Power Set is Algebra of Sets
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Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Then $\powerset S$ is an algebra of sets where $S$ is the unit.
From Power Set is Closed under Intersection and Power Set is Closed under Symmetric Difference, we have that:
- $(1): \quad \forall A, B \in \powerset S: A \cap B \in \powerset S$
- $(2): \quad \forall A, B \in \powerset S: A * B \in \powerset S$
From the definition of power set:
- $\forall A \in \powerset S: A \subseteq S$
and so $S$ is the unit of $\powerset S$.
Thus we see that $\powerset S$ is a ring of sets with a unit.
Hence the result, by definition of an algebra of sets.