Intersection with Complement

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


 * $S \cap \complement \left({S}\right) = \varnothing$

Proof
Substitute $\mathbb U$ for $S$ and $S$ for $T$ in $T \cap \complement_S \left({T}\right) = \varnothing$ from Intersection with Relative Complement.

Also see

 * Union with Complement

Notice the similarity with the Principle of Noncontradiction.

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