Union is Associative

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This article is a landmark page. It was the 3rd proof on ProofWiki!

Theorem

$A \cup \left({B \cup C}\right) = \left({A \cup B}\right) \cup C$

$\displaystyle $	$\displaystyle x \in A \cup \left({B \cup C}\right)$	$\iff$	$\displaystyle x \in A \lor \left({x \in B \lor x \in C}\right)$	$\displaystyle $	Definition of Union
$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle \left({x \in A \lor x \in B}\right) \lor x \in C$	$\displaystyle $	Rule of Association
$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle x \in \left({A \cup B}\right) \cup C$	$\displaystyle $	Definition of Union

Therefore, $x \in A \cup \left({B \cup C}\right)$ iff $x \in \left({A \cup B}\right) \cup C$.

Thus it has been shown that $A \cup \left({B \cup C}\right) = \left({A \cup B}\right) \cup C$.

$\blacksquare$

\(\displaystyle \)	\(\displaystyle x \in A \cup \left({B \cup C}\right)\)	\(\iff\)	\(\displaystyle x \in A \lor \left({x \in B \lor x \in C}\right)\)	\(\displaystyle \)	Definition of Union
\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle \left({x \in A \lor x \in B}\right) \lor x \in C\)	\(\displaystyle \)	Rule of Association
\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle x \in \left({A \cup B}\right) \cup C\)	\(\displaystyle \)	Definition of Union