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.

Explanations

L1 human

The subsets of a set form a Boolean algebra: intersection is and, union is or, and complement is not. Venn diagrams show the laws of this Boolean algebra.