# Intersection of Events is Event

## Theorem

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

The event space $\Sigma$ of $\EE$ has the property that:

$A, B \in \Sigma \implies A \cap B \in \Sigma$

That is, the intersection of two events is also an event in the event space.

## Proof

 $\ds A, B$ $\in$ $\ds \Sigma$ $\ds \leadsto \ \$ $\ds \Omega \setminus A, \ \Omega \setminus B$ $\in$ $\ds \Sigma$ Definition of Event Space: Axiom $(\text {ES} 2)$ $\ds \leadsto \ \$ $\ds \paren {\Omega \setminus A} \cup \paren {\Omega \setminus A}$ $\in$ $\ds \Sigma$ Definition of Event Space: Axiom $(\text {ES} 3)$ $\ds \leadsto \ \$ $\ds \Omega \setminus \paren {A \cap B}$ $\in$ $\ds \Sigma$ De Morgan's Laws: Difference with Intersection $\ds \leadsto \ \$ $\ds \Omega \setminus \paren {\Omega \setminus \paren {A \cap B} }$ $\in$ $\ds \Sigma$ Definition of Event Space: Axiom $(\text {ES} 2)$ $\ds \leadsto \ \$ $\ds A \cap B$ $\in$ $\ds \Sigma$ Relative Complement of Relative Complement

$\blacksquare$