1. **Problem Statement:** Given sets \( \varepsilon = \{p, q, r, s, t, u, v\} \), \( A = \{p, q, r, s\} \), \( B = \{r, t, u, v\} \), and \( C = \{v, s, u, v\} \), find: a. \( n(A \cap C) \) b. List the elements of: i. \( (B \cup C)' \) ii. \( (A \cup C) \cap B \) 2. **Formula and Rules:** - Intersection \( A \cap C \) is the set of elements common to both \( A \) and \( C \). - Union \( B \cup C \) is the set of elements in either \( B \) or \( C \) or both. - Complement \( (B \cup C)' \) is the set of elements in \( \varepsilon \) not in \( B \cup C \). - Cardinality \( n(S) \) is the number of elements in set \( S \). 3. **Step-by-step Solution:** **a. Find \( n(A \cap C) \):** - \( A = \{p, q, r, s\} \) - \( C = \{v, s, u, v\} = \{v, s, u\} \) (duplicates removed) - Intersection \( A \cap C = \{s\} \) (only \( s \) is common) - Therefore, \( n(A \cap C) = 1 \) **b.i. Find \( (B \cup C)' \):** - \( B = \{r, t, u, v\} \) - \( C = \{v, s, u\} \) - Union \( B \cup C = \{r, t, u, v, s\} \) - Complement \( (B \cup C)' = \varepsilon - (B \cup C) = \{p, q\} \) **b.ii. Find \( (A \cup C) \cap B \):** - \( A \cup C = \{p, q, r, s, v, u\} \) - \( B = \{r, t, u, v\} \) - Intersection \( (A \cup C) \cap B = \{r, u, v\} \) 4. **Final answers:** - \( n(A \cap C) = 1 \) - \( (B \cup C)' = \{p, q\} \) - \( (A \cup C) \cap B = \{r, u, v\} \)