nLab
proper subset

Proper subsets

Idea

A subset AA of a set SS is proper if it is not the improper subset of SS (SS as a subset of itself). We may also say (in the context of concrete sets) that SS is a proper superset of AA or that SS properly contains? AA.

Definition

We may state that AA is proper in any of these equivalent ways:

  1. AA is not equal (as a subset) to SS;
  2. There exists an element xx of SS such that xAx \notin A;
  3. Given any way of expressing AA as the intersection of a family of subsets of SS, this family is inhabited.

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 XX, thought of as a subset of the power set of XX.) 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.

Revised on August 28, 2013 06:48:09 by Toby Bartels (141.0.11.201)