Final answer:
To show that a nonempty set has the same number of subsets with an odd number of elements as subsets with an even number of elements, we can use combinatorial reasoning.
Step-by-step explanation:
To show that a nonempty set has the same number of subsets with an odd number of elements as subsets with an even number of elements, we can use combinatorial reasoning.
Let's consider a set with n elements, where n is a positive integer. Each element in the set can either be included in a subset or not included. Therefore, for each element, there are 2 choices.
Using the multiplication principle, the total number of subsets of the set is 2^n. Now, let's look at the subsets with an odd number of elements.
Since there are 2 choices for each element, the subsets with an odd number of elements can be formed by choosing an odd number of elements from the set.
Similarly, for the subsets with an even number of elements, we can choose an even number of elements from the set.
This means that the number of subsets with an odd number of elements is equal to the number of subsets with an even number of elements, since the number of odd and even combinations is the same.