A power set is basically a set that contains all the possible subsets of the original given set, including the null or empty set. If we have a set A, then the power set of A contains all the subsets of A, including the empty set.
Other example: Set A = {1, 2, 9}.
Then its power set will be {∅, {1}, {2}, {9}, {1, 2}, {1, 9}, {2, 9}, {1, 2, 9}}
{1}, {2}, {9} this represents all the subsets with one elements.
{1, 2}, {1, 9}, {2, 9} this represents all the subsets with two elements.
Lastly, {1, 2, 9}, which represents the set itself.
Power sets are used in various fields where a list of all possibilities from some finite number of elements is required, such as computer science, data analysis, and even artificial intelligence.
Mathematically, if S is a set, then the power set P(S) is defined as:
P(S) = {T | T is a subset of S}
Where,
T represents a subset of the set S.
"|" denotes "such that."
The curly braces i.e., {} indicate a set.
Let see an example for a clear and better understanding, consider a set A = {a, e, i, o, u}, therefore power set of A is given by P(A), i.e.
{a, e, i}, {a, e, o}, {a, e, u}, {a, i, o}, {a, i, u}, {a, o, u}, {e, i, o}, {e, i, u}, {e, o, u}, {i, o, u},
{a, e, i, o}, {a, e, i, u}, {a, e, o, u}, {a, i, o, u}, {e, i, o, u},
{a, e, i, o, u}}
Here ∅ represents a Null set or Empty set.
How to Find Power Set?
In order to find a power set, follow these steps:
Start with a null or empty set.
Then add all combinations of subsets with one element.
Then add all combinations of subsets with two elements.
Do this till you reach the subsets with N-1 elements (where N is the total number of elements in the original set).
Then add the original set.
Cardinality of Power Set
Cardinality (cardinality of a set means the number of elements of a set) of a power set denotes the number of elements present in the power set. It is denoted by |P(A)|. Thus, number of elements in the power set is given by:
|P(A)| = 2n
Where "n" is the number of elements of Set A.
Example: Find the cardinality of the Power Set of A, where A = {1,2,9}.
Answer:
As |A| = 3, thus number of elements in Power Set of A = 2|A|
Thus, |P(A)| = 23 = 8
Therefore, there are 8 elements in the power set of A.
Properties of Power Set
There are several properties of the power set, some of which are listed as follows:
Total number of elements of a power set is 2n ( where n is the total number of elements of the original Set).
Power Set always contains an empty set and the original set as its members.
The elements of the power set are always greater than the elements of the original set (since it has 2n elements of the original set).
The power set of an empty or null set is the set itself.
An empty or null set's power set is the set itself. Following distributive rules, power sets can be utilized for set operations like union, intersection, and complement.
Power set size is always 2n, where n is the size of the initial set.
Each member of the original set's subsets makes is always a member of power set too.
