Definition:Algebra of Sets

Definition 1

Given a set $X$ and a collection of subsets of $X$, $\mathcal S \subset \mathcal P \left({X}\right)$, $\mathcal S$ is called an algebra of sets if, given that $A, B \in \mathcal S$:

$(1): \quad A \cup B \in \mathcal S$

$(2): \quad \complement_X \left({A}\right) \in \mathcal S$

where $\complement_X \left({A}\right)$ is the relative complement of $A$ in $X$.

Definition 2

An algebra of sets is a ring of sets with a unit.

The two definitions are equivalent.

Examples

Power Set

The Power Set is Algebra of Sets.

Null Set and Set Itself

Let $S$ be any non-empty set.

Then $\left\{{S, \varnothing}\right\}$ is (trivially) an algebra of sets, where $S$ is the unit.

Historical Note

The concept of an algebra of sets was invented by George Boole, after whom Boolean algebra was named.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 1.5$