Confusion about contained in and subset What is the difference between saying that something is a subset of and something is contained in? I am studying basic set theory on my own and this is one of the finer points I feel is important but am unable to grasp.  
 A: That a set is contained another set usually means
it is a subset.  That an element is contained
in a set means it is in the set or a member of
the set.  Much to be prefered is the expression
"is in".  To be avoided is using in to mean subset.
A: In modern mathematics, the English language is only used to make our (math) statements in a more comfortable way. But it can always be made more formal with symbols, signs, operators, relations, etc.
So, if you feeling confused about English statements, see if you can transcribe them into more formal expressions.
So if you see "belongs to", "is a subset of", "is contained in" in most contexts you have to set up formal statements using either $\in$ or $\subset$. Also, you might at times be looking at $\subseteq$, $\subsetneq$, $\supset$, or $\notin$.
Now, to be honest, I really don't know what is the difference between $\subset$ and $\subseteq$. But if I am reading something and the author likes to examine proper subsets, then I will adjust accordingly. If you are reading different books on set theory you might have to get a feel for 'different tempos' and blend it all together into understandable chunks.
