Final answer:
a) There are 512 subsets of the set {1,2,3,4,5,6,7,8,9}. b) There are 126 subsets of the set {1,2,3,4,5,6,7,8,9} that contain exactly 5 elements. c) There are 16 subsets of the set {1,2,3,4,5,6,7,8,9} that contain only even numbers. d) There are 8 subsets of the set {1,2,3,4,5,6,7,8,9} that contain an even number of elements.
Step-by-step explanation:
a. The number of subsets of a set with n elements is 2^n. Therefore, the number of subsets of {1,2,3,4,5,6,7,8,9} is 2^9 = 512. This includes the empty set and the set itself.
b. To find the number of subsets of {1,2,3,4,5,6,7,8,9} that contain exactly 5 elements, we need to choose 5 elements from the 9 available. This can be done in (9 choose 5) = 126 ways.
c. To find the number of subsets of {1,2,3,4,5,6,7,8,9} that contain only even numbers, we need to consider the power set of the even numbers {2,4,6,8}. Since there are 2^4 = 16 subsets of the even numbers, there are 16 subsets of {1,2,3,4,5,6,7,8,9} that contain only even numbers.
d. To find the number of subsets of {1,2,3,4,5,6,7,8,9} that contain an even number of elements, we can use the same reasoning as in part c. The even numbers subset has 16 subsets. Half of these subsets contain an even number of elements, so there are 8 subsets that contain an even number of elements.