Union with Complement

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Theorem

The union of a set and its complement is the universe:

$S \cup \map \complement S = \mathbb U$


Proof

Substitute $\mathbb U$ for $S$ and $S$ for $T$ in $T \cup \relcomp S T = S$ from Union with Relative Complement.

$\blacksquare$


Law of the Excluded Middle

This theorem depends on the Law of the Excluded Middle, by way of Union with Relative Complement.

This is one of the axioms of logic that was determined by Aristotle, and forms part of the backbone of classical (Aristotelian) logic.

However, the intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom. This in turn invalidates this theorem from an intuitionistic perspective.


Also see


Sources