I was just introduced to set theory and was given two problems. Both basically g

Question

I was just introduced to set theory and was given two problems. Both basically give you an interval for A-of-x, i.e. (x, 2x). Then it says to find, with proof, the union and intersection with x being an element of all positive rational numbers of A-of-x.

here are the problems. Exact answers would be great but I really just need to understand the process. Thanks!

With Ax = (x,2x), for all x Q+, find, with proof, x Q+Ax and x Q+Ax. With Ax = (0, for all x R+, find, with proof, x R +AX and x R+Ax. [Here Q+ stands for the set of all positive rational numbers and R+ for the set of all positive real numbers.]

Explanation / Answer

1)

As x belongs to Q+, x can be as small as 0 but not zero. x can be 1/n with n belongs to N

Intersection of all (x,2x) will converge to singleton zero i.e., {0} but not any definite interval

Now, union will be (0,infinite) as x can be upto infinite and can be as small as zero.

2)

Ax = (0,2x/(x^2 + 1))

x belongs to R+

2x/(x^2 + 1) is increasing from 0 to 1 and decreasing from 1 to infinite (By differention and Maxima,minima method)

=> maximum is at x = 1 i.e., 1

Therefore, union of Ax = (0,1) and intersection converges to {0}

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