Let \( S \) be the universal set, where: \( S=\{1,2,3, \ldots, 18,19,20\} \) Let sets \( A \) and \( B \) be subsets of \( S \), where: Set \( A=\{1,3,4,8,10,12,14\} \) Set \( B=\{2,3,4,6,7,9,10,12,13,14,15,17,18\} \) Find the following: The number of elements in the set \( (A \cup B) \) : \( n(A \cup B)=\square \) The number of giements in the set \( (A \cap B) \) : \( n(A \cap B) \) is

To find the number of elements in the set \( A \cup B \) (the union of sets \( A \) and \( B \)), we first identify all unique elements from both sets. Set \( A = \{1, 3, 4, 8, 10, 12, 14\} \) Set \( B = \{2, 3, 4, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18\} \) The union \( A \cup B \) includes all elements from both sets, without duplicates: - From \( A \): 1, 3, 4, 8, 10, 12, 14 - From \( B \): 2, 3, 4, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18 Combining these, we find: \( A \cup B = \{1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18\} \) Counting the unique elements, we have 15 different elements in total. So, \[ n(A \cup B) = 15 \] Next, to find the number of elements in the set \( A \cap B \) (the intersection of sets \( A \) and \( B \)), we list the common elements in both sets. The common elements are: - \( 3 \) (in both) - \( 4 \) (in both) - \( 10 \) (in both) - \( 12 \) (in both) - \( 14 \) (in both) Thus, \[ A \cap B = \{3, 4, 10, 12, 14\} \] Counting these, we find 5 elements. So, \[ n(A \cap B) = 5 \] In summary, we have: \[ n(A \cup B) = 15 \] and \[ n(A \cap B) = 5 \]