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Zorluk: ZorVenn Diagrams and Set-Based Data

A district administration conducted a specialized training audit for a cadre of 300300 officers across three administrative domains: E-Governance (EE), Disaster Management (DD), and Financial Administration (FF). The data gathered from the audit is as follows:

- 140140 officers are trained in E-Governance.
- 130130 officers are trained in Disaster Management.
- 120120 officers are trained in Financial Administration.
- 5050 officers are trained in both E-Governance and Disaster Management.
- 4040 officers are trained in both Disaster Management and Financial Administration.
- 4545 officers are trained in both E-Governance and Financial Administration.
- 2020 officers are trained in all three domains.

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

  1. The number of officers trained in exactly two subjects is 7575.Cevap
  2. The number of officers trained in at most one subject is 205205.Cevap
  3. C
    The number of officers trained in E-Governance only is 7575.
  4. D
    The number of officers trained in both Disaster Management and Financial Administration but not E-Governance is 4040.

Cevap

The correct statements are that the number of officers trained in exactly two subjects is 75, and the number of officers trained in at most one subject is 205.
The statement declaring that 75 officers are trained in exactly two subjects is accurate because summing the disjoint exclusive intersections (50−20)+(40−20)+(45−20)(50-20) + (40-20) + (45-20) yields 30+20+25=7530 + 20 + 25 = 75. Furthermore, the statement claiming that 205 officers are trained in at most one subject is correct because adding the single-domain officers (65+60+55=18065 + 60 + 55 = 180) to those trained in none of the three domains (300−275=25300 - 275 = 25) gives exactly 205205.

Adım Adım Çözüm

1
Calculate the region cardinalities for exactly two domains.
E-Governance and Disaster Management only = 50−20=3050 - 20 = 30; Disaster Management and Financial Administration only = 40−20=2040 - 20 = 20; E-Governance and Financial Administration only = 45−20=2545 - 20 = 25. Total in exactly two domains = 30+20+25=7530 + 20 + 25 = 75.
The intersection of any two sets includes elements present in all three sets; subtracting the three-set intersection yields the count for 'only two'.
2
Calculate single-domain region cardinalities.
Only E-Governance = 140−(30+25+20)=65140 - (30 + 25 + 20) = 65; Only Disaster Management = 130−(30+20+20)=60130 - (30 + 20 + 20) = 60; Only Financial Administration = 120−(25+20+20)=55120 - (25 + 20 + 20) = 55.
Each total set count contains its single-domain region plus three overlapping sub-regions.
3
Calculate total officers trained in at least one domain and those trained in none.
Total trained in at least one domain = 65+60+55+30+20+25+20=27565 + 60 + 55 + 30 + 20 + 25 + 20 = 275. Officers trained in none = 300−275=25300 - 275 = 25.
Summing all 7 disjoint set regions gives the union cardinality; subtracting from total population yields the complement.
4
Evaluate the statement regarding 'at most one subject'.
Officers in at most one domain = (Only E + Only D + Only F) + None = 65+60+55+25=20565 + 60 + 55 + 25 = 205.
'At most one' includes both the zero-domain category and single-domain categories.

Anahtar Kavram

Three-Set Principle of Inclusion-Exclusion and Region Partitioning
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Konular

Venn Diagrams and Set-Based Data

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