Set Union

Last updated Nov 7, 2022

Let $A$, $B$ be Sets. Then $A \cup B := {x : x \in A \vee x \in B}$.

By Axiom of Pairing, we can make the Set ${A, B}$. Then, by Axiom of Union we have $$\begin{align*} \bigcup\limits_{} {A, B} &= {x : \exists D \in {A, B} (x \in D)}\\ &={x : x \in A \vee x \in B} \end{align*}$$ $\blacksquare$

Let $\mathcal{A}$ be a collection of Sets. Then we define $$\bigcup\limits_{A \in \mathcal{A}} A := \bigcup\limits_{} \mathcal{A}$$ Existence follows directly from the Axiom of Union.

• Enderton - Elements of Set Theory
• Lian - Fundamentals of ZFC