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Zorluk: OrtaLogical Venn Diagrams

An environmental agency reviewed the compliance of 380380 industrial plants across three pollution control standards: Air Emissions (AA), Water Discharge (WW), and Solid Waste (SS). The review revealed the following data:
- 170170 plants met standard AA.
- 150150 plants met standard WW.
- 160160 plants met standard SS.
- 6060 plants met both AA and WW.
- 5555 plants met both WW and SS.
- 5050 plants met both AA and SS.
- 2020 plants failed to meet any of the three standards.

Based on this information, how many industrial plants met EXACTLY TWO of the pollution control standards?

  1. 30Cevap
  2. B
    45
  3. C
    120
  4. D
    165

Cevap

The number of industrial plants that met exactly two pollution control standards is 30.
The number of plants that met at least one standard is 380−20=360380 - 20 = 360. Using the inclusion-exclusion principle for three sets, 360=170+150+160−60−55−50+N(all three)360 = 170 + 150 + 160 - 60 - 55 - 50 + N(\text{all three}), which yields N(all three)=45N(\text{all three}) = 45. The intersection values given (60,55,5060, 55, 50) include these 4545 plants. To isolate the plants meeting *exactly two* standards, we subtract 4545 from each intersection and sum the remaining values: (60−45)+(55−45)+(50−45)=15+10+5=30(60 - 45) + (55 - 45) + (50 - 45) = 15 + 10 + 5 = 30.

Adım Adım Çözüm

1
Calculate the total number of plants that met at least one standard.
380−20=360380 - 20 = 360
To find the union of the three sets, we must subtract the plants that met none of the standards from the total number of surveyed plants.
2
Calculate the number of plants that met all three standards using the inclusion-exclusion principle.
N(A∩W∩S)=45N(A \cap W \cap S) = 45
Substituting the known values into the formula N(A∪W∪S)=N(A)+N(W)+N(S)−N(A∩W)−N(W∩S)−N(A∩S)+N(A∩W∩S)N(A \cup W \cup S) = N(A) + N(W) + N(S) - N(A \cap W) - N(W \cap S) - N(A \cap S) + N(A \cap W \cap S) gives 360=170+150+160−60−55−50+N(A∩W∩S)360 = 170 + 150 + 160 - 60 - 55 - 50 + N(A \cap W \cap S), which simplifies to 360=315+N(A∩W∩S)360 = 315 + N(A \cap W \cap S).
3
Calculate the number of plants that met exactly two standards.
(60−45)+(55−45)+(50−45)=15+10+5=30(60 - 45) + (55 - 45) + (50 - 45) = 15 + 10 + 5 = 30
The 'both' values provided in the problem include the plants that met all three standards. We must subtract the 'all three' value (4545) from each of the three intersections and sum the results.

Anahtar Kavram

Logical Venn Diagrams and the Principle of Inclusion-Exclusion for three sets.
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Konular

Logical Venn Diagrams

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