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Zorluk: KolayLogical Venn Diagrams

In a regional sports academy, a group of 7070 trainees were surveyed about their participation in two sports: Cricket and Football. 4545 trainees play Cricket, 3030 trainees play Football, and 1010 trainees play neither sport. How many trainees play both Cricket and Football?

  1. 1515Cevap
  2. B
    55
  3. C
    2525
  4. D
    1010

Cevap

15 trainees play both Cricket and Football.
The total number of trainees participating in at least one sport is found by subtracting the 1010 non-participants from the total 7070, giving 6060. Summing the individual sport counts gives 45+30=7545 + 30 = 75. The overlap represents those counted twice, which is 75−60=1575 - 60 = 15.

Adım Adım Çözüm

1
Calculate the total number of trainees who play at least one sport.
Trainees playing at least one sport = 70−10=6070 - 10 = 60.
Subtract trainees playing neither sport from the total group size.
2
Apply the Principle of Inclusion-Exclusion for two sets.
N(Cricket∪Football)=N(Cricket)+N(Football)−N(Cricket∩Football)N(\text{Cricket} \cup \text{Football}) = N(\text{Cricket}) + N(\text{Football}) - N(\text{Cricket} \cap \text{Football}).
The sum of individual sets double-counts the intersection.
3
Substitute the known values into the equation and solve for the intersection.
60=45+30−N(Cricket∩Football)  ⟹  N(Cricket∩Football)=75−60=1560 = 45 + 30 - N(\text{Cricket} \cap \text{Football}) \implies N(\text{Cricket} \cap \text{Football}) = 75 - 60 = 15.
Algebraic simplification yields the number of trainees playing both sports.

Anahtar Kavram

Two-Set Inclusion-Exclusion Principle
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Konular

Logical Venn Diagrams

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