Definition of set containment Let $X$ and $Y$ be sets. Does "$X$ contains $Y$" mean $Y \subseteq X$ or $Y\in X$, or is it ambiguous?
 A: In general it is ambiguous, and the sense must be determined from context. In this case the fact that $X$ and $Y$ are the same kind of letter, both being upper case Latin letters, suggests that they represent similar kinds of mathematical objects, or at least ones at the same level of complexity, and hence that the intended sense is $Y\subseteq X$: if $Y$ were an element of $X$, one would be somewhat more likely to see $y\in X$. (This is by no means a hard and fast rule, however.)
A: The symbols $\subset$ and $\subseteq$ are used for sets which are subsets of another, which can also be described as containment. The former is used when the sets are not equal, the latter when they can be.
The symbol $\in$ is used for elements of a set, rather than whole sets. For example, you could say: $$y\in Y\subseteq X \implies y\in X$$
A: It is ambiguous. For example, I can say the real numbers contain the rationals and this would mean $\mathbb{Q} \subseteq \mathbb{R}$. I can also say the real numbers contains $3$ and this would mean $3 \in \mathbb{R}$.
