<p>Which of the following has an element that is less than any other element in that sete?</p>
<p>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 r such that r^2 > 4</p>
<p>The answer is None. Why?</p>
<p>the lowest for each is 1</p>
<p>All have a non-inclusive lower limit:</p>
<p>I - 0 is lower limit, but 0 is a non-positive rational
II - SQRT(2) is lower limit, but this is irrational
III - 2 is lower limit, but it 2^2 ~> 4, so is not included in the set</p>
<p>Because the rationals are dense (pick any two rationals, there is always another in between) the lower limits can be approached arbitrarily closely (any rational chosen can be "beaten" by a still lower rational) - there is no "first element" of the set.</p>
<p>Note that if III had specified r^2 >= 4, it would qualify.</p>
<p>for I, there isn't any one rational number that is smaller than all other, so its not true. for example, 1/10 is smaller than 1/9, and 1/11 is smaller than 1/10, and 1/2039920840 is smaller than them all, yet 1/2048209358093485098345 is smaller yet...</p>
<p>for II, the smallest that r could be is sqrt2, but sqrt2 isn't rational. because the minumum isn't absolutely defined, one could find rational numbers closer and closer to sqrt2... its like trying to find the endpoint of an open-ended endpoint... get it? lol thats the best i can do</p>
<p>same for III, 2.01^2 > 4, 2.001^2>4, 2.000001^2>4.... u can go on and on</p>
<p>edit: i was too slow lol</p>
<p>hmmm the question is about infinums, i'm not sure is that how they're called in english... anyway, it's a thing that u leear before the limes thingie (pre-calculus i supose).. </p>
<p>I well it's obvious right?
II well if r is such a number that r^2>=2 then the least value for r^2 is 2.. if r^2=2 then r=sqrt(2)... but sqrt(2) is not a rational number => so it's not in the set...
III well if r^2>4 then r>2 => there is no infinum here either, because if you find a very small number close to 2, there is always some smaller number close to 2</p>
<p>so the correct answer is none :)
but as i said this is a bit tough question because it's either u know the whole pre-calculs thingie good, or u have to really start your brain :P</p>
<p>eidt: lol i was even slower :DDD</p>