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Zorluk: KolaySet Theory Concepts and Venn Diagrams

At a technology conference attended by 8080 software engineers, 5050 engineers write code in Python, 4040 write code in Java, and 1515 write code in neither Python nor Java. How many of the software engineers write code in both Python and Java?

  1. A
    10
  2. B
    15
  3. 25Cevap
  4. D
    30
  5. E
    35

Cevap

The number of software engineers who write code in both Python and Java is 25.
Out of 8080 engineers, 1515 write neither language, which means 80−15=6580 - 15 = 65 engineers write Python, Java, or both. Using the inclusion-exclusion principle, ∣P∪J∣=∣P∣+∣J∣−∣P∩J∣|P \cup J| = |P| + |J| - |P \cap J|, substituting the values gives 65=50+40−∣P∩J∣65 = 50 + 40 - |P \cap J|. Solving for the overlap gives ∣P∩J∣=90−65=25|P \cap J| = 90 - 65 = 25. Thus, 25 engineers write code in both languages.

Adım Adım Çözüm

1
Find the number of engineers who write code in at least one of the two languages (Python or Java).
∣P∪J∣=80−15=65|P \cup J| = 80 - 15 = 65
Subtracting the engineers who write neither language from the total gives the union of the two sets.
2
Apply the Principle of Inclusion-Exclusion formula for two sets.
∣P∪J∣=∣P∣+∣J∣−∣P∩J∣|P \cup J| = |P| + |J| - |P \cap J|
The total number of engineers in the union is equal to the sum of the individual sets minus their intersection.
3
Substitute the known values into the equation to solve for the intersection ∣P∩J∣|P \cap J|.
65=50+40−∣P∩J∣  ⟹  65=90−∣P∩J∣  ⟹  ∣P∩J∣=2565 = 50 + 40 - |P \cap J| \implies 65 = 90 - |P \cap J| \implies |P \cap J| = 25
Solving the linear equation yields the number of engineers writing both languages.

Anahtar Kavram

Principle of Inclusion-Exclusion for Two Sets
Tahmini Süre:1m 0s
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Set Theory Concepts and Venn Diagrams

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