Union with Empty Set

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The union of any set with the empty set is the set itself:

$S \cup \O = S$

Proof 1

\(\ds S\) \(\subseteq\) \(\ds S\) Set is Subset of Itself
\(\ds \O\) \(\subseteq\) \(\ds S\) Empty Set is Subset of All Sets
\(\ds \leadsto \ \ \) \(\ds S \cup \O\) \(\subseteq\) \(\ds S\) Union is Smallest Superset
\(\ds \leadsto \ \ \) \(\ds S\) \(\subseteq\) \(\ds S \cup \O\) Set is Subset of Union
\(\ds \leadsto \ \ \) \(\ds S \cup \O\) \(=\) \(\ds S\) Definition 2 of Set Equality


Proof 2

From Empty Set is Subset of All Sets:

$\O \subseteq S$

From Union with Superset is Superset‎:

$S \cup \O = S$


Also see