Order, lattices, ordered algebraic structures Constructions of ordered sets Example

Power set Boolean algebra

Example

Let $X$ be a set, and let $\mathcal{P}(X)$ be the set of all subsets of $X$ . With $A \wedge B = A \cap B$ , $A \vee B = A \cup B$ , $\neg A = X \setminus A$ , $0 = \varnothing$ and $1 = X$ , the set $\mathcal{P}(X)$ is a Boolean algebra. Its order is inclusion $\subseteq$ .

Remarks

If $X$ has $n$ elements, this Boolean algebra has $2^n$ elements. A field of sets on $X$ is a subset of $\mathcal{P}(X)$ that contains $\varnothing$ and is closed under complement and binary union. It is a Boolean subalgebra. Every finite Boolean algebra is isomorphic to a power set Boolean algebra, by Stone's representation theorem for finite Boolean algebras.

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Stone's representation theorem for finite Boolean algebras

Tag	`004J7ZEFAT`
URI	`m2://004J7ZEFAT`
Kind	example
Writing	W0 stub
MSC	06E05
Instance of	Boolean algebra

Power set Boolean algebra

Example

Remarks

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