I'm supposed to describe the situation using set theoretical operators $(\cup, \cap,\setminus,$ etc.). So we have $A_1,..,A_n$ events and only one of them occurs. I know that $A^c_n$ means $A_n$ does not occur. So I'm supposed to combine, let's say, $A_1$ with $n$ events that do not occur, correct? I'm not so sure about how to express that. I know for example, $(A_1 \cup … \cup A_n)$ means at least one event occurs.
Does that mean the solution here is $(A_1 \cup A^c_2\cup … \cup A^c_n )$?
Best Answer
You can read $\cup$ like "or", $\cap$ like "and" and $\cdot^c$ like "not".
So what you want to say is "$A_1$ happens and none of $A_2,\dots, A_n$ happens or $A_2$ happens and none of $A_1, A_3,\dots, A_n$ happens or $\dots$ or $A_n$ happens and none of $A_1,\dots, A_{n-1}$ happens."
Translate this into the set-theoretical writing and you obtain $$(A_1\cap (A_2\cup \dots \cup A_n)^c) \cup (A_2\cap (A_1\cup A_3\cup \dots \cup A_n)^c \cup\dots \cup (A_n\cap (A_1\cup \dots \cup A_{n-1})^c).$$
You can of course rewrite this using the usual rules of set-theory.