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I've been studying elementary set theory from my textbook. I'm confused about what exactly an improper subset is.

I know that if we say $A \subset B$, it means all the elements of $A$ are also elements of $B$ but $A \neq B$. So $A$ is a proper subset of $B$.

But if I say that $A \subseteq B$, does that mean (i) $A$ must be equal to $B$ or (ii) $A$ may be equal to $B$

I tried looking up on google but some websites agree to (i) some agree to (ii) so it didn't clear up my question.

I also found this somewhat related question which was somewhat helpful, but didn't exactly clear my doubt.

SemiballisticS
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Raknos13
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4 Answers4

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$A\subseteq B$ means that all elements of $A$ are also elements of $B$, without any further restriction. With $\subset$ defined as you write$^1$, you may also view $\subseteq$ as "$\subset$ or $=$" just as the graphical composition may suggest (compare with $\le $ being the same as "$<$ or $=$").

$^1$ Personally, I prefer to use $\subseteq$ for not necessarily proper subset and $\subsetneq$ for proper subset. While this may seem a bit redundant, it is often clearer because the meaning of $\subset$ seems to vary between different authors and texts and hence may be lead to confusion.

Hagen von Eitzen
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  • I was only taught the $\subset$ operator ("in the old days"), denoting improper subsets. So I use to be careful about the meaning. –  Jul 25 '17 at 06:36
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A proper subset (usually denoted as $A\subset B$) is such that $A\ne B$, undisputably.

An improper subset (usually denoted as $A\subseteq B$) is such that $A=B$ is allowed (but not mandated), hence (ii). The option (i) is simply stated as $A=B$.

Anyway, you could find sentences such as "for this reason, the subset $A$ is improper" in proofs, stressing that in the case at hand $A=B$ indeed holds (or conversely, proper to stress that the sets are indeed different).

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    There is a whole school of notation preferring $\subsetneq$ or even $\subsetneqq$ for proper subsets, since there is also a school of notation using $\subset$ for $\subseteq$. So clarity is key. – Asaf Karagila Jul 25 '17 at 08:51
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$A\subseteq B$ means that all the elements of $A$ are elements of $B$. And unlike your definition for $A\subset B $, in this case $A$ may equal $B.$

If it must be equal (i.e. is equal), we just say $A=B.$

spaceisdarkgreen
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I know that if we say ⊂, that means all the elements of are also the elements of but ≠. So is a proper subset of .

Right.

The thing that makes it a proper subset is that ≠.

⊆ is a subset. It's not necessarily a proper subset. It can be a proper subset. But there's no rule that ≠. So A can equal B.

There's only one A for a given B such that A is a subset of B, but not a proper subset of B.

"Improper subset" is a funny, uncommon way of describing two sets that are equal.

The even numbers are an "improper subset" of the multiples of two.

Silly. Not a practical way to speak about sets in most situations.

Daniel
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