# Proper subsets

## Idea

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

## Definition

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

1. $A$ is not equal (as a subset) to $S$;
2. There exists an element $x$ of $S$ such that $x \notin A$;
3. Given any way of expressing $A$ as the intersection of a family of subsets of $S$, 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 $X$, thought of as a subset of the power set of $X$.) 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)