of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition. The set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinite, but the latter set has a larger cardinality (or ) than the former set. Another example in an Euler diagram: Other properties of inclusion Inclusion is the canonical partial order, in the sense that every partially ordered set is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example: if each ordinal n is identified with the set of all ordinals less than or equal to n, then if and only if For the power set of a set S, the inclusion partial order is—up to an order isomorphism—the Cartesian product of (the cardinality of S) copies of the partial order on for which This can be illustrated by enumerating , and associating with each subset (i.e., each element of ) the k-tuple from of which the ith coordinate is 1 if and only if is a member of T. See also Convex subset Inclusion order Region Subset sum problem Subsumptive containment Total subset References Bibliography External links Basic concepts in set theory