**Other Properties of Inclusion**

Inclusion is the canonical partial order in the sense that every partially ordered set (*X*, ) 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 *a* ≤ *b* 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 *k* = |*S*| (the cardinality of *S*) copies of the partial order on {0,1} for which 0 < 1. This can be illustrated by enumerating *S* = {*s*_{1}, *s*_{2}, …, *s*_{k}} and associating with each subset *T* ⊆ *S* (which is to say with each element of 2*S*) the *k*-tuple from {0,1}*k* of which the *i*th coordinate is 1 if and only if *s*_{i} is a member of *T*.

Read more about this topic: Subset

### Famous quotes containing the words properties and/or inclusion:

“A drop of water has the *properties* of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”

—Ralph Waldo Emerson (1803–1882)

“Belonging to a group can provide the child with a variety of resources that an individual friendship often cannot—a sense of collective participation, experience with organizational roles, and group support in the enterprise of growing up. Groups also pose for the child some of the most acute problems of social life—of *inclusion* and exclusion, conformity and independence.”

—Zick Rubin (20th century)