6. Let A = {2,4,6,8}. Suppose B is a set with |B| = 5. The smallest possible value of |A ∪ B| is 2, and the largest possible value is 8. The smallest possible value of |A ∩ B| is 1, and the largest

Let's analyze the problem step by step. Sets A and B are given, where: - \( A = \{2, 4, 6, 8\} \) - \( |A| = 4 \) (since A has 4 elements) - \( |B| = 5 \) (since B has 5 elements) ### Part (a) **What are the smallest and largest possible values of \( |A \cup B| \)?** The union \( A \cup B \) represents all distinct elements in either A or B or both. **Smallest possible value of \( |A \cup B| \):** - The smallest value occurs when all elements of B are already in A. Since \( A \) has 4 elements, the smallest union, in this case, happens if there is an overlap of 4 elements and B contributes 1 new element. - Hence, \( |A \cup B| = 5 \) (when maximum overlap occurs). **Largest possible value of \( |A \cup B| \):** - The largest value happens when there is no overlap between A and B. Since A has 4 elements and B has 5 elements: - Hence, \( |A \cup B| = 4 + 5 = 9 \) (when there is no overlap). So, the smallest value is 5, and the largest value is 9. ### Part (b) **What are the smallest and largest possible values of \( |A \cap B| \)?** The intersection \( A \cap B \) represents all elements that are both in A and B. **Smallest possible value of \( |A \cap B| \):** - The smallest value occurs when there is no overlap between A and B. - Hence, \( |A \cap B| = 0 \). **Largest possible value of \( |A \cap B| \):** - The largest value is determined by the smaller of the two sets. Since \( |A| = 4 \): - Hence, \( |A \cap B| = 4 \) (when B includes all elements of A). So, the smallest value is 0, and the largest value is 4. ### Part (c) **What are the smallest and largest possible values of \( |A \times B| \)?** The Cartesian product \( A \times B \) represents all possible ordered pairs (a, b) where \( a \in A \) and \( b \in B \). **Smallest and largest possible values of \( |A \times B| \):** - The size of the Cartesian product depends directly on the sizes of A and B. - Regardless of the overlap in elements, each element in A pairs with each element in B. Thus, the number of ordered pairs remains consistent. - \( |A \times B| = |A| \times |B| = 4 \times 5 = 20 \). So, both the smallest and largest value of \( |A \times B| \) is 20.

(a) The smallest possible value of |A ∪ B| is 8, which is the same as the number of elements in A. The largest possible value of |A ∪ B| is 9, since B has 5 elements that could potentially be added to A. (b) The smallest possible value of |A ∩ B| is 0, since B contains elements that may not exist in A. The largest possible value of |A ∩ B| is 4, which is the same as the number of elements in A. (c) The smallest possible value of |A × B| is 0, since B could contain elements that are not related to any of the elements in A. The largest possible value of |A × B| is 40, since A contains 4 elements and B contains 5 elements, and each element in A could have a relationship with each element in B.