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So far I know that if $A \subset B$ then every element that is in $A$ will be also in $B$ but there will be at least one element in $B$ that is not in $A$

What about $A \subseteq B$? This site says that $A \subseteq B$ says that "$A$ is a subset of $B$. set $A$ is included in set $B$." But well, it's pretty much the same definition as $A \subset B$.

So my question is, $A \subset B$ and $A \subseteq B$, what is the difference?

cmk
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Stokolos Ilya
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    Sometimes people use $A\subset B$ to mean that $A$ is a proper subset of $B$ (i.e. all elements of $A$ are in $B$, but $B$ has some element(s) not in $A$). However, $A\subset B$ is usually used to mean that $A$ is a subset of $B$ (not necessarily proper). The notation $A\subseteq B$ means $A$ is a subset of $B$, and a better way to denote a proper subset is $A\subsetneq B$. – Dave Jul 24 '19 at 18:40
  • $A\subseteq B$ always means that $A=B$ is allowed. In some instances, $\subset$ is ambiguous - some authors use $A\subset B$ to exclude $A=B,$ others use it to mean the same as $\subseteq.$ (In that case, these authors will write $A\subsetneq B$ to ensure that $A\neq B.$ – Thomas Andrews Jul 24 '19 at 18:46

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It depends on convention. Some people use $A\subseteq B$ to mean $A$ is a subset of $B$ or $A=B.$ These people often use $A\subset B$ to denote proper subset. However, some use $A\subset B$ to mean that $A$ is a subset of $B$ or $A=B.$ These people typically use $A\subsetneq B$ to denote a proper subset.

In general, $A\subseteq B$ allows for the case of $A=B,$ and $A\subsetneq B$ means that $A$ is a strict subset of $B$, while $A\subset B$ is the primary source of ambiguity.

cmk
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  • +1 yeah, that's the right way to frame it. – rschwieb Jul 24 '19 at 18:42
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    I must be in another camp of mathematicians: I use $A\subseteq B$ to mean $A$ is a subset of $B$ (possibly being equal to $B$), while $A\subsetneq B$ means $A$ is a proper subset of $B$. – Clayton Jul 24 '19 at 18:45
  • @Clayton That's probably the best way to do it, since it's the least ambiguous (in my opinion). – cmk Jul 24 '19 at 18:47
  • I mean, I guess it's just convention, but it seems like a terrible choice when every other order uses $<$ and $\leq$. I'm leaving my answer up in protest. :) – Michael Biro Jul 24 '19 at 18:47
  • @MichaelBiro I typically use that notation myself (although I understand the confusion; it's just how I learned it) – cmk Jul 24 '19 at 18:54
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Annoyingly, these two symbols do mean exactly the same thing, unless the writer states otherwise. If you want to say that $A$ is a proper subset of $B$, you have to write $A\subsetneq B$.

TonyK
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ETA: Apparently, I'm fighting against the tide of convention. Use whatever definition the author of your book uses.

$A \subseteq B$ is $A \subset B$ or $A = B$. It's analogous to $a < b$ vs $a \leq b$.

Michael Biro
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    This is wrong. $A\subset B$ is usually taken to mean exactly the same as $A\subseteq B$. – TonyK Jul 24 '19 at 18:38
  • @TonyK I disagree with the notion that they are "usually synonymous," but understand that they are occasionally. I've certainly seen authors use it as described in this post, and I would have judged it was more often the case. – rschwieb Jul 24 '19 at 18:39
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    Maybe some people use $A \subset B$ to mean $A \subseteq B$, but that seems like an abuse of the notation – Michael Biro Jul 24 '19 at 18:39
  • @MichaelBiro In this, the "abuse" is usually due to an old standard. It is a more ancient view, I believe, due to (1) not really thinking of $\subset$ as a partial order, so not using that notion, and (2) typesetting was actual physical objects, so it was cheaper to have lots of $\subset$ and only a few $\subsetneq$ characters, and (3) $\subset$ in the "possibly equal" sense was a much more common usage. – Thomas Andrews Jul 24 '19 at 18:52
  • @ThomasAndrews Fair enough. – Michael Biro Jul 24 '19 at 18:59