Power Set of a Set

The power set of a set is the set of all possible subsets of the given set. For example, if the set is {1, 2}, then the power set is {{}, {1}, {2}, {1, 2}}.

Order of the Power Set

The order of a set is the number of elements in the set. The order of the power set of a set with n elements can be found using the formula:

2^n

Explanation

To understand why the order of the power set is 2^n, consider the following:

- For each element in the original set, there are two possibilities: either the element is included in a subset, or it is not.
- Therefore, for a set with n elements, there are 2 choices for the first element, 2 choices for the second element, and so on, up to 2 choices for the nth element.
- The total number of possible subsets is then the product of all these choices, which is 2^n.

Example

Let's take an example to illustrate this. Consider the set {1, 2, 3}. The power set of this set is:

{{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

The original set has 3 elements, so the order of the power set is:

2^3 = 8

Conclusion

The order of the power set of a set with n elements is 2^n. This formula can be used to quickly find the number of possible subsets of a set, which can be useful in combinatorics and other areas of mathematics.