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Difficulty: MediumSets, Set Operations, and Venn Diagrams

In a sports academy of 8585 athletes, 5252 participate in track events, 4343 participate in field events, and 1212 participate in neither track nor field events. How many athletes participate in both track and field events?

Answer: 22 athletes

Answer

The number of athletes participating in both track and field events is 22.
To find the number of athletes in both events, first calculate the total number of athletes who take part in at least one event by subtracting the 12 non-participants from 85, giving 73. Adding the 52 track athletes and 43 field athletes yields 95, which double-counts those who participate in both. The difference between 95 and 73 is 22, representing the athletes in the intersection.

Step-by-Step Solution

1
Determine the cardinality of the union of track and field athletes
N(T∪F)=85−12=73N(T \cup F) = 85 - 12 = 73
Athletes participating in neither event are outside the union of track and field sets.
2
Formulate the two-set inclusion-exclusion equation
N(T∪F)=N(T)+N(F)−N(T∩F)N(T \cup F) = N(T) + N(F) - N(T \cap F)
Adding individual set cardinalities double-counts the intersection.
3
Substitute values and solve for the intersection
N(T∩F)=52+43−73=22N(T \cap F) = 52 + 43 - 73 = 22
Rearranging the equation yields the number of athletes in both sets.

Key Concept

Cardinality of Sets and Principle of Inclusion-Exclusion
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Topics

Sets, Set Operations, and Venn Diagrams

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