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Zorluk: Çok zorSet Theory Concepts and Venn Diagrams

A market research firm surveyed 150150 venture capital investors regarding their investments in three emerging sectors: Artificial Intelligence (AA), Biotechnology (BB), and Clean Energy (CC). The survey revealed the following data:

- 1515 investors did not invest in any of the three sectors.
- 7878 investors invested in Artificial Intelligence.
- 6565 investors invested in Biotechnology.
- 6262 investors invested in Clean Energy.
- 1010 investors invested in all three sectors.

If 5050 investors invested in exactly two of the three sectors, how many investors invested in exactly one of the three sectors?

  1. A
    65
  2. B
    70
  3. 75Cevap
  4. D
    85
  5. E
    90

Cevap

75 investors invested in exactly one of the three sectors.
The total number of investors who invested in at least one of the three sectors is 150−15=135150 - 15 = 135. In a 3-set Venn diagram, the total union is the sum of those who belong to exactly one sector (E1E_1), exactly two sectors (E2E_2), and all three sectors (E3E_3). We are given E2=50E_2 = 50 and E3=10E_3 = 10. Setting up the region equation E1+E2+E3=135E_1 + E_2 + E_3 = 135 gives E1+50+10=135E_1 + 50 + 10 = 135, which yields E1=75E_1 = 75.

Adım Adım Çözüm

1
Calculate the total number of investors who invested in at least one sector (the union of sets A, B, and C).
∣A∪B∪C∣=150−15=135|A \cup B \cup C| = 150 - 15 = 135
Subtracting the investors who invested in none of the sectors from the total population yields the size of the union.
2
Express the union as the sum of non-overlapping region counts.
∣A∪B∪C∣=E1+E2+E3|A \cup B \cup C| = E_1 + E_2 + E_3, where E1E_1 is the number of investors in exactly 1 sector, E2E_2 is the number in exactly 2 sectors, and E3E_3 is the number in all 3 sectors.
Any element in the union belongs to exactly one, exactly two, or all three sets.
3
Substitute the known values into the partition equation and solve for E1E_1.
135=E1+50+10  ⟹  E1=135−60=75135 = E_1 + 50 + 10 \implies E_1 = 135 - 60 = 75
We are given E2=50E_2 = 50 and E3=10E_3 = 10, leaving E1E_1 as the remaining unknown value.

Anahtar Kavram

Three-Set Inclusion-Exclusion and Region Partitioning
Tahmini Süre:2m 0s
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Konular

Set Theory Concepts and Venn Diagrams

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