Intersection with Complement

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Theorem

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

$S \cap \relcomp {} S = \O$


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.


Sources