Cardinality Math Example 4

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Example 4

medium
In a class of 40 students, 25 play football, 20 play basketball, and 10 play both. Use cardinality formulas to find how many play at least one sport and how many play neither.

Solution

  1. 1
    Let FF = football players, BB = basketball players. Given ∣F∣=25|F|=25, ∣B∣=20|B|=20, ∣F∩B∣=10|F \cap B|=10.
  2. 2
    By inclusion-exclusion: ∣F∪B∣=25+20−10=35|F \cup B| = 25 + 20 - 10 = 35.
  3. 3
    Students playing neither: 40−35=540 - 35 = 5.

Answer

35 play at least one,5 play neither35 \text{ play at least one},\quad 5 \text{ play neither}
The inclusion-exclusion principle prevents double-counting students who play both sports. Subtracting from the total class size gives those who play neither.

About Cardinality

The cardinality of a finite set is the number of distinct elements it contains, written ∣A∣|A| — it measures the size of the set without regard to element order or identity.

Learn more about Cardinality →

More Cardinality Examples

Example 1 easy

Find the cardinality of: (a) [formula], (b) [formula], (c) [formula].

Example 2 medium

Let [formula] and [formula]. Verify the formula [formula].

Example 3 easy

A set [formula] has 3 elements. How many subsets does [formula] have? How many proper subsets?

← All Cardinality Examples Cardinality Concept

Related Concepts

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