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.