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Zorluk: KolayLogical Venn Diagrams

In a municipal office survey of 100100 administrative officers, 5050 officers speak English, 4040 officers speak Hindi, and 2020 officers speak both English and Hindi. Based on this information, which of the following statements are correct?

  1. The number of officers who speak only English is 3030.Cevap
  2. The number of officers who speak neither English nor Hindi is 3030.Cevap
  3. C
    The number of officers who speak only Hindi is 4040.
  4. D
    The total number of officers who speak at least one of the two languages is 9090.

Cevap

The correct statements are that the number of officers who speak only English is 3030, and the number of officers who speak neither English nor Hindi is 3030.
The number of officers speaking only English is obtained by subtracting the shared region (2020) from the total English speakers (5050), giving 3030. The officers speaking at least one language equals 30 (only English)+20 (both)+20 (only Hindi)=7030 \text{ (only English)} + 20 \text{ (both)} + 20 \text{ (only Hindi)} = 70. Subtracting 7070 from the total 100100 officers gives 3030 officers who speak neither language.

Adım Adım Çözüm

1
Calculate the number of officers who speak only English.
Only English=50−20=30\text{Only English} = 50 - 20 = 30
Subtract the intersection (both languages) from the total English speakers.
2
Calculate the number of officers who speak only Hindi.
Only Hindi=40−20=20\text{Only Hindi} = 40 - 20 = 20
Subtract the intersection (both languages) from the total Hindi speakers.
3
Calculate the total number of officers speaking at least one language (the union of sets).
At least one language=30+20+20=70\text{At least one language} = 30 + 20 + 20 = 70
Sum the exclusive counts and the shared intersection count.
4
Calculate the number of officers speaking neither language.
Neither language=100−70=30\text{Neither language} = 100 - 70 = 30
Subtract the number of officers speaking at least one language from the total group size.

Anahtar Kavram

Inclusion-Exclusion Principle for Two Sets
Tahmini Süre:1m 0s
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Konular

Logical Venn Diagrams

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