Intersection with Universe

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Theorem

The intersection of a set with the universe is the set itself:

$\mathbb U \cap S = S$


Proof

\(\displaystyle S\) \(\subseteq\) \(\displaystyle \mathbb U\) $\quad$ Definition of Universe $\quad$
\(\displaystyle \iff \ \ \) \(\displaystyle \mathbb U \cap S\) \(=\) \(\displaystyle S\) $\quad$ Intersection with Subset is Subset‎ $\quad$

$\blacksquare$


Also see


Sources