If a set has n elements, then the total number of elements in its power set is equal to:

Question

Accepted Answer

To determine the total number of elements in the power set of a set with $ n $ elements, we need to understand the concept of a power set. A power set is the set of all possible subsets of a given set, including the empty set and the set itself.

1. **Explanation for Option 2 ($2^n$):**

The power set of a set with $ n $ elements contains all possible combinations of those elements. For each element in the original set, there are two choices: either the element is included in a subset, or it is not. Therefore, for each of the $ n $ elements, there are 2 possibilities (include or exclude). This results in a total of $ 2 \times 2 \times \ldots \times 2 $ (n times), which is mathematically expressed as $ 2^n $. Hence, the total number of subsets, or the number of elements in the power set, is $ 2^n $. This is why Option 2 is correct.

2. **Explanation for why other options are incorrect:**

- **Option 1 ($2^{n-1}$):** This option suggests that the number of subsets is $ 2^{n-1} $. However, this would imply that each element contributes less than two choices (include or exclude), which contradicts the fundamental principle of forming subsets. Therefore, this option underestimates the number of subsets.

- **Option 3 ($2^{n+1}$):** This option suggests that the number of subsets is $ 2^{n+1} $. This would imply that there are more subsets than possible combinations of including or excluding each element, which is not possible. This option overestimates the number of subsets.

- **Option 4 ($2^{2n}$):** This option suggests an even larger number of subsets, $ 2^{2n} $. This would imply an exponential increase far beyond the actual number of combinations possible with $ n $ elements, making it an incorrect estimation.

In conclusion, the correct number of elements in the power set of a set with $ n $ elements is $ 2^n $, which is why Option 2 is the correct answer.