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 dancers but not singers are represented by?
Show Hint
Look for the region that lies inside both the College Students and the Dancers shapes but stays completely outside the Singers shape.
Step 1: Read what each labelled region stands for.
The diagram has three overlapping rectangles: College Students, Singers and Dancers. Region P is the part of College Students that touches neither of the other two rectangles, so P means "college student only". Region S is where College Students overlaps Singers alone, so S means "college student and singer, not dancer". Region R is where College Students overlaps Dancers alone, so R means "college student and dancer, not singer". Region T sits inside all three rectangles at once, so T means "college student, singer and dancer, all three". Region U is shared only between Singers and Dancers, outside College Students, region Q is Singers only (not a college student at all), and region V is Dancers only (not a college student at all).
Step 2: Translate the question into set language.
We want college students who dance but do not sing. This needs three things at once: inside College Students, inside Dancers, and outside Singers.
Step 3: Test each region against these three conditions.
Region S is college student and singer, so it fails the "not a singer" condition, it is ruled out. Region T is college student, singer and dancer together, so it also fails the "not a singer" condition. Region P is college student only, with no dancing at all, so it fails the "dancer" condition. Region R is exactly college student and dancer, with no singing, which satisfies all three conditions at once.
Final Answer:
College students who are dancers but not singers are represented by R.
\[ \boxed{R} \]
