Power Set is Closed under Intersection
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Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
- $\forall A, B \in \powerset S: A \cap B \in \powerset S$
Let $A, B \in \powerset S$.
Then by the definition of power set, $A \subseteq S$ and $B \subseteq S$.
From Intersection is Subset we have that $A \cap B \subseteq A$.
It follows from Subset Relation is Transitive that $A \cap B \subseteq S$.
Thus $A \cap B \in \powerset S$ and closure is proved.
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.1$. Binary operations on a set: Example $60$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): algebra of sets