What does $X \subsetneqq Y$ mean What does $X \subsetneqq Y$ mean? I cannot find it on Google, I suppose this means, it is a subset and at the same time it is not the same. I.e. it could be simply written as $\subset$. Pardon my ignorance.
 A: Some people use $\subsetneq$ for strict and $\subset$ for non-strict. This is OK, but may confuse people accustomed to other conventions.
Some people use $\subset$ for strict and $\subseteq$ for non-strict. This is OK, but it will confuse many people.
To avoid confusion, you might think to use $\subsetneq$ and $\subseteq$, but those symbols are visually very similar.
Thus $\subsetneqq$ and $\subseteq$ would be the safe approach.
Note: There is also a $\subseteqq$ for symmetry.
Also note: $X\not\subsetneqq Y$ is a bad idea, which gives an advantage to the $\subset$/$\subseteq$ crowd—an advantage unlikely to help their cause.
A: There are religious wars about the. I like to use $\subset$ for strict include and $\subseteq$ for nonstrict.  But I am an oddity, so beware.
A: In some places $\subset$ is used to denote a proper subset, but in others $\subset$ is used to denote all sort of inclusions (proper and improper).
This is why some people (like me) prefer to use $\subsetneq$ for proper inclusion, and $\subseteq$ for general inclusion.
Other people, like one of my teachers in undergrad, prefers to make sure that no one misses the one tiny line with the diagonal on it, and uses $\subsetneqq$ which no one can mistake for anything but proper inclusion.
A: It probably means $X$ is a proper subset of $Y$, i.e. $X\subset Y$ but with $X \neq Y$. Just to make sure, the text probably defines it somewhere.
A: Yep. Some people choose to use $\subset$ to include the case of equality, and some use it to mean strict proper subset, opting for $\subseteq$ in the case of possible equality. Unfortunately it's an ambiguous natation nowadays and has to be given in context. Often if $\subset$ is being used to denote any subset, regardless of equality, an author may use some variant of $\subsetneqq$ or $\subsetneq$ to denote proper subset.
