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Zorluk: OrtaLogical Venn Diagrams

A cybersecurity agency audited 250250 government web portals and found vulnerabilities across three categories: Data Breach (DD), Injection Attacks (II), and Authentication Flaws (AA). The audit report stated:
- 110110 portals were vulnerable to Data Breach
- 130130 portals were vulnerable to Injection Attacks
- 100100 portals were vulnerable to Authentication Flaws
- 4545 portals had both DD and II vulnerabilities
- 5555 portals had both II and AA vulnerabilities
- 4040 portals had both DD and AA vulnerabilities
- 2020 portals had all three types of vulnerabilities

Based on this data, how many portals had exactly one type of vulnerability?

  1. 120120Cevap
  2. B
    6060
  3. C
    220220
  4. D
    8080

Cevap

There are 120120 portals with exactly one type of vulnerability.
To find the number of portals with exactly one vulnerability, we isolate the non-overlapping portion of each set. For set DD, this is 110−45−40+20=45110 - 45 - 40 + 20 = 45. For set II, it is 130−45−55+20=50130 - 45 - 55 + 20 = 50. For set AA, it is 100−40−55+20=25100 - 40 - 55 + 20 = 25. Adding these exclusive regions gives 45+50+25=12045 + 50 + 25 = 120.

Adım Adım Çözüm

1
Calculate the number of portals vulnerable ONLY to Data Breach (DD).
∣Donly∣=∣D∣−∣D∩I∣−∣D∩A∣+∣D∩I∩A∣=110−45−40+20=45|D_{only}| = |D| - |D \cap I| - |D \cap A| + |D \cap I \cap A| = 110 - 45 - 40 + 20 = 45
Subtracting the pairwise intersections removes the portals with multiple vulnerabilities, but it subtracts the triple intersection twice, so it must be added back once.
2
Calculate the number of portals vulnerable ONLY to Injection Attacks (II).
∣Ionly∣=∣I∣−∣D∩I∣−∣I∩A∣+∣D∩I∩A∣=130−45−55+20=50|I_{only}| = |I| - |D \cap I| - |I \cap A| + |D \cap I \cap A| = 130 - 45 - 55 + 20 = 50
Applying the same region isolation principle to set II.
3
Calculate the number of portals vulnerable ONLY to Authentication Flaws (AA).
∣Aonly∣=∣A∣−∣D∩A∣−∣I∩A∣+∣D∩I∩A∣=100−40−55+20=25|A_{only}| = |A| - |D \cap A| - |I \cap A| + |D \cap I \cap A| = 100 - 40 - 55 + 20 = 25
Applying the same region isolation principle to set AA.
4
Sum the three isolated regions to find the total number of portals with exactly one vulnerability.
45+50+25=12045 + 50 + 25 = 120
These regions represent portals that have one and only one type of vulnerability.

Anahtar Kavram

Applying set theory and the inclusion-exclusion principle to isolate specific regions within a three-set Venn diagram.
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Konular

Logical Venn Diagrams

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