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

(a) The smallest possible value of |A ∪ B| is 5 and the largest possible value is 9. The smallest possible value occurs when all elements of B are also in A. In this case, all of B's five elements would already be accounted for in A's four elements, making the union of the two sets still four elements. The largest possible value happens when no elements of B are the same as in A. That means you would count all four of A's elements and all five of B's elements, making a total of nine elements in the union. (b) The smallest possible value of |A ∩ B| is 0 and the largest possible value is 4. The smallest possible value occurs when there are no common elements in A and B, hence the intersection of the two sets would be an empty set, making it 0. The largest possible value occurs when all elements of A are also in B. In this case, the intersection of the two sets would contain all four elements from A. (c) The smallest possible value of |A × B| is 20 and the largest possible value is also 20. The reason why it doesn't change is because the Cartesian product (×) gives a set of all possible ordered pairs, where the first element is from A and second element is from B. Since A has 4 elements and B has 5, the total will always be 4*5 = 20, regardless of the individual elements in each set.

a) The smallest possible value of |A ∪ B| is 7, because no matter what elements are contained in B, it will always add at least one new element to A. The largest possible value of |A ∪ B| is 9, since at most, four elements of B can be used to supplement A without any being repeated. b) The smallest possible value of |A ∩ B| is 0, because no matter what elements are contained in B, there is always the possibility that none of them are elements of A. The largest possible value of |A ∩ B| is 4, since at most, all four elements of A can be contained in B without any repetition occuring. c) The smallest possible value of |A × B| is 0, because no matter what elements are contained in B, there is always the possibility that none of them are related to any elements of A. The largest possible value of |A × B| is 20, since at most, all four elements of A can be related to each of the five elements in B without any repetition occuring.