The collection of all the subsets of a set is called the power set. For example, the power set of {a,b,c} has eight elements: Ø, {a}, {b}, {c}, {a,b}, {a,c}, {b,c} and {a,b,c}.

🍫 Method to Write the Power Set of a Given Set

Let a set having n elements is given, then for writing its power set, we use the following steps

Step I.	Write all the possible subsets having single element of given set.
Step II.	Write all the possible subsets having two elements at a time of given set.
Step III.	Write all the possible subsets having three elements at a time of given set. Repeat this process for writing all possible subsets having n elements at a time, as the given set has n elements.
Step IV.	Form a set, with the help of the subsets obtained from steps I, II and III and element Ø. This set will give the power set of given set.

⛲ Example 1. Worked out Problem
If D={1,2,3} then find the power set of D.
✍ Solution:

Step I.	Write all the possible subsets having single element of given set. All possible subsets of a given set having single element are {1}, {2}, {3}.
Step II.	Write all the possible subsets of a given set having two elements at a time. All possible subsets of a given set having two elements at a time are {1,2}, {2,3}, {3,1}.
Step III.	Write all the possible subsets of a given set having three elements at a time. All possible subsets of a given set having three elements at a time is {1,2,3}.
Step IV.	Form a set with the help of the subsets obtained from steps I, II and Ill and element Ø. Hence, the required power set is {{1}, {2}, {3}, {1,2}, (2,3}, {3,1}, {1,2,3}, Ø}.

The Power Set
[Definition] The set of all subsets of a set A is called the power set of A.
The power set of A is denoted by þ(A).
Hence

þ(A)={x|x⊆A }
If A has n elements in it, then þ(A) has 2ⁿ elements:
For example, if A={a,b} then
þ(A)={Ø, {a}, {b}, {a,b}}
The empty set Ø, has only subset, therefore þ(Ø)={Ø}.
[Note] A set is never equal to its power set. In the programming language Pascal, the notion power set is used to define data type in the language.

⛲ Ex2. Write down the power set of the following sets.
(i) B={0,1,3} (ii) C={1,{2}}
✍ Solution:
(i) Given, B={0,1,3}

∴þ(B) ={Ø, {0}, {1}, {3}, {0,1}, {0,3}, {1,3}, {0,1,3}}
(ii) Given, C={1,{2}}
þ(C)={ Ø, {1}, {{2}}, {1,{2}}}

Given a set, you can form a new set with the power set operation, defined as follows.
[Definition] If E is a set, the power set of E is another set, denoted as þ(E) and defined to be the set of all subsets of E. In symbols, þ(E)={x:x⊆E}.

the number of Distinct Subsets of a Set — Power Set