# Set Difference Equals First Set iff Empty Intersection

## Theorem

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

## Proof

Assume $S, T \subseteq \Bbb U$ where $\Bbb U$ is a universal set.

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

$\blacksquare$