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Zorluk: KolayOverlapping Sets and Venn Diagrams

In a corporate workshop of 8080 executives, 5252 executives attended the Leadership module and 4040 executives attended the Negotiation module. If 1818 executives attended neither module, how many executives attended both modules?

Cevap: 30 executives

Cevap

30 executives attended both modules.
Applying the formula Total=Group A+Group B−Both+Neither\text{Total} = \text{Group A} + \text{Group B} - \text{Both} + \text{Neither} yields 80=52+40−Both+1880 = 52 + 40 - \text{Both} + 18. Simplifying gives 80=110−Both80 = 110 - \text{Both}, so the number of executives who attended both modules is 3030.

Adım Adım Çözüm

1
Determine the number of executives who attended at least one of the two modules.
Executives attending at least one module = 80−18=6280 - 18 = 62.
Subtracting those who attended neither module from the total gives the union of the two sets.
2
Apply the inclusion-exclusion principle to find the intersection of the two sets.
Both=Leadership+Negotiation−At least one=52+40−62=30\text{Both} = \text{Leadership} + \text{Negotiation} - \text{At least one} = 52 + 40 - 62 = 30.
Summing the participants of each module double-counts those who attended both, so subtracting the number attending at least one module isolates the overlap.

Anahtar Kavram

Two-Set Overlapping Sets Formula (Inclusion-Exclusion Principle)
Tahmini Süre:1m 0s
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Konular

Overlapping Sets and Venn Diagrams

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