Here are some examples of pullbacks of sets.
Proof of Example 2.1.4.1.2.
Item 1: Unions via Intersections
Indeed, we have
\begin{align*} A\times _{A\cup B}B & \cong \webleft\{ \webleft (x,y\webright )\in A\times B\ \middle |\ x=y\webright\} \\ & \cong A\cap B. \end{align*}
This finishes the proof.
