Definition:The Algebra of Sets

Definition

Let $E$ be a universal set.

The power set $\powerset E$, together with:

the binary operations union $\cup$ and intersection $\cap$

the unary operation complement $\complement$

is referred to as the algebra of sets on $E$.

Also defined as

Note that the concept of an algebra of sets is a more specific concept that is applied to a subset of $\powerset E$ that is closed under union, intersection and complement, and also has a unit.

So while the algebra of sets is an algebra of sets, the reverse is not necessarily true.

Also see

• Power Set is Algebra of Sets, justifying the definition

• Definition:Algebra of Sets

Historical Note

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

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.5$. The algebra of sets
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): algebra of sets
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): algebra of sets