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What are the values of $n$ that satisfy the condition $1/|n| > n$?

(a) $0<n<1$ (and) $ -\infty < n < 0$

(b) $0 < n < \infty$ (or) $-\infty < n < -1$

(c) $0 < n < 1$ (and) $-1 < n < 0$

(d) $-\infty < n < 0$ (or) $0 < n < 1$

(e) $0<n<1$ (or) $-\infty < n < 0$

My problem is not: why the answer is option-a. My question is why we should choose an option that has AND.

To me, option e is correct. In option e, we find OR instead of AND. Option e seems correct to me because $0<n<1$ alone can satisfy the $1/|n| > n$. And $-\infty<n<0$ alone can satisfy the condition as well. To me AND seems appropriate when neither $0<n<1$ nor $-\infty<n<0$ alone can satisfy the condition. If in order to satisfy the condition, both must be applied at a time, only then usage of AND seems correct to me.

The fact why AND are appropriate here was discussed in the site from which I have collected this question. But their explanations don't make any sense to me. I cannot understand their explanation.

Another question is: what are the differences between option d and option e?

TQFT
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user64814
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    Welcome to MSE. Please use MathJax to format math on this site. To begin with, enclose all math expressions (including numbers) in $ signs. For example, $x_1^2$ will give you $x_1^2$. You'll get a much better response if your posts are easy to read. – saulspatz May 01 '21 at 17:04
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    In my opinion, this is a badly written question. I would probably choose (e) also. It would have been better to write the mathematical symbols $\cap$ or $\cup$ instead of $AND$, $OR$, then the meaning would be unambiguous. – Ted May 01 '21 at 17:10

1 Answers1

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The question asks us to consider the set $A =\{n:\frac{1}{|n|} > n\}$. It would have been helpful for the question to state that $n \neq 0$ but we will assume this.

By considering the cases $n>0$ (which leads to $n^2 < 1$ so $0 < n < 1$) and $n<0$ (which leads to $n^2 > -1$ which is true for all $n$), we can see that:

$A = B \cup C$ where $B=\{n: 0 < n <1 \}$ and $C=\{n:n < 0 \}$

It seems slightly odd to write "$-\infty < n < 0$". If we are assuming $n$ is real that is normally just written $n <0$.

Set $A$ is made up of set $B$ and set $C$. So, to consider the options:

The values that $n$ can take are those in set $B$ and in set $C$. So that is option (a).

If the question had said "what is the possible value of $n$" then we would say

$n \in B \cup C$ so $n \in B$ or $n \in C$, and (e) would be the better option.

But I feel it's a bit of nit-picking distinction!

MilesB
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  • Thanks a lot for your explanation. Will you please say what is the difference between option d and option e? – user64814 May 02 '21 at 01:32
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    I didn't even look at (d), as your question focused on (a) and (e). But (d) is the same as (e), which rather gives the game away, that those can't be the "right" answer, and (a) is expected. – MilesB May 03 '21 at 11:22