For three circular regions, labeled as P, Q and R, shown in the figure, select the Boolean operator, which returns the shaded area.
Show Hint
Notice the shaded area is only the part of circle R that does not touch P or Q, then check which Boolean expression selects exactly that region within the extent of the three circles.
Step 1: Label the seven basic regions of a three-set Venn diagram. Any three overlapping sets $P$, $Q$, $R$ divide the plane into 7 regions: $P$ only, $Q$ only, $R$ only, $P\cap Q$ only, $P\cap R$ only, $Q\cap R$ only, and $P\cap Q\cap R$. Step 2: Identify which region is shaded. The shaded crescent, being inside $R$ but not touching $P$ or $Q$, corresponds exactly to the region $R$ only, which in set notation is $R \wedge \lnot P \wedge \lnot Q$, that is, $R \wedge \lnot(P \vee Q)$. Step 3: Simplify using the restricted domain. Because the whole figure only shows points inside $P \cup Q \cup R$, any point that satisfies $\lnot(P \vee Q)$ within this domain is automatically forced to lie in $R$. So within this domain, $R \wedge \lnot(P \vee Q)$ and $\lnot(P \vee Q)$ select the same set of points. Step 4: Match to the given options. $\lnot(P \vee Q)$ is written in the options as NOT (P OR Q), option (A). The other three options correspond to different Venn regions: NOT (P AND Q) is everything except $P \cap Q$ (too large), NOT (P OR Q OR R) is the empty region outside all circles, and NOT ((P OR Q) AND R) removes only $R$ only from the whole domain, leaving $P$ only plus $Q$ only plus overlaps, none of which match the small crescent. Step 5: Conclude. The shaded $R$ only crescent corresponds to option (A), NOT (P OR Q). \[ \boxed{\text{Option (A), NOT (P OR Q)}} \]
