A subset of a set is proper if it is not the improper subset of ( as a subset of itself). We may also say (in the context of concrete sets) that is a proper superset of or that properly contains? .
We may state that is proper in any of these equivalent ways:
Actually, these three definitions are equivalent only if we accept the principle of excluded middle; in constructive mathematics, we usually prefer (3). (For example, consider the notion of proper filter on a set , thought of as a subset of the power set of .) However, (3) is not predicative; see positive element for discussion of this in the dual context. Also, (2) may be strengthened using an inequality relation other than the denial inequality.
Last revised on August 28, 2013 at 06:48:09. See the history of this page for a list of all contributions to it.