Set Intersections

Why

We can consider intersections of more than two sets.

Definition

Let $\mathcal{A} $ denote a set of sets. In other words, every element of $\mathcal{A} $ is a set. And suppose that $\mathcal{A} $ has at least one set (i.e., $\mathcal{A} \neq \varnothing$). Let $C$ denote a set such that $C \in \mathcal{A} $. Then consider the set,

\[ \Set{x \in C}{(\forall A)(A \in \mathcal{A} \implies x \in A)}. \]

This set exists by the principle of specification (see Set Specification). Moreover, the set does not depend on which set we picked. So the dependence on $C$ does not matter. It is unique by the axiom of extension (see Set Equality). This set is called the intersection of $\mathcal{A} $.

Notation

We denote the intersection of $\mathcal{A} $ by $\bigcap \mathcal{A} $.

Equivalence with pair intersections

As desired, the the set denoted by $\mathcal{A} $ is a pair (see Unordered Pairs) of sets, the pair intersection (see Pair Intersections) coincides with intersection as we have defined it in this sheet.¹

$\bigcap \set{A, B} = A \cap B$

A full account of the proof will appear in future editions. ↩︎