Inclusion-Exclusion Principle/Examples/3 Events in Event Space

Examples of Use of Inclusion-Exclusion Principle
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $A, B, C \in \Sigma$.

Then:

Proof
The Inclusion-Exclusion Principle is applicable for an additive function on an algebra of sets.

We have that $\Sigma$ is a $\sigma$-algebra.

Hence by definition, $\Sigma$ is an algebra of sets which is closed under countable unions.

Hence, a fortiori, $\Sigma$ is an algebra of sets.

We have by definition of probability measure, that $\Pr$ is an additive function fulfilling certain conditions.

The result then follows as a special case of the Inclusion-Exclusion Principle.