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Zorluk: OrtaSet Theory Concepts and Venn Diagrams

An aerospace engineering team evaluated telemetry logs from a constellation of 150150 satellites to monitor three types of sensor anomalies: Power Fluctuation (PP), Thermal Spike (TT), and Signal Attenuation (SS). The evaluation revealed the following:

- 6565 satellites exhibited Power Fluctuation.
- 5858 satellites exhibited Thermal Spike.
- 4242 satellites exhibited Signal Attenuation.
- 2222 satellites exhibited both Power Fluctuation and Thermal Spike.
- 1818 satellites exhibited both Thermal Spike and Signal Attenuation.
- 1515 satellites exhibited both Power Fluctuation and Signal Attenuation.
- 88 satellites exhibited all three anomalies.

How many satellites exhibited exactly one of these three anomalies?

  1. A
    55
  2. 79Cevap
  3. C
    110
  4. D
    118
  5. E
    125

Cevap

79 satellites exhibited exactly one anomaly.
To find the number of satellites exhibiting exactly one anomaly, break down each set into its mutually exclusive regions. First, find the satellites in exactly two sets by subtracting the triple intersection (8) from each pairwise intersection: Power and Thermal only = 22 - 8 = 14; Thermal and Signal only = 18 - 8 = 10; Power and Signal only = 15 - 8 = 7. Next, subtract all overlapping regions from each individual set total: Power only = 65 - (14 + 7 + 8) = 36; Thermal only = 58 - (14 + 10 + 8) = 26; Signal only = 42 - (7 + 10 + 8) = 17. Adding these single-anomaly counts together gives 36 + 26 + 17 = 79.

Adım Adım Çözüm

1
Determine the number of satellites exhibiting ONLY pairwise anomalies (exactly two anomalies).
P and T only = 22 - 8 = 14; T and S only = 18 - 8 = 10; P and S only = 15 - 8 = 7.
The given pairwise intersection values include the 8 satellites that exhibited all three anomalies.
2
Calculate the number of satellites exhibiting exactly one anomaly for each set.
P only = 65 - (14 + 7 + 8) = 36; T only = 58 - (14 + 10 + 8) = 26; S only = 42 - (7 + 10 + 8) = 17.
Subtracting all elements that belong to two or three sets from each total set size leaves only those in the single set.
3
Sum the single-anomaly totals.
36 + 26 + 17 = 79.
The regions corresponding to exactly one anomaly are mutually exclusive.

Anahtar Kavram

Principle of Inclusion-Exclusion and Venn Diagram Region Decomposition
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Konular

Set Theory Concepts and Venn Diagrams

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