# Intersection with Complement

## Theorem

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

$S \cap \map \complement S = \O$

## Proof

Substitute $\mathbb U$ for $S$ and $S$ for $T$ in $T \cap \relcomp S T = \O$ from Intersection with Relative Complement is Empty.

$\blacksquare$

## Also see

Notice the similarity with the Principle of Non-Contradiction.

The complement of a set is similar to the negation of a proposition, and intersection is similar to conjunction.