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I have a set (of related concepts) that I wish to divide (classify) into an arbitrary number of subsets (categories), but when I divide the set (into categories) there will always be one subset that is equal to the original "unabridged" set (e.g. the category tree root)... What's the English term for the set that is the union of all subsets?

The reason that I am asking is because I need a term for a UML class diagram where there is an object type from which a group of derived types need to be created, but due to the variability of the derived types of the derived types the unifying term of the "super-parent" (e.g. the logical union of all subsets) is abstract to the point that I simply want to use a term from set theory to abstract this meaning of the ultimate "super-parent"... short of calling the super parent something a vanilla as "object" or "class"...

Neoheurist
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    If $A$ is a set, the $B= A$ is the only non-proper subset of A. – amWhy Oct 04 '18 at 20:04
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    Given a set $A$, every proper subset $B$ of $A$ can be written $B\subsetneq A$. – amWhy Oct 04 '18 at 20:05
  • ;-)

    I've received feedback that this question needs details or clarity... within the domain of software architecture there aren't any additional details that I can add... perhaps the question of clarity stems from the fact that my inquiry comes from an intellectual domain with terminology and logical concerns that don't align well with the formal language of set theory... At this point I am embarrassed by the ugly turn that things have taken and would like to delete the question to have this stain on my stackexchange reputation expunged (please help)

    ;-)

     – Neoheurist Apr 26 '20 at 03:03

1 Answers1

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Not clear what you mean by "dividing" a given set $A$.

However, you are correct that given a set $A$, there is one subset which is equivalent to $A$. This is the non-proper "subset" $A= A$.

The powerset of $A$, $\mathcal P(A)$, is the set of all subsets of $A$. Indeed, there will be one element that is the non-proper subset of $A$, which is the set $A$ itself, such that $A \in\mathcal P(A)$.

Every other element of $\mathcal P(A)$, will be a proper subset of $A$.

amWhy
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  • Is $\emptyset$ proper? – Surb Oct 07 '18 at 21:03
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    Yes, provided $A \neq \emptyset$. But the empty set is often referred to as the "trivial subset", as well. – amWhy Oct 07 '18 at 21:29
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    Indeed, @Surb, oftentimes when wants to speak of subsets of $A$ without referring to $A$ itself, nor to $\emptyset$, we will see written something to the effect: "For all proper, non-trivial subsets of $A$, foo and ..." – amWhy Oct 10 '18 at 18:15
  • improper subset! – Mikasa Feb 13 '22 at 08:06