# Union equals Intersection iff Sets are Equal

## Theorem

Let $S$ and $T$ be sets.

Then:

$\paren {S \cup T = S} \land \paren {S \cap T = S} \iff S = T$

where:

$S \cup T$ denotes set union
$S \cap T$ denotes set intersection.

## Proof

$S \subseteq T \iff S \cap T = S$
$S \subseteq T \iff S \cup T = T$

That is:

$T \subseteq S \iff S \cup T = S$

Thus:

$\paren {S \cup T = S} \land \paren {S \cap T = S} \iff S \subseteq T \subseteq S$

By definition of set equality:

$S = T \iff S \subseteq T \subseteq S$

Hence the result.

$\blacksquare$