\( 0 / 2 \) pts \( 3 \rightleftarrows 99 \) (i) Details Let the Universal Set be S. Let A and B are subsets of S. Set A contains 45 elements and Set B contains 23 elements. Sets A and \( B \) have 21 elements in common. If there are 23 elements that are in S but not in A nor B, how many elements are in S? Answer \( = \) elements

To find the number of elements in the universal set \( S \), we can use the principle of inclusion-exclusion. The principle of inclusion-exclusion states that the number of elements in the union of two sets is equal to the sum of the number of elements in each set, minus the number of elements in their intersection. In this case, we have: \(|A \cup B| = |A| + |B| - |A \cap B|\) Substituting the given values, we have: \(|A \cup B| = 45 + 23 - 21\) \(|A \cup B| = 47\) Now, we need to add the number of elements that are in \( S \) but not in \( A \) or \( B \). We are given that there are 23 such elements. \(|S| = |A \cup B| + |S \setminus (A \cup B)|\) \(|S| = 47 + 23\) \(|S| = 70\) Therefore, there are 70 elements in the universal set \( S \).