What exactly is an improper subset?
I've been studying elementary set theory from my textbook. I'm confused about what exactly is an improper subset.
I know that if we say $A \subset B$, that means all the elements of $A$ are also the 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 cleared by doubt.
$\endgroup$3 Answers
$\begingroup$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).
$\endgroup$ 1 $\begingroup$$A\subseteq B$ means that all elements of $A$ are also elements of $B$, without any further restriction. With $\subset$ defined as you write$^1$, you may also view $\subseteq$ as "$\subset$ or $=$" just as the graphical composition may suggest (compare with $\le $ being the same as "$<$ or $=$").
$^1$ Personally, I prefer to use $\subseteq$ for not necessarily proper subset and $\subsetneq$ for proper subset. While this may seem a bit redundant, it is often clearer because the meaning of $\subset$ seems to vary between different authors and texts and hence may be lead to confusion.
$\endgroup$ 1 $\begingroup$$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 $A=B.$
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