Are the events A and overlineA+B+C compatible (can they…

Are the events $A$ and $\overline{A+B+C}$ compatible (can they occur simultaneously)?

Step 1: Compatible and incompatible events

Step 2: Rewrite $\overline{A+B+C}$

By De Morgan's law for three events, "none of $A$, $B$, $C$ occurs" is the opposite of "at least one occurs":

\[ \overline{A+B+C} = \overline{A}\,\overline{B}\,\overline{C}. \]

Step 3: Multiply by $A$

\[ A\cdot\overline{A+B+C} = A\,\overline{A}\,\overline{B}\,\overline{C} = (A\overline{A})\,\overline{B}\,\overline{C} = \varnothing\cdot\overline{B}\,\overline{C} = \varnothing. \]

We used $A\overline{A}=\varnothing$ (an event and its opposite never occur together) and $\varnothing\cdot X=\varnothing$.

Step 4: Conclusion

\[ \boxed{A\cdot\overline{A+B+C}=\varnothing,\ \text{so the events are incompatible.}} \]

Step 5: The same reasoning in words

If $A$ occurs, then "at least one of $A$, $B$, $C$ occurs" is certainly true, i.e. $A+B+C$ occurs. Then its opposite $\overline{A+B+C}$ cannot occur. So whenever $A$ occurs, $\overline{A+B+C}$ does not.

More generally: if $A\subseteq S$, then $A$ and $\overline{S}$ are always incompatible. Here $S=A+B+C$.

Step 6: Proof by outcomes

Suppose some outcome $\omega$ belonged to both events. From $\omega\in\overline{A+B+C}=\overline{A}\,\overline{B}\,\overline{C}$ we get $\omega\in\overline{A}$, i.e. $\omega\notin A$. This contradicts $\omega\in A$. Hence no such outcome exists.

Step 7: Example with a die

$\Omega=\{1,\dots,6\}$, $A=\{1,2\}$, $B=\{3\}$, $C=\{4\}$. Then $A+B+C=\{1,2,3,4\}$ and $\overline{A+B+C}=\{5,6\}$. Its product with $A$ is $\{1,2\}\cap\{5,6\}=\varnothing$. ✓

Step 8: Remark — incompatible, but not necessarily opposite

$A$ and $\overline{A+B+C}$ are always incompatible, but they are opposite events (one of them always occurs) only if $A+B+C=A$, i.e. $B\subseteq A$ and $C\subseteq A$. In the example above neither occurs when the die shows $3$ or $4$.