# Definition:The Algebra of Sets

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## 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

## 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**