Notation for much smaller subset Similar to $x \ll y$ or $x \lll y$, I'm wondering if there is a corresponding symbol for subsets.
For example:
$$x \subset \subset y$$
Something to represent $x$ is subset of $y$, but much smaller in size than $y$. 
I realize one could simply say $x \subset y \wedge |x| \ll |y|$, but this seems a bit verbose and I'm also genuinely curious if this symbol exists. 
 A: As far as I know, there is no commonly adopted symbol for the scenario you describe. It is always best practice to use words where symbols could suffice for maximum legibility. There are exceptions to this rule, of course, and if you want to introduce notation, you should feel free to do so only after you've defined it for your readers.
In your example, if you want to make a symbol to mean "a set has cardinality much less than another," first take care to define precisely what you mean by that, and then pick your favorite symbol for the shorthand, being sure to spell it out for your readers.
For example, let's define what it means for a finite set $A$ to have cardinality much less than that of $B$ and invent our own notation for it:

Definition: Let $A$ and $B$ be two finite sets. If $|A| < |B|-5$ then we say the cardinality of $A$ is much less than the cardinality of $B$, and in this case we write $A\preccurlyeq B$.

Now whenever we want to use this shorthand, we can do so in the confidence that our readers know exactly what we mean.

Edit: If you are just trying to impart some intuition to your readers, it would probably be fine to write something like $|A|\ll|B|$ to convey the idea that the cardinality of $A$ is much less than the cardinality of $B$. But even then, it would be better to just say "$A$ has cardinality much less than that of $B$" rather than risk your readers misinterpreting the intuition you are trying to give them because they don't know how to read a symbol that doesn't exist for this exact purpose.
A: The notion of size in a set is given by the cardinality in a set. As I commented, mathematics deals with arbitrary degrees of precision. For finite sets, a simple counting argument could show which set is larger, ie: which one has larger cardinality, but this type of argument fails for infinite sets. One has to characterize "how many" elements there are in a set using a different concept, and that concept is functions.
To illustrate the idea of how it's done, imagine a set $A$ of $7$ elements and another set $B$ of $5$. Any function $A\to B$ will necessarily be not injective; you can start by assigning the first $5$ elements to distinct elements in $B$, but the last $2$ will always be mapped non-injectively. Similarly, you can't find a function $B\to A$ that is surjective.
Cardinality is then extended to infinite sets by the definition that two sets have the same cardinality if there exists a bijection between them. Unintuitively, we find that the integers have the same cardinality as the rationals, and again, that the rationals don't have the same cardinality as the real numbers. We define the lowest infinity to be that of the integers, and call  a set countably infinite set if there exists a bijection with the integers.
We can therefore quantify how large a set sits within another set by having some sort of measure on their cardinality.
A term that is used in measure theory that is the "equivalent" to $\ll$ is the rigorous term "almost all". The definition for almost all is "all but countably many". For example, within the reals, almost all elements are irrational numbers, since there are only countably many rationals.
