Study the diagram below, which shows three overlapping groups, College Students, Singers and Dancers, divided into the labelled regions P, Q, R, S, T, U and V, and answer the question that follows.
College students who are neither dancers nor singers are represented by?
Show Hint
Look for the region that lies inside the College Students shape but stays completely outside both the Singers shape and the Dancers shape.
Step 1: Read what each labelled region stands for.
The diagram overlaps three rectangles: College Students, Singers and Dancers. Region P is the part of College Students that touches neither Singers nor Dancers. Region S is College Students overlapping Singers only. Region Q sits inside the Singers rectangle but completely outside the College Students rectangle, so Q means "singer, not a college student". Region V sits inside the Dancers rectangle but completely outside College Students, so V means "dancer, not a college student".
Step 2: Translate the question into set language.
We want college students who are neither dancers nor singers. This needs the region to sit inside College Students, but outside both Singers and Dancers.
Step 3: Test each option.
Region S involves singing, so it fails the "not a singer" condition. Region V and region Q are both drawn completely outside the College Students rectangle in this diagram, so neither one can represent "college students" at all, regardless of the singing or dancing condition, this makes them impossible answers to a question that specifically asks about college students. Region P is the only region that is inside College Students while staying outside both Singers and Dancers.
Step 4: Note on the answer.
Some circulated answer keys list option (d), Q, for this question. That cannot be correct here, because Q lies outside the College Students rectangle in the diagram (it is the "singer only, not a college student" region), the same way V lies outside it and is correctly rejected as the answer to the previous question about dancers and singers together. A region drawn outside College Students cannot answer a question restricted to college students. Applying the exact same reading of the diagram that correctly gives R, S and T as the answers to the three previous questions gives P as the only consistent answer here.
Final Answer:
College students who are neither dancers nor singers are represented by P.
\[ \boxed{P} \]
