Which of the following Venn diagram best represents the relationship among Americans, Professors and Men?
Show Hint
Whenever three groups are independent categories and no subset relationship is given, use the Venn diagram with three intersecting circles.
Examples:
\[
\text{Teachers, Indians, Women}
\]
\[
\text{Doctors, Athletes, Men}
\]
Such cases are usually represented by three overlapping circles with a common intersection region.
Concept:
To solve Venn diagram questions, we must analyze the relationship among the given groups.
The three groups are:
\[
\text{Americans}, \quad \text{Professors}, \quad \text{Men}
\]
There is no statement indicating that one group is completely contained within another.
Also, it is possible that:
• Some Americans are Professors.
• Some Professors are Men.
• Some Americans are Men.
• Some individuals may belong to all three groups simultaneously.
Therefore, all three sets can overlap partially.
Step 1: Analyze Diagram A.
In Diagram A, one circle lies completely inside another circle.
This implies:
\[
\text{One set is a subset of another set.}
\]
But:
\[
\text{Professors} \nsubseteq \text{Men}
\]
and
\[
\text{Americans} \nsubseteq \text{Men}
\]
Hence Diagram A does not correctly represent the relationship.
Step 2: Analyze Diagram B.
Diagram B contains three circles intersecting one another.
It allows:
• Partial overlap between every pair of sets.
• A common region shared by all three sets.
• Independent portions belonging exclusively to each set.
This perfectly matches the relationship among Americans, Professors and Men.
Step 3: Analyze Diagram C.
Diagram C shows both smaller circles completely inside a larger circle.
This implies:
\[
\text{Americans and Professors are subsets of Men}
\]
which is clearly incorrect.
Step 4: Analyze Diagram D.
Diagram D allows overlap only between adjacent circles.
The first and third circles do not intersect.
However, some Americans can be Men.
Therefore all three sets should be allowed to overlap.
Hence Diagram D is not correct.
Step 5: Select the correct diagram.
The only diagram that correctly represents the possible relationships among all three groups is:
\[
\boxed{\text{Diagram B}}
\]
Therefore,
\[
\boxed{\text{Only B}}
\]
Hence option (D) is correct.
