# Intersection with Complement is Empty iff Subset

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## Theorem

$S \subseteq T \iff S \cap \map \complement T = \O$

where:

$S \subseteq T$ denotes that $S$ is a subset of $T$
$S \cap T$ denotes the intersection of $S$ and $T$
$\O$ denotes the empty set
$\complement$ denotes set complement.

### Corollary

$S \cap T = \O \iff S \subseteq \relcomp {} T$

## Proof

 $\displaystyle S$ $\subseteq$ $\displaystyle T$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle S \setminus T$ $=$ $\displaystyle \O$ Set Difference with Superset is Empty Set‎ $\displaystyle \leadstoandfrom \ \$ $\displaystyle S \cap \map \complement T$ $=$ $\displaystyle \O$ Set Difference as Intersection with Complement

$\blacksquare$