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Hello did a few exercises about supermum and infimum but im not sure if my solutions are correct. The following set are given:

(i) $\{n \text{ is element of the whole numbers } | n³ < 10\}$ Here was a little mistake. I think my answer is right now ?

(ii) {(3/n) +4 |n is element of the natural numbers (without 0)} Same translation mistake.

(iii) $\{ x\text{ element of real numbers } | x² < 5\}$

(iv) {(k-2)² < 10 | k { element of integers }

(v) { 3 + ((-1)^n)/(n) | n element of natural numbers (without 0) } Sorry i made a translation mistake.

My solutions are:

(i) $\sup = 2$

(ii) $\sup = 7$ ? inf = 4

(iii) $\sup = \sqrt{5}$

(iv) $\sup = 5, \inf = -1$

(v) $\sup = 3,5 \inf = 2$

My problem with exercises like these is : if you take (i) for instance. I know for this set n = 2. But is the supremum now 2 or 8 because 2³ = 8 ?

Question regarding (iii)

Im not exactly sure why it has to be $\sqrt 5$. ($\sqrt 5$)² < 5 is not true ?

One question regarding syntax:

Is this right: {n is element of the whole numbers | n³ < 10}; sup = 2

but

{n³ < 10 |n is element of the whole numbers}; sup = 8

hprandom
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1 Answers1

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After question was edited 1 min ago:

Your main problem is that you're not figuring out what the sets are before attempting to find their supremum or infimum.

(i) correct answer

(ii) correct answer; what about infimum?

(iii) correct answer

(iv) correct answer

(v) wrong answer; try $n = -1$?

As for your problems, well the thing is you have to first figure out what exactly the sets are before even attempting to figure out its supremum and infimum. That would answer your first problem. The second problem is answered by thinking about what is the supremum of the open interval $(0,1)$. It is $1$, even though $1$ is not an element in $(0,1)$.

For your query about syntax, "$\{ n^3 < 10 : n \in \mathbb{Z}_{\ne 0} \}$" is invalid syntax. You would have to write $\{ n^3 : n \in \mathbb{Z}_{\ne 0} \text{ and } n^3 < 10 \}$, and this set does have supremum $8$.

user21820
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  • Sorry for the formatting. Im new here and i dont know how to use fractions etc :/ I tried to edit my first post – hprandom Dec 11 '15 at 14:26
  • @hprandom: I didn't say anything about formatting. I can interpret fractions even if you write it linearly as "a/b", but your (ii) and (iv) are simply syntactically invalid. Go read the wikipedia article on set-builder notation. Also I said about working, of which you didn't show any. No one can read your mind to know why you're getting the wrong answers. – user21820 Dec 11 '15 at 14:29
  • I tried to update, if there are still syntactically invalid i dont know why. I got them from a textbook – hprandom Dec 11 '15 at 14:39
  • @hprandom: Well you did change (v) and now it is syntactically valid although you didn't fix your spelling mistake. Similarly you should know how the syntactically correct version for (ii). In mathematics you cannot anyhow write things if you want to convey your thoughts to others accurately. "a - b" is not "b - a", same with all other notation. – user21820 Dec 11 '15 at 14:42
  • @hprandom: I've updated my answer according to your new question. Notation is now valid, though you should also learn to write it in symbol form which is more precise. For example (ii) would be ${ \frac{3}{n}+4 : n \in \mathbb{Z}_{\ne 0} }$. – user21820 Dec 11 '15 at 14:50
  • i updated my answers, i hope it right now – hprandom Dec 11 '15 at 15:02