Subset: A subset A is a set that is contained in another set B in the sense that every element of A is an element of B. The set theoretical notation and definition comes to this: (A a B) iff (x)(xeA > xeB).
In the sense given, which did not refer to any characteristic property of a subset other than that it contains only elements of the set it is a subset of, this may be somewhat problematic.
The reason is that it is easy to prove that for any finite set with n elements the number of its subsets must be 2^{n}  and even for rather small sets this gets quickly quite large. Thus, if n=10, 2^{n} =1024, and if n=100 then 2^{n}= 1267650600228229401496703205376. Hence, there are that many distinct subsets of 100 elements  and it would be quite impossible, in practice, to list them all or characterize them all by some property that would differ from a mere explicit list of all and only elements of the set.
Hence, while the notion of a subset is quite easy to grasp and define, and seems quite sensible, it easily leads to possibilities that are hard to fathom since they are impossible to survey in a finite human life, other than by formal means  which in the case of n=100 mentioned above might be to refer to all the distinct subsets of a given 100 elements by the numbers of 1 to 2^{n}.
For more, see Powerset.
