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 |

**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

*A*)={

*x*|

*x*⊆

*A*}

If

*A*has

*n*elements in it, then þ(

*A*) has 2

^{n}elements:

For example, if

*A*={a,b} then

*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}

*B*) ={Ø, {0}, {1}, {3}, {0,1}, {0,3}, {1,3}, {0,1,3}}

(ii) Given,

*C*={1,{2}}

*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