In arithmetic, the power set is described as the set of all subsets which includes the null set and the unique set itself. it's far donated through P(X). In simple phrases, this is the set of the aggregate of all subsets which include an empty set of a given set.
For example, X = {1, 2, 3} is a fixed set.
Then all subsets {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} are the elements of the power set, such as:
Power set of X, P(X) = {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}.
Where P(X) denotes the power set.
If the set has n elements, then its power set will maintain 2n elements. It also gives the cardinality of the electricity set.
Power Set Example:
Assume a set X = {1, 2, 3, 4}
n = Number of elements = 4
Therefore, according to the power set formula, the number of subsets is 2^4 = 16
Power Set (List of All Subsets):
Empty set: $$\{\}$$
Single element subsets: $$\{1\}, \{2\}, \{3\}, \{4\}$$
Two-element subsets: $$\{1, 2\}, \{1, 3\}, \{1, 4\}, \{2, 3\}, \{2, 4\}, \{3, 4\}$$
Three-element subsets: $$\{1, 2, 3\}, \{1, 2, 4\}, \{1, 3, 4\}, \{2, 3, 4\}$$
Four-element subset: $$\{1, 2, 3, 4\}$$
Therefore, the complete power set of the set X = {1, 2, 3, 4} contains 16 subsets:
$$\{\}$$
$$\{1\}$$
$$\{2\}$$
$$\{3\}$$
$$\{4\}$$
$$\{1, 2\}$$
$$\{1, 3\}$$
$$\{1, 4\}$$
$$\{2, 3\}$$
$$\{2, 4\}$$
$$\{3, 4\}$$
$$\{1, 2, 3\}$$
$$\{1, 2, 4\}$$
$$\{1, 3, 4\}$$
$$\{2, 3, 4\}$$
$$\{1, 2, 3, 4\}$$
Normally, the range of digits of a power set may be written as |X|, if X has n values then:
$$|P(X)| = 2^n$$
Houses of Notation:
A null set has no element. therefore, the electricity set of a null set { }, may be stated as;
The power set need to include at the least one wide variety. The subset of empty set is (2^0 = 1). it's miles the smallest powerset and proper subset of each powerset.
The null set is taken into consideration as a finite set, and its cardinality cost is 0.
