In infinite sets, a part can be exactly as big as the whole
Every natural number is also a rational number, but plenty of rationals, such as fractions, are not natural numbers. So the naturals form a strict part of the rationals. Yet the two collections have exactly the same size. With infinite sets, taking away elements does not always make a set smaller, which runs against everyday intuition.
The underlying idea is simple. A set A is a subset of B when every element of A also belongs to B, and B is then called a superset of A. The two may be equal; if B has at least one extra element, A is a proper subset. So {1, 2} is a proper subset of {1, 2, 3}, while {1, 2, 3} is a subset of itself but not a proper one. The empty set counts as a subset of every set, because it has no elements that could fail the test. Mathematicians call this vacuous truth.
Proving that one set sits inside another usually follows a standard pattern called the element argument: pick an arbitrary element of the first set and show it must belong to the second. Because the element was chosen without any special features, the conclusion holds for all of them. There are handy equivalent tests too: A is inside B precisely when their overlap equals A, or when combining them gives back B.
Infinite examples are where subsets get strange. The prime numbers above 10 form a proper subset of the odd numbers above 10. The points of a line segment form a proper subset of the points of a full line, yet, like the naturals and rationals, the part matches the whole in size. Not every case works out that way, though: the rationals are a proper subset of the real numbers, and the reals form a strictly larger infinity.
Collected together, all the subsets of a set form its power set. Ordered by inclusion, they make a Boolean algebra, the same structure that underlies logic, with overlap and combination as its basic operations. Inclusion is in fact the model for every partial ordering: any partially ordered collection can be mirrored by sets ranked by which contains which.
Source: Subset
