Pair Intersections

Why

Does a set exist containing the elements shared between two sets? How might we construct such a set?

Definition

Let $A$ and $B$ denote sets. Consider the set $\Set{x \in A}{x \in B}$. This set exists by the principle of specification (see Set Specification). Moreover $(y \in \Set{x \in A}{x \in B}) \iff (y \in A \land y \in B)$. In other words, $\Set{x \in A}{x \in B}$ contains all the elements of $A$ that are also elements of $B$.

We can also consider $\Set{x \in B}{x \in A}$, in which we have swapped the positions of $A$ and $B$. Similarly, the set exists by the principle of specification (see Set Specification) and again $y \in \Set{x \in B}{x \in A} \iff (y \in B \land y \in B)$. Of course, $y \in A \land y \in B$ means the same as¹ $y \in B \land y \in A$ and so by the principle of extension (see Set Equality)

\[ \Set{x \in A}{x \in B} = \Set{x \in B}{x \in A}. \]

Notation

We denote the intersection fo the set denoted by $A$ with the set denoted by $B$ by $A \cap B$. We read this notation aloud as “A intersect B”.

Basic properties

All the following results are immediate.²

1. Future editions will name and cite this rule. ↩︎
2. Proofs of these results will appear in the next edition. ↩︎