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Let the probability space of an experiment $\mathcal E$ be $\left({\Omega, \Sigma, \Pr}\right)$.

Let $A, B \in \Sigma$, so that $A \subseteq \Omega$ and $B \subseteq \Omega$.

Let the outcome of the experiment be $\omega \in \Omega$.

Then the following real-world interpretations of the occurrence of events can be determined:

If $\omega \in A$, then $A$ occurs.
If $\omega \notin A$, i.e $\omega \in \Omega \setminus A$, then $A$ does not occur.

If $\omega \in A \cup B$, then either $A$ or $B$ occur.
If $\omega \in A \cap B$, then both $A$ and $B$ occur.

If $\omega \in A \setminus B$, then $A$ occurs but $B$ does not occur.
If $\omega \in A * B$, where $*$ denotes symmetric difference, then either $A$ occurs or $B$ occurs, but not both.

Also known as

The word happen is often used for occur, and it can be argued that it is easier to understand what is meant.