The number of subsets of a set with n elements is 2^n. Here, n = 2, so 2^2 = 4.

The subsets containing 'a' are {a}, {a, b}, so there are 2 subsets.

If 1 is included, we can choose from the remaining 4 elements (2, 3, 4, 5). The number of subsets is 2^4 = 16.

The total number of subsets is 2^4 = 16. Proper subsets exclude the set itself, so there are 16 - 1 = 15 proper subsets.

The power set of a set with n elements has 2^n elements. Here, n = 2, so 2^2 = 4.

The subsets containing 1 are {1}, {1, 2}, and the empty set is not included.

The total number of subsets is 2^4 = 16. Proper subsets exclude the set itself, so 16 - 1 = 15.

If 'a' is included, we can choose from the remaining elements {b, c}. The subsets containing 'a' are {a}, {a, b}, {a, c}, and {a, b, c}, totaling 4 subsets.

The union of two sets includes all unique elements from both sets. Here, the union is {a, b, c, d}.

The power set of a set with n elements has 2^n subsets. For E, n = 2, so the size of the power set is 2^2 = 4.

The prime numbers less than 10 are 2, 3, 5, and 7. Therefore, E = {2, 3, 5, 7}.

The prime numbers less than 10 are {2, 3, 5, 7}. The power set has 2^4 = 16 elements.

If 1 is included, we can choose from the remaining elements {2, 3, 4}, which has 2^3 = 8 subsets.

The total number of subsets of a set with n elements is 2^n. For F, n = 4, so there are 2^4 = 16 subsets. Proper subsets are total subsets minus the set itself, so 16 - 1 = 15.

The power set of a set with n elements has 2^n elements. For F, n = 3, so the power set has 2^3 = 8 elements.

The subsets containing exactly 2 elements are {a, b}, {a, c}, and {b, c}. So, there are 3 such subsets.

The even numbers between 1 and 10 are 2, 4, 6, and 8. Therefore, F = {2, 4, 6, 8}.

The size of the power set is 2^n where n is the number of elements in the set. Here, n = 3, so the size is 2^3 = 8.

The subsets of F that do not contain 'y' are {∅, {x}}, totaling 2 subsets.

The number of ways to choose 3 elements from 5 is given by the combination formula C(5, 3) = 10.

The number of subsets of a set with n elements is 2^n. Here, n = 5, so 2^5 = 32.

The subsets containing '1' can be formed by including '1' and choosing from the remaining elements {2, 3}. There are 2^2 = 4 subsets, but we need to exclude the empty subset, so there are 4 - 1 = 3 subsets containing '1'.

The subsets with exactly 2 elements are {1, 2}, {1, 3}, and {2, 3}. So, there are 3 such subsets.

The subsets containing exactly 2 elements are {x, y}, {x, z}, and {y, z}. So, there are 3 such subsets.

The number of subsets of a set with n elements is 2^n. Here, n=2, so the number of subsets is 2^2 = 4.

The number of ways to choose 2 elements from 3 is given by the combination formula C(3, 2) = 3.

The total number of subsets is 2^3 = 8. Subtracting the empty set gives 8 - 1 = 7.

The power set of H is {∅, {x}, {y}, {x, y}}. The subsets of H are {∅, {x}, {y}}.

If 'a' is excluded, we can form subsets from {b, c}, which has 2^2 = 4 subsets.

The intersection of two sets includes only the elements that are present in both sets. Here, the intersection is {2, 3}.