Question

For three circular regions, labeled as P, Q and R, shown in the figure, select the Boolean operator, which returns the shaded area.

• A. NOT (P OR Q)
• B. NOT (P AND Q)
• C. NOT (P OR Q OR R)
• D. NOT ((P OR Q) AND R)

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.

Answer 1

The Correct Option is A

Step 1: Identify the shaded region in the figure.
The shaded area is a crescent-shaped piece that lies entirely inside circle \(R\), on the side away from circles \(P\) and \(Q\), that is, the part of \(R\) that does not overlap either \(P\) or \(Q\).
Step 2: Take the analysis extent as the union of the three circles.
Since only the three circles \(P\), \(Q\), \(R\) are shown and nothing is shaded outside them, the effective universe for the Boolean overlay is the union \(P \cup Q \cup R\); nothing outside this union is being queried.
Step 3: Test option (A), NOT (P OR Q).
Within the union \(P \cup Q \cup R\), not in \(P\) and not in \(Q\) is exactly the part of \(R\) that lies outside both \(P\) and \(Q\) (any point in the union that is not in \(P\) or \(Q\) must be in \(R\) alone). This is precisely the shaded crescent.
Step 4: Rule out the other options.
NOT (P AND Q) would shade everything except the small overlap of \(P\) and \(Q\), far larger than the shaded crescent. NOT (P OR Q OR R) would shade the region outside all three circles, which is empty within the shown extent. NOT ((P OR Q) AND R) removes only the part of \(R\) that overlaps \(P\) or \(Q\) from the whole extent, so it would shade all of \(P\), all of \(Q\), and the non-overlapping part of \(R\), again much larger than the crescent shown.
Step 5: Conclude.
Only option (A), NOT (P OR Q), matches the shaded region. \[ \boxed{\text{Option (A), NOT (P OR Q)}} \]