Symmetric Difference 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:

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

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

Proof

\(\ds A, B\)

\(\in\)

\(\ds \Sigma\)

\(\ds \leadsto \ \ \)

\(\ds A \cup B\)

\(\in\)

\(\ds \Sigma\)

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

$\blacksquare$

Also see

• Elementary Properties of Event Space

Sources

• 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $1$: Events and probabilities: $1.2$: Outcomes and events: Exercise $3$