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Zorluk: Çok zorVenn Diagrams and Set-Based Data

In a state administrative academy, a batch of 300300 probationary officers were surveyed regarding their enrolment in three specialized training modules: Cyber Security (CC), Public Policy (PP), and Financial Management (FF). The survey revealed the following data:
- Total officers enrolled in Cyber Security: 160160
- Total officers enrolled in Public Policy: 140140
- Total officers enrolled in Financial Management: 130130
- Officers enrolled in both Cyber Security and Public Policy: 6565
- Officers enrolled in both Public Policy and Financial Management: 5555
- Officers enrolled in both Cyber Security and Financial Management: 5050
- Officers enrolled in all three modules: 2020

Based on the data provided above, which of the following statements are correct?

  1. The total number of officers enrolled in exactly one training module is 150150.Cevap
  2. B
    The number of officers enrolled in at least two training modules is 110110.
  3. The ratio of officers enrolled in Cyber Security only to those enrolled in Financial Management only is 13:913 : 9.Cevap
  4. D
    The number of officers who did not enroll in any of the three modules is 3030.

Cevap

The correct statements are that the total number of officers enrolled in exactly one training module is 150150, and the ratio of officers enrolled in Cyber Security only to those enrolled in Financial Management only is 13:913 : 9.
The statements confirming that 150150 officers are enrolled in exactly one module and that the ratio of Cyber Security only to Financial Management only is 13:913 : 9 are both mathematically true based on region decomposition.

Adım Adım Çözüm

1
Identify the 3-set intersection region
The number of officers in all three modules (C∩P∩FC \cap P \cap F) is given as 2020.
The central intersection is the foundation for calculating all non-overlapping regions in a 3-set Venn diagram.
2
Calculate the regions corresponding to exactly two modules
C∩P only=65−20=45C \cap P \text{ only} = 65 - 20 = 45; P∩F only=55−20=35P \cap F \text{ only} = 55 - 20 = 35; C∩F only=50−20=30C \cap F \text{ only} = 50 - 20 = 30. Total in exactly two modules = 45+35+30=11045 + 35 + 30 = 110.
Subtracting the 3-set intersection from each 2-set intersection isolates the elements belonging exclusively to two sets.
3
Calculate the regions corresponding to exactly one module
C only=160−(45+30+20)=65C \text{ only} = 160 - (45 + 30 + 20) = 65; P only=140−(45+35+20)=40P \text{ only} = 140 - (45 + 35 + 20) = 40; F only=130−(30+35+20)=45F \text{ only} = 130 - (30 + 35 + 20) = 45. Total in exactly one module = 65+40+45=15065 + 40 + 45 = 150.
Subtracting all double-counted and triple-counted intersections from total set counts yields single-set cardinalities.
4
Evaluate the complement (neither set) and statement conditions
Total in at least one module = 150+110+20=280150 + 110 + 20 = 280. Neither = 300−280=20300 - 280 = 20. At least two modules = 110+20=130110 + 20 = 130. Ratio C only:F only=65:45=13:9C \text{ only} : F \text{ only} = 65 : 45 = 13 : 9.
Verifying each statement against calculated set region cardinalities confirms which statements are true.

Anahtar Kavram

Principle of Inclusion-Exclusion for Three Sets
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Konular

Venn Diagrams and Set-Based Data

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