Number of Proper Subsets of a Finite Set with n Elements

To find the number of proper subsets of a given finite set with n elements, we need to understand the concept of subsets and proper subsets.

Subsets:
A subset of a set is a collection of elements that are all contained within the original set. For example, if we have a set A = {1, 2, 3}, then the subsets of A are:
- The empty set {}.
- The individual elements: {1}, {2}, and {3}.
- The pairs of elements: {1, 2}, {1, 3}, and {2, 3}.
- The entire set itself: {1, 2, 3}.

Proper Subsets:
A proper subset of a set is a subset that does not include all the elements of the original set. In other words, it is a subset that is not equal to the original set itself. For example, in the set A = {1, 2, 3}, the proper subsets are:
- The empty set {}.
- The individual elements: {1}, {2}, and {3}.
- The pairs of elements: {1, 2}, {1, 3}, and {2, 3}.

Finding the Number of Proper Subsets:
To find the number of proper subsets of a set with n elements, we need to consider the following:
- Each element in the original set can either be included or excluded from a subset.
- For each element, we have two choices: either include it or exclude it.

Based on this, for a set with n elements, we have 2 choices for each element, giving us a total of 2^n possible subsets. However, this count includes the original set itself, which is not a proper subset.

To find the number of proper subsets, we need to subtract 1 from the total count of subsets. Therefore, the number of proper subsets of a set with n elements is 2^n - 1.

Applying this to the given options:
a) 2n - 1 - This is not the correct answer as it is missing the exponentiation of 2^n.
b) 2n - 2 - This is also not the correct answer as it subtracts 2 instead of 1.
c) 2n - 1 - This is the correct answer as it subtracts 1 from 2^n, giving us the number of proper subsets.
d) 2n - 2 - This is not the correct answer as it subtracts 2 instead of 1.

Therefore, the correct answer is option 'c': 2n - 1.