Does a set exist containing the elements shared between two sets? How might we construct such a set?
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)
\[
\Set{x \in A}{x \in B} = \Set{x \in B}{x \in A}.
\]
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”.
All the following results are immediate.2