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I've searched a lot but couldn't help myself with the distinction between an improper subset and equal sets. I would appreciate if someone could help me here.

Mauro ALLEGRANZA
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Sharjeel Faiq
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2 Answers2

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The starting point is the rigorous definition of the subset relation:

$A \subseteq B \leftrightarrow \forall x (x \in A \to x \in B)$.

According to this definition, we have that every set is subset of itself and that the empty set is subset of every set. That's all.

But we can exploit the analogy with the $\le$ relation and we may say that $A$ is a proper subset of $B$ just in case that $A \subseteq B$ and $A \ne B$ [in symbol: $\subsetneq$].

If we want, we can use the - IMO useless - notion of "improper" subset, i.e. of a subset that is not a proper subset, that means that the only "improper subset" of $A$ is $A$ itself.

And finally, it is worth noting that in set theory equlity can be defined as mutual inclusion.

Mauro ALLEGRANZA
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A proper subset of a set S is any subset of S besides S itself and the empty set. So an improper subset would be either $S$ or $\emptyset$. Equality, of course, would be $S$.

Paul Tanenbaum
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  • Why besides the empty set? – Damian Oct 17 '23 at 13:43
  • @Damian, that’s how I’ve most often seen improper subset used. This notion excludes the two (dual) extreme cases—all or nothing—leaving behind only the interesting ones. – Paul Tanenbaum Oct 17 '23 at 13:47
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    I've always seen "proper subset" only to mean not equal to full set, i.e. just like a strict inequality. I'm sure I've seen "non-empty proper subset" used several times. – Jaap Scherphuis Oct 17 '23 at 14:26
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    I've only seen proper subset refer to strict subsets, so the empty set is a proper subset unless the set is also itself empty. – Michael Carey Oct 17 '23 at 22:01