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.

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.