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Zorluk: OrtaVenn Diagrams and Set-Based Data

In a rural development assessment of 200200 villages in a district, data regarding access to three basic infrastructure facilities—Piped Water, Solar Power, and Broadband Internet—was recorded as follows:
- 110110 villages have access to Piped Water.
- 100100 villages have access to Solar Power.
- 9090 villages have access to Broadband Internet.
- 5050 villages have access to both Piped Water and Solar Power.
- 4040 villages have access to both Solar Power and Broadband Internet.
- 4545 villages have access to both Piped Water and Broadband Internet.
- 2020 villages have access to all three facilities.

Which of the following statements are correct?

  1. Exactly 7575 villages have access to exactly two infrastructure facilities.Cevap
  2. The number of villages having access to at least two infrastructure facilities is 9595.Cevap
  3. C
    The number of villages having access to none of the three facilities is 2525.
  4. Exactly 3535 villages have access to Piped Water only.Cevap

Cevap

The correct statements are those indicating that 7575 villages have access to exactly two facilities, 9595 villages have access to at least two facilities, and 3535 villages have access to Piped Water only.
Evaluating each region using set theory shows: exactly two facilities = 30+20+25=7530 + 20 + 25 = 75; at least two facilities = 75+20=9575 + 20 = 95; Piped Water only = 110−(30+25+20)=35110 - (30 + 25 + 20) = 35. All three statements accurately reflect the set cardinality.

Adım Adım Çözüm

1
Determine the exclusive two-set overlap regions.
Piped Water & Solar Power only = 50−20=3050 - 20 = 30; Solar Power & Broadband Internet only = 40−20=2040 - 20 = 20; Piped Water & Broadband Internet only = 45−20=2545 - 20 = 25.
The given pairwise intersections include the 2020 villages that have access to all three facilities.
2
Determine the single-set only regions.
Piped Water only = 110−(30+25+20)=35110 - (30 + 25 + 20) = 35; Solar Power only = 100−(30+20+20)=30100 - (30 + 20 + 20) = 30; Broadband Internet only = 90−(25+20+20)=2590 - (25 + 20 + 20) = 25.
Subtract all overlapping regions containing that facility from its total.
3
Calculate total villages with at least one facility using the Principle of Inclusion-Exclusion.
Total with at least one facility = 35+30+25+30+20+25+20=18535 + 30 + 25 + 30 + 20 + 25 + 20 = 185.
Summing all 7 disjoint regions yields the union of the three sets.
4
Find villages with none of the facilities.
Villages with none = 200−185=15200 - 185 = 15.
Subtract total union from the universe of 200200 villages.

Anahtar Kavram

Three-Set Principle of Inclusion-Exclusion and Venn Diagram Region Cardinality
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Konular

Venn Diagrams and Set-Based Data

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