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Difficulty: MediumSet Theory Concepts and Venn Diagrams

An agricultural research station evaluated a sample of 120120 organic farm plots for contamination by three types of crop fungi: *Fusarium* (FF), *Rhizoctonia* (RR), and *Sclerotinia* (SS). The audit revealed the following data:

- 5555 plots were contaminated with *Fusarium*.
- 5050 plots were contaminated with *Rhizoctonia*.
- 4545 plots were contaminated with *Sclerotinia*.
- 2020 plots were contaminated with both *Fusarium* and *Rhizoctonia*.
- 1515 plots were contaminated with both *Rhizoctonia* and *Sclerotinia*.
- 1818 plots were contaminated with both *Fusarium* and *Sclerotinia*.
- 88 plots were contaminated with all three fungi.

Which of the following statements must be true? Select all such statements.

  1. Exactly 15 farm plots were free from all three types of crop fungi.Answer
  2. Exactly 29 farm plots were contaminated by exactly two of the three crop fungi.Answer
  3. C
    Exactly 37 farm plots were contaminated by *Fusarium* only.
  4. D
    Exactly 53 farm plots were contaminated by at least two of the three crop fungi.
  5. E
    Exactly 50 percent of all plots contaminated by *Rhizoctonia* were also contaminated by *Fusarium*.

Answer

The correct statements are that exactly 15 farm plots were free from all three types of crop fungi, and exactly 29 farm plots were contaminated by exactly two of the three crop fungi.
The statement regarding 15 farm plots being free from all three fungi is correct because the total number of plots contaminated by at least one fungus is calculated using inclusion-exclusion: 55 + 50 + 45 - 20 - 15 - 18 + 8 = 105, leaving 120 - 105 = 15 plots clean. The statement regarding 29 farm plots contaminated by exactly two fungi is correct because summing the mutually exclusive regions containing exactly two fungi yields (20 - 8) + (15 - 8) + (18 - 8) = 12 + 7 + 10 = 29.

Step-by-Step Solution

1
Calculate the non-overlapping regions for each intersection.
Triple intersection |F ∩ R ∩ S| = 8.
Region |F ∩ R only| = 20 - 8 = 12.
Region |R ∩ S only| = 15 - 8 = 7.
Region |F ∩ S only| = 18 - 8 = 10.
Determining exact two-set regions requires removing elements that belong to all three sets.
2
Calculate single-fungus-only regions.
|F only| = 55 - (12 + 10 + 8) = 25.
|R only| = 50 - (12 + 7 + 8) = 23.
|S only| = 45 - (10 + 7 + 8) = 20.
Subtract all double and triple overlaps from total individual set counts.
3
Determine union of all three sets and the neither region.
Total contaminated = 25 + 23 + 20 + 12 + 7 + 10 + 8 = 105.
Plots free from all fungi = 120 - 105 = 15.
Subtracting the total contaminated plots from the overall sample size gives the plots outside all sets.
4
Verify statement validity.
Plots free of all fungi = 15 (Valid).
Plots with exactly two fungi = 12 + 7 + 10 = 29 (Valid).
Match calculated regional counts against the presented options.

Key Concept

Three-Set Principle of Inclusion-Exclusion and Venn Diagram Partitioning
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Topics

Set Theory Concepts and Venn Diagrams

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