Power Set

In this chapter we will understand the concept of power set with properties and examples.

To understand power sets, you should have good understanding about the subsets and its properties.

Read about the subsets here.

What are Power sets ?

Power set is a collection of all subset of a given set.

If A is a set. Then collection of all subset of A is called Power set of A.

Power set is denoted by P(A)

How to find Power set of given set ?

Follow the below steps;

(a) Find all the possible subsets of given set.

(b) Make new set of all the subsets

(c) This new set will be represented by P (set name)

Let us understand the above process with the help of examples.

Examples of Power set

Example 01
If set A = { 1, 2 }, then find power set P(A).

For the given set A, first find all the possible subsets.

Subsets of A are;
⟹ {𝜙}, {1}, {2}, {1, 2}

Form a new set with all above subsets;
P (A) = {{𝜙}, {1}, {2}, {1, 2}}

Hence, the collection of all subset is the Power set.

Example 02
If set X = { a, b, c }, then find the proper set P (X}

Find all the possible subset of X.

Subset of X are;
⟹ {𝜙}, {a}, {b}, {a, b}, {b, c}, {c, a}, {a, b, c}

Now form the set of above subsets
P(X) = {{𝜙}, {a}, {b}, {a, b}, {b, c}, {c, a}, {a, b, c}}

Hence the above set P(X) is the power set of set X.

Example 03
If A = { a, {b} }, then find the power set P(A)

Here the set A is given as;
A = { a, {b} }

Here inside the set A, we have another set in form of { b }.

Suppose that;
X = { b }

A = { a, X }

Now find all the subset of set A.
⟹ {𝜙}, {a}, {X}, {a, X}

Put the value of X in all subsets;
⟹ {𝜙}, {a}, {{b}}, {a, {b}}

Now form the set of all the subsets.
P (A) = { {𝜙}, {a}, {{b}}, {a, {b}} }

Hence, P(A) is the power set of given set A.

Example 04
If A = { x : x is prime number less than 7 }.
Find he power set P (A)

First write the set A in Roster form.
A = { 2, 3, 5 }

Now write all the possible subsets of A
⟹ {𝜙}, {2}, {3}, {5}, {2, 3}, {3, 5}, {5, 2}, {2, 3, 5}

Now form the new set with all the above elements
P (A) = { {𝜙}, {2}, {3}, {5}, {2, 3}, {3, 5}, {5, 2}, {2, 3, 5} }

Cardinality of Power set

The number of unique element present in given set is called cardinality.

If A is the given set, then;

Power set cardinality = Number of subsets of A

Number of subset in set A is given by formula;
Total subset = \mathtt{2^{n}}

n = number of unique element in set A

Hence, the cardinality of power set is given as;
Power set Cardinality = \mathtt{2^{n}}

Let us understand the concept with examples.

Example 01
A = { 2, 4 }
Find cardinality of P (A)

Number of subsets of A = \mathtt{2^{2} =\ 4}

Listing all the 4 subsets
⟹ {𝜙}, {2}, {4}, {2, 4}

For power set, we will form the set of all subsets.
P (A) = { {𝜙}, {2}, {4}, {2, 4} }

There are 4 unique elements in set P(A).
Hence, cardinality of power set is 4.

To find the cardinality of power set just apply below formula.

Power set cardinality = \mathtt{2^{n}}

Example 02
Set A = { 9, 7, 10 }
Find the cardinality of power set A.

Cardinality of power set is given by following formula;

Power set Cardinality = \mathtt{2^{n}}

Power set cardinality = \mathtt{2^{3} =\ 8}

Hence, there are 8 unique elements in Power set.

Properties of Power set

Given below are important properties of power set;

(a) Power set is the set of subsets.

(b) Power set of empty set has one element.

Empty set is the one which contains 0 elements.

Empty set is expressed as follows;
If A = { 𝜙 };

Subset of A will be { 𝜙 }

Then P (A) = { { 𝜙 } }

Hence, power set of empty set is not empty, it contain the null set A.

(c) Cardinality of Power set is given by \mathtt{2^{n}}

(d) Power set of finite set contain finite number of elements and is much larger than the original set.

(e) Power set of infinite set is also infinite.

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