# Intersection with Relative Complement is Empty

## Theorem

The intersection of a set and its relative complement is the empty set:

$T \cap \relcomp S T = \O$

## Proof

 $\displaystyle T \cap \relcomp S T$ $=$ $\displaystyle \paren {S \setminus T} \cap T$ Definition of Relative Complement $\displaystyle$ $=$ $\displaystyle \O$ Set Difference Intersection with Second Set is Empty Set

$\blacksquare$