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List the elements of each of the following sets: $\{x | x ∈ \mathbb{Z}^+ , x | 36\}$.

So, I understand what $\mathbb{Z}^+$ and $x | 36$ means; however, confusion has risen when I saw two slightly different solutions to this question. A tutor has said that the number 36 would NOT be included in the set, and in another resource, I've seen that $36$ is indeed included in the set. I'd think that 36 is an element of the set as it satisfies both set conditions. So, which one is correct (with 36 or without), and why?

ShyamalSayak
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Talal Farhoud
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    Done! What else do you want me to do? – José Carlos Santos Nov 05 '23 at 10:51
  • I'm new to this forum, and from my end I don't see any answers posted. So could you please provide me with the answer again? – Talal Farhoud Nov 05 '23 at 11:03
  • If you want answers, then you should take into account the fact that your question does not match many users' quality standards, and so it may attract downvotes, or be closed. To prevent that, please [edit] the question. This will help you recognize and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Nov 05 '23 at 11:06
  • Do you know what $x\mid36$ means, Talal? Do you know what ${\bf Z}+$ means? Do you know how to understand set notation? Where exactly are you getting stuck? – Gerry Myerson Nov 05 '23 at 11:16
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    Hopefully now the question fulfills the quality standards. Appreciate your patience. – Talal Farhoud Nov 05 '23 at 11:40
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    I see no argument for leaving $36$ out. Perhaps who ever said that was looking at a somewhat different definition. – lulu Nov 05 '23 at 12:02

1 Answers1

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The standard definition of $a|b$ is that there exists an integer $n$ such that $b = an$. By this definition, $36|36$ because $36 = 1 \times 36$, and so $36$ would be a member of this set.

In number theory, we do sometimes distinguish between the divisors of a number and the proper divisors. The divisors of $n$ are all the numbers $m$ such that $m | n$, whereas the (positive) proper divisors are all of the positive divisors that are less than $n$, i.e. everything except for $n$ itself. So while set of positive divisors of $6$ is $\{1, 2, 3, 6\}$, the set of positive proper divisors of $6$ is $\{1, 2, 3\}$.

It's possible your tutor is confusing the given set definition with the definition of proper divisors of $36$.

ConMan
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