rational numbers r such that r^2 > 4</li>
</ol>
<p>A. none
B. I only
C. II only
D. III only
E. I and III</p>
<p>Apparently, the answer is A. none, but I don’t understand why. I think I may be misreading the question… not sure. An explanation would be greatly appreciated!</p>
<p>I. For any positive rational number you can think of, there can always be one that is less than it. So that's not right. </p>
<p>II. If r-squared = 2, then r has to be root 2, and there is no other element less than or greater than root 2 whose square equals 2. </p>
<p>III. Basically the same as II. r has to be 2, there is no number less than or greater than 2 whose square is 4.</p>
<p>wait a sec...i misread I. It would seem to me that that one is correct. Like one-third and one-fourth are in the set of positive rational numbers, but one-third is less than one-fourth.</p>
<p>No, I think you got it right, there is always a lesser element because a real number is a number that can be expressed as the quotient of two integers. For example, if i started at 1/5, something smaller would be 1/10, and something even smaller than that would be 1/9999999999999999999999. The denominator has infinite possibilities. Thanks! don't second guess yourself :-)</p>
<p>wait, uhhhhhh Im really confused now. For 2, if the smallest possible element is root 2, then wouldnt that make it finite... as in... it would be correct. Also, for 3, the same thing, if the smallest number to make r squared greater than 4 is 2.... it would be correct also. GREAT. lol</p>
<p>Root 2 is NOT a rational number.</p>
<p>ah, what was i thinking, it all makes sense now</p>