gen-define-the-inclusion-exclusion-principle-and-give-one-short-cbc6

Define the inclusion-exclusion principle and give one short worked example.

# Inclusion-Exclusion Principle ## Definition The **inclusion-exclusion principle** is a counting technique that calculates the size of the union of multiple sets by alternately adding and subtracting their intersections. For two sets A and B: $$|A \cup B| = |A| + |B| - |A \cap B|$$ For three sets A, B, and C: $$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$$ ## Worked Example **Problem:** In a class of 30 students, 18 play soccer, 12 play basketball, and 5 play both sports. How many students play at least one sport? **Solution:** Let S = students who play soccer, B = students who play basketball Given: - |S| = 18 - |B| = 12 - |S ∩ B| = 5 Using inclusion-exclusion: $$|S \cup B| = |S| + |B| - |S \cap B|$$ $$|S \cup B| = 18 + 12 - 5 = 25$$ **Answer:** 25 students play at least one sport. (Note: This means 30 - 25 = 5 students play neither sport.)