Use the following information. In a school of 310 students,...

Use the following information. In a school of 310 students, 80 participate in academic clubs, 115 participate in sports, and 20 students participate in both.
How many students do not participate in either academic clubs or sports?

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Key Concepts

Intersection of Sets

The intersection of sets is the set containing only the elements that are common to all the involved sets. Recognizing the intersection is important in situations where there may be overlapping membership, as it ensures that these elements are not overcounted in union calculations.

Complement of a Set

The complement of a set refers to all elements in the universal set that are not part of a specific subset. In problems involving counting, identifying the complement helps determine how many subjects do not have a particular attribute or are not participants in any of the groups being considered.

Inclusion-Exclusion Principle

This principle is a method used in combinatorics to correctly count the total number of elements in the union of two or more sets by compensating for elements that are present in more than one set. It involves adding the sizes of individual sets and then subtracting the sizes of all pairwise intersections, ensuring that no element is counted more than once.

Union of Sets

The union of sets refers to the set that contains all elements from the involved sets, with any duplicate elements included only once. In counting problems, calculating the union is essential when determining the total number of distinct elements that belong to at least one of several groups.

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