Question

Let x = 1, 2, ...,9. 1. Find the number of subsets a of x. 2. Find the number of pairs (a, b) of subsets a and b of x such that a ⊆ b. 3. Find the number of pairs (a, b) of subsets a and b of x such that a ∩ b =∅. 4. Find the number of pairs (a, b) of subsets a and b of x such that a ∪ b = x. 5. Find the number of pairs (a, b) of subsets a and b of x such that a ∩ b =∅. 6. Find the number of pairs (a, b) of subsets a and b of x such that a ∪ b = x and a ∩ b =∅.

Answer 1

1. The number of subsets of x = 9 is 512. 2. The number of pairs (a, b) of subsets a and b of x such that a ⊆ b is also 512. 3. The number of pairs (a, b) of subsets a and b of x such that a ∩ b = ∅ is 19683. 4. The number of pairs (a, b) of subsets a and b of x such that a ∪ b = x is 19683. 5. The number of pairs (a, b) of subsets a and b of x such that a ∩ b = ∅ and a ∪ b = x is 262144.

Step-by-step explanation:

1. To find the number of subsets of x, we can use the formula 2^x, where x is the number of elements in the set. In this case, the number of subsets of x = 9 is 2^9 = 512.
2. To find the number of pairs (a, b) of subsets a and b of x such that a ⊆ b, we need to consider that each element in x can either be included in a subset or not. So, for each element, there are 2 possibilities. Since there are 9 elements, the total number of pairs (a, b) is 2^9 = 512.
3. To find the number of pairs (a, b) of subsets a and b of x such that a ∩ b = ∅, we need to consider that no elements are common between the subsets a and b. For each element, there are 3 possibilities: it can be included in subset a only, in subset b only, or not included in either subset. So, the total number of pairs (a, b) is 3^9 = 19683.
4. To find the number of pairs (a, b) of subsets a and b of x such that a ∪ b = x, we need to consider that the union of subsets a and b should include all elements of x, which means that all elements must be included in at least one of the subsets. For each element, there are 3 possibilities: it can be included in subset a only, in subset b only, or in both subsets. So, the total number of pairs (a, b) is 3^9 = 19683.
5. To find the number of pairs (a, b) of subsets a and b of x such that a ∩ b = ∅ and a ∪ b = x, we need to consider that the subsets a and b should have no common elements and should together include all elements of x. For each element, there are 4 possibilities: it can be included in subset a only, in subset b only, in both subsets, or in neither subset. So, the total number of pairs (a, b) is 4^9 = 262144.