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If we write $A \subset B$ and $B \subset A$ then we can assert that $A = B$ and the same goes for $A \subseteq B$ , $B \subseteq A$ ...

So then what is the essential difference between these two notations?

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  • $\begingroup$ As they already mention below, some authors prefer $\subset$ for strict inclusion. If you wish to be sure that you are understood, you can use \subsetneq for $\subsetneq$ to denote strict inclusion in an unambiguous manner. $\endgroup$ – JMoravitz Mar 16 '15 at 1:28
  • $\begingroup$ Related / possible duplicate: $\subset$ vs $\subseteq$ when *not* referring to strict inclusion $\endgroup$ – epimorphic Mar 16 '15 at 1:38
  • $\begingroup$ @epimorphic is there a particular way to search for terms containing ⊂ or other latex notation? If I google " ⊂ math stackexchange", it will just suggest the frontpage of this website, but no related results $\endgroup$ – piman314 Mar 16 '15 at 1:42
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    $\begingroup$ I find it's best to use the latex code for the symbol while searching. I came across that post by entering \subset \subseteq in the site's search box in the bar at the top of the page. On Google you can use \subset \subseteq site:math.stackexchange.com. If you ever need to find the code for a symbol Detexify is really convenient. $\endgroup$ – epimorphic Mar 16 '15 at 1:53
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Some sources make no difference between $\subset$ and $\subseteq$.

However, in other sources, $A\subseteq B$ allows the possibility that $A=B$, while $A\subset B$ specifically excludes that possibility.

From what you have said, it appears that in your text/course/whatever, there is no difference between the notations. You should check with the front of the book, or with your teacher/instructor.

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It depends on your convention. Some use $\subset$ for strict inclusion, some do not. It depends on the author. I'd prefer $\subset$ for strict inclusion and $\subseteq$ for non-strict. It is consistent with the usage for $<$ and $\le$.

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  • $\begingroup$ This is what I initially assumed, however in my lecture notes I saw the assertion that $A \subset B$ and $B \subset A \implies A = B$ and was caught off-guard, so wanted to double check. $\endgroup$ – piman314 Mar 16 '15 at 1:23

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