Set operators vs. Logical operators while discussing probability theory. Let $S = \\{1, 2, \dots, 10\\}$ Let $A$ be the event that a number selected from $S$ is even. Let $B$ be the event that a number selected from $S$ is a multiple of $3$. If any number is equally likely to be selected from $S$, the probability that a number selected from the set $S$ is neither even nor a multiple of $3$ is $\frac{3}{10}$. I write it as $P(\overline A \cap \overline B) = \frac{3}{10}$. But I have some people writing it as $P(\neg A \text{ and } \neg B) = \frac{3}{10}$. I don't understand the reason for this notation. They seem to be using logical operators which apply to statements in propositional logic. But $A$ and $B$ are not statements. They are events, and events are sets. Shouldn't we stick strictly to set theory notation while discussing probability? If we have to use logical operators, shouldn't we use $P(\neg A \land \neg B)$ instead of $P(\neg A \text{ and } \neg B)$?

Events are sets only because we have _defined_ them to be sets, as part of the general $20$-th century pattern that attempts to frame all of mathematics set-theoretically. However, such events as "is red-haired," "has freckles," "had an allergic reaction," and many others have natural interpretations as predicates. And even in mathematical discussions of probability, connectives of a logical nature (and, or, not) are used. We _say_ "and" and _write_ $\cap$.