# Set Difference with Disjoint Set

## Theorem

Let $S, T$ be sets.

Then:

$S \cap T = \varnothing \iff S \setminus T = S$

where:

$S \cap T$ denotes set intersection
$\varnothing$ denotes the empty set
$S \setminus T$ denotes set difference.

## Proof

 $\displaystyle S \cap T$ $=$ $\displaystyle \varnothing$ $\displaystyle \iff \ \$ $\displaystyle S$ $\subseteq$ $\displaystyle \complement \left({T}\right)$ Intersection with Complement is Empty iff Subset $\displaystyle \iff \ \$ $\displaystyle S \cap \complement \left({T}\right)$ $=$ $\displaystyle S$ Intersection with Subset is Subset‎‎ $\displaystyle \iff \ \$ $\displaystyle S \setminus T$ $=$ $\displaystyle S$ Set Difference as Intersection with Complement

$\blacksquare$