Example 2.1.4.1.4 (001A): Examples of Pullbacks of Sets

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Table of Contents
Part 1: Sets
Chapter 2: Constructions With Sets
Section 2.1: Limits of Sets
Subsection 2.1.4: Pullbacks
Example 2.1.4.1.4: Examples of Pullbacks of Sets (cite)

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Example 2.1.4.1.4 ▶ Examples of Pullbacks of Sets

Here are some examples of pullbacks of sets.

Unions via Intersections. Let $X$ be a set. We have
for each $A,B\in \mathcal{P}\webleft (X\webright )$.

Proof of Example 2.1.4.1.4.

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.

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