Symmetric Difference of Events is Event

From ProofWiki
Jump to navigation Jump to search


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

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

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

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


\(\ds A, B\) \(\in\) \(\ds \Sigma\)
\(\ds \leadsto \ \ \) \(\ds A \cup B\) \(\in\) \(\ds \Sigma\) Definition of Event Space: Axiom $(\text {ES} 3)$
\(\, \ds \land \, \) \(\ds A \cap B\) \(\in\) \(\ds \Sigma\) Intersection of Events is Event
\(\ds \leadsto \ \ \) \(\ds \paren {A \cup B} \setminus \paren {A \cap B}\) \(\in\) \(\ds \Sigma\) Set Difference of Events is Event
\(\ds \leadsto \ \ \) \(\ds A \ast B\) \(\in\) \(\ds \Sigma\) Definition 2 of Symmetric Difference


Also see