Set Difference of Events is Event

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

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

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

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


\(\ds A, B\) \(\in\) \(\ds \Sigma\)
\(\ds \leadsto \ \ \) \(\ds A, \Omega \setminus B\) \(\in\) \(\ds \Sigma\) Definition of Event Space: Axiom $(\text {ES} 2)$
\(\ds \leadsto \ \ \) \(\ds A \cap \paren {\Omega \setminus B}\) \(\in\) \(\ds \Sigma\) Intersection of Events is Event
\(\ds \leadsto \ \ \) \(\ds A \setminus B\) \(\in\) \(\ds \Sigma\) Set Difference as Intersection with Relative Complement


Also see