Theoretical Concepts of Calculus

[angelet]

Find the interior int (S), the boundary bd (S), the closure cl (S), and the set accumulation points S' of each subset S of R.
Classify the set S as open, closed, neither, or both. Is S a compact set? Justify your answers.


For each subset S of R, fi…nd its supremum sup S, maximum max S, infi…mum inf S, and minimum min S, if exists. Otherwise, write "DNE". Justify your answers.

a. S = { (-1)^n + (-1)^n/n | n is an element of N }


b. S = U [ -2+1/(n^2), 2-1(2n+1) ] when n is an element of N


c. S = nQ \ {pi}


d. S = { x is an element of R | 0 < x^2 <= 3 }



please answer any of the following questions, if not all.

 

[Deleted member 4993]

Find the interior int (S), the boundary bd (S), the closure cl (S), and the set accumulation points S' of each subset S of R.
Classify the set S as open, closed, neither, or both. Is S a compact set? Justify your answers.


For each subset S of R, fi…nd its supremum sup S, maximum max S, infi…mum inf S, and minimum min S, if exists. Otherwise, write "DNE". Justify your answers.

a. S = { (-1)^n + (-1)^n/n | n is an element of N }


b. S = U [ -2+1/(n^2), 2-1(2n+1) ] when n is an element of N


c. S = nQ \ {pi}


d. S = { x is an element of R | 0 < x^2 <= 3 }



please answer any of the following questions, if not all.

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[HallsofIvy]

Have you calculated what numbers are in each of these sets?
For example, for \(\displaystyle \{(-1)^n+ \frac{(-1)^n}{n}| n\in N\}\)
we have, for n= 1, -1- 1= -2; for n= 2; 1+ 1/2= 3/2; for n= 3, -1- 1/3= -4/3; for n= 4, 1+ 1/4= 5/4; for n= 5, -1- 1/5= -6/5; for n= 6, 1+ 1/6= 7/6, etc. Those are the numbers in the set.

For \(\displaystyle \cup_{n\in N} \left(-2+ \frac{1}{n^2}, 2- \frac{1}{2n+1}\right)\)
for n= 1, (-2+ 1, 2- 1/3)= (-1, 5/3); for n= 2, (-2+ 1/4, 2- 1/5)= (-7/4, 9/5); for n= 3, (-2+ 1/9, 2- 1/7)= (-17/9, 13/7),etc Take the intersections of those.


I have no clue what "\(\displaystyle nQ\ \{\pi\}\)" means! Q is the set of rational numbers but what is nQ? And \(\displaystyle \pi\) is not in Q so how can you remove it?

\(\displaystyle \{ x\in R| 0< x^2\le 3\}\) is just all numbers larger than of equal to \(\displaystyle \sqrt{3}\) and less than 0 together with all numbers larger than 0 and less than or equal to \(\displaystyle \sqrt{3}\): \(\displaystyle \left[\sqrt{3}, 0\right)\cup \left(0, \sqrt{3}\right]\).