subset of element This seems really basic but for some reason I am struggling to get my head around it. I am currently reading about sets and categories, and have found a statement in the first few pages which says: 
taking $x \in U$ and $y \subseteq x$ 
But I don't quite get how an element can have a subset coming off it. It's been a while since I worked in set theory and I cant remember the conditions this satisfies.
Please could someone give me an example of how this works in practice or try to explain it to me a bit more?
Thanks
 A: A set can contain other sets. Indeed, in the most commonly used set theory, ZFC, that's all a set can contain.
Of course, every set can have a subset; sets that are elements of other sets are not excluded.
As a concrete example, take some arbitrary set, say $x=\{a,b,c\}$. Now be $U$ its power set, that is, the set of all its subsets:
$$U=\{\emptyset,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\}$$
Now in particular, $x\in U$, as you can easily verify. But just as clearly, there exist subsets of $x$ (we just defined $U$ to be the set of all of them!). For example, be $y=\emptyset.$ Then clearly $y\subseteq x$.
So we have $x\in U$ and $y\subseteq x$.
A: Suppose $U$ is a set of sets.  Then $x$ is a set and $y$ is a subset of $x$.
For example, some sets, such as the set of rubies, contain only red things.  Other sets, such as the set of all potatoes, do not contain only red things.  
Suppose $U$ is the set of all the sets that contain only red things.  One of $U$'s elements is the set of all rubies, but the set of potatoes is not one of $U$'s elements.  Say $x$ is the set of all rubies.  Then $x\in U$.
Now say $y$ is the set of all rubies that belong to the Queen of England.  Then $y\subseteq x$, and $x\in U$.
(For another example, note also that $\emptyset\subseteq x$ and $x\in U$, or that $x\subseteq x$ and $x\in U$.)
