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