empty subset

Given a set $A$, the **empty subset** of $A$, denoted $\empty_A$, is the subset of $A$ defined by the property that, for every element $x$ of $A$, it is false that $x$ belongs to $\empty_A$.

The underlying set (or *shadow*) of any empty subset is the? empty set. That is, if we interpret $\empty_A$ as an injective function $S \hookrightarrow A$, then the source $S$ of this function is the empty set.

In the usual framework of material set theory, every empty subset is identical to the empty set. For this reason, it is common to write simply $\empty$ instead of $\empty_A$. Even from a structural perspective, this is an abuse of language that is unlikely to cause any confusion.

Created on March 20, 2010 19:36:46
by Toby Bartels
(75.88.69.30)