Union of Event with Complement is Certainty

Theorem

Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.

Let $A \in \Sigma$ be an events of $\EE$, so that $A \subseteq \Omega$.

Then:

$A \cup \overline A = \Omega$

where $\overline A$ is the complementary event to $A$.

That is:

$A \cup \overline A$ is a certainty

or:

$\map \Pr {A \cup \overline A} = 1$

Proof

By definition:

$A \subseteq \Omega$

and:

$\overline A = \relcomp \Omega A$

From Union with Relative Complement:

$A \cup \overline A = \Omega$

We then have from Kolmogorov axiom $(2)$ that:

$\map \Pr \Omega = 1$

The result follows by definition of certainty.

$\blacksquare$

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 logical axioms 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

• Intersection of Event with Complement Can't Happen

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): complementary (for probability)