TITLE: Is there any difference between $\subset$ and $\subseteq$
QUESTION [5 upvotes]: 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?

REPLY [6 votes]: 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.

REPLY [4 votes]: 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 $\le$.