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.
Another example: Set A = {1, 2, 9}.
Then its power set will be {∅, {1}, {2}, {9}, {1, 2}, {1, 9}, {2, 9}, {1, 2, 9}}
Let's break down the above Power set:-
- Here ∅ represents the Null or Empty Set.
- {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.
P(A) = {∅,
{a}, {e}, {i}, {o}, {u},
{a, e}, {a, i}, {a, o}, {a, u}, {e, i}, {e, o}, {e, u}, {i, o}, {i, u}, {o, u},
{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.