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
B) iff (x)(xeA --> xeB).
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 2n - and even for
rather small sets this gets quickly quite large.
Thus, if n=10, 2n =1024, and if n=100 then 2n=
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 2n.
For more, see Powerset.