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Difficulty: MediumLogical Venn Diagrams

A city library conducted a reading preferences survey among 600600 of its registered members. The survey focused on three genres: Mystery (MM), Sci-Fi (SS), and Biography (BB). The results showed that 120120 members do not read any of these three genres.

Furthermore, the survey revealed the following:
- 250250 members read Mystery.
- 210210 members read Sci-Fi.
- 180180 members read Biography.
- 7070 members read both Mystery and Biography.
- 6060 members read both Sci-Fi and Biography.
- 4040 members read all three genres.

Based on the survey results, how many members read both Mystery and Sci-Fi, but NOT Biography?

  1. A
    70
  2. B
    110
  3. 30Answer
  4. D
    150

Answer

The correct answer is 30, which is the number of members who read both Mystery and Sci-Fi, but not Biography.
By applying the inclusion-exclusion principle, we find that the total number of readers for both Mystery and Sci-Fi is 70. However, this 70 includes the 40 readers who also read Biography. Since the question specifies readers who do NOT read Biography, we must subtract the 40 who read all three, leaving 30 readers who read exactly Mystery and Sci-Fi.

Step-by-Step Solution

1
Determine the total number of members who read at least one of the three genres.
600−120=480600 - 120 = 480 members read at least one genre.
The union of all three sets (∣M∪S∪B∣|M \cup S \cup B|) is equal to the total surveyed population minus those who belong to none of the sets.
2
Use the inclusion-exclusion principle for three sets to find the total number of members who read both Mystery and Sci-Fi (∣M∩S∣|M \cap S|).
480=250+210+180−(∣M∩S∣+70+60)+40480 = 250 + 210 + 180 - (|M \cap S| + 70 + 60) + 40. Simplifying this yields 480=550−∣M∩S∣480 = 550 - |M \cap S|, which means ∣M∩S∣=70|M \cap S| = 70.
The formula ∣M∪S∪B∣=∣M∣+∣S∣+∣B∣−∣M∩S∣−∣M∩B∣−∣S∩B∣+∣M∩S∩B∣|M \cup S \cup B| = |M| + |S| + |B| - |M \cap S| - |M \cap B| - |S \cap B| + |M \cap S \cap B| links all individual totals, pairwise intersections, and the central three-way intersection.
3
Isolate the members who read exactly Mystery and Sci-Fi by excluding those who also read Biography.
70−40=3070 - 40 = 30 members.
The total intersection ∣M∩S∣|M \cap S| includes the members who read all three genres. Subtracting ∣M∩S∩B∣|M \cap S \cap B| leaves only those who read exactly Mystery and Sci-Fi.

Key Concept

Applying the inclusion-exclusion principle for three sets to solve for an unknown pairwise intersection, then isolating a specific exclusive region in a logical Venn diagram.
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Topics

Logical Venn Diagrams

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