Forty-two cities were researched to determine whether they had a protessional sports team, a symphony, or a childrer's museum, of these cties, 21 had a professional sports team, 24 had a symphory, 16 had a children's museum, 15 had a professional sports leam and a symphony, 8 had a professional sports team and a childrer's muceum, 11 had a symphony and a childrer's muceum, and 6 had at tries adivities. Complete parts a) through e) below. a) How many of the cities surveyed had only a professional sports team? 4 (Simpility your ancwer) b) How many of the cities surveyed had a protessional sports team and a symphony, but not a children's museum? 9 (Simplity your answer) c) How many of the cities surveyed had a professional sports team or a symphony? 30 (Simplify your answer) d) How many of the cities surveyed had a professional sports team or a symphony, but not a childrer's museum? \( 17^{7} \) (Simplify your answer.) e) How many of the cries surveyed had exactly two of the activities? \( 16^{\circ} \) (Simplify your answer.)
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Bonus Knowledge
To tackle these questions effectively, we can rely on the principle of inclusion-exclusion, which is like a superhero of set theory! It helps us systematically count different group intersections without double-counting elements. To simplify problems involving sets, it's crucial to clearly define each variable. For instance, let A be the set of cities with a professional sports team, B the cities with a symphony, and C those with a children's museum. Creating a visual tool like a Venn diagram can also provide a quick and engaging reference, allowing you to visualize the overlaps while making calculations more manageable and fun!
