2.4 Surveys and Cardinal Numbers Example: A survey of 132 people asking what sports they regularly watch found: 88 like football 70 like baseball 32 like hockey 20 like all three 20 like football and baseball only 25 like baseball and hockey 3 like football & hockey, but not baseball (a) How many people like only football? (b) How many people don’t like any of the sports? 132 people. 88 like F 20 like F and B only 70 like B 25 like B and H 32 like H 3 like F and H but not B 20 like all three F B H U F B 20 45 25 20 3 5 4 H 10 (a) How many people like only football? (b) How many people don’t like any of the sports? Example. Let U = {p, q, r, s, t, u}, A = {p, q, r, s}, and B = {s, t, u}. Find: (a) đ(đ´) (b) đ(đĩ) (c) đ(đ´ ∩ đĩ) (d) đ(đ´ ∪ đĩ) Cardinal Number Formula Definition: For any sets A and B, the cardinality of the union of A with B can be found using nī¨ A B īŠ īŊ n( A) īĢ n( B ) ī n( A B ). Example. (a) Find đ(đ´) if đ(đ´ ∪ đĩ) = 50, đ(đ´ ∩ đĩ) = 8, and đ(đĩ) = 38. Recall: đ(đ´ ∪ đĩ) = 50, đ(đ´ ∩ đĩ) = 8, and đ(đĩ) = 38. (b) Draw a Venn diagram to illustrate sets A and B and fill in the cardinalities for each region. Example. Preferences and ages of breakfast patrons are summarized in the following table: Find the number of people in the following sets. (a) đ ∪ đļ (b) đ ∩ đ′ (a) đ(đ ∪ đļ) (b) đ(đ ∪ đ′) đ(đ ∪ đ′) = 55 + 77 + 24 = 156 people. Or, 15 + 22 + 18 + 30 + 25 + 22 + 24 = 156 people.