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POWER SET

A power set includes all the subsets of a given set including the empty set. The
power set is denoted by the notation P(S) and the number of elements of the
power set is given by 2n. A power set can be imagined as a place holder of all
the subsets of a given set, or, in other words, the subsets of a set are the
members or elements of a power set.

A set, in simple words, is a collection of distinct objects. If there are two
sets A and B, then set A will be the subset of set B, if all the elements of set
A are present in the set B. Let us learn more about the properties of power set,
the cardinality of a power set, and the power set of an empty set, with the help
of examples, FAQs.

1. Power Set Definition 2. Cardinality of a Power Set 3. Power Set Properties 4.
Power Set Proof 5. Power Set of Empty Set 6. FAQs on Power Set


POWER SET DEFINITION

A power set is defined as the set or group of all subsets for any given set,
including the empty set, which is denoted by {}, or, ϕ. A set that has 'n'
elements has 2n subsets in all. For example, let Set A = {1,2,3}, therefore, the
total number of elements in the set is 3. Therefore, there are 23 elements in
the power set. Let us find the power set of set A.
Set A = {1,2,3}
Subsets of set A = {}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}
Power set P(A) = { {}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3} }




CARDINALITY OF A POWER SET

The cardinality of a set is the total number of elements in the set. A power set
contains the list of all the subsets of a set. The total number of subsets for a
set of 'n' elements is given by 2n. Since the subsets of a set are the elements
of a power set, the cardinality of a power set is given by |P(A)| = 2n. Here, n
= the total number of elements in the given set.

Example: Set A = {1,2}; n = 2
|P(A)| = 2n = 22 = 4.

Subsets of A = {}, {1},{2},{1,2}
Therefore, |P(A)| = 4.


POWER SET PROPERTIES

A power set is a set that has a list of all the subsets of a given set. The
power set which is denoted by P(A) with 'n' elements has the following
properties:

 * The total number of elements of a set is 2n.
 * An empty set is a definite element of a power set.
 * The power set of an empty set has only one element.
 * The power set of a set with a finite number of elements is finite. For
   example, if set X = {b,c,d}, the power sets are countable.
 * The power set of an infinite set has infinite number of subsets. For example,
   if Set X has all the multiples of 5 starting from 5, then we can say that Set
   X has an infinite number of elements. Though there is an infinite number of
   elements, a power set still exists for set X, in this case, it has infinite
   number of subsets. 
 * These the power set exists for both finite and infinite sets.


POWER SET PROOF

Let us see how a set containing 'n' elements has a power set that has 2n
elements. In other words, the cardinality of a finite set A with 'n' elements is
|P(A)| = 2n.

The proof of the power set follows the pattern of mathematical induction. To
start with, let us consider the case of a set with no elements or an empty set.

Case 1: A set with no elements. Let A = {}.
Here, the power set of A, which is denoted by P(A) = {} and the cardinality of
the power set of A = |P(A)| = 1, since there is only one element, which is the
empty set. Also, by the formula of the cardinality of a power set, there will be
2n power sets, which are equal to 20 or 1.

Case 2:
This is an inductive step. It is to be proved that P(n) → P(n+1). This means, if
a set that has 'n' elements has 2n subsets, then a set that has 'n+1' elements
will have 2n+1 subsets.
To prove this, let us assume two sets 'X' and 'Y' with the following elements.
X = {a1a1, a2a2, a3a3,a4a4, anan} and
Y = {a1a1, a2a2, a3a3,a4a4, anan, an+1an+1}

The cardinality of the two sets 'X' and 'Y' are,
|X| = n , which means there are 2n subsets for the set 'X'.
|Y| = n+1
We can write that Y = X U {an+1an+1}, this means, every subset of set 'X' is
also a subset of set 'Y'.
A subset of set Y may or may not contain the element an+1an+1.
If an element of set 'Y' does not contain the element an+1an+1, then it is clear
that it is an element of set 'X'.
Also, if the subset of 'Y' has the element an+1an+1, this means that the element
an+1an+1 is included in any of the 2n subsets of the set 'X'. So we can conclude
that, set 'Y' has 2n subsets with the element an+1an+1. Therefore, set Y has 2n
subsets with element an+1an+1 and 2n subsets without the element an+1an+1.
An example of this proof is as follows.
Example:
Let X = {1,2}
Let Y = {1,2,3}
Here, the |X| = 2, so there will be 22 subsets for set X.
and |Y| = 3. We will prove that set Y has 23 subsets.
Subsets of X are = {ϕ}, {1}, {2}, {1,2}
Subsets of Y are = {ϕ}, {1}, {2}, {3}, {1,2} ,{2,3}, {1,3}, {1,2,3}
Here, '3' is the extra element in set Y that is not in set X. Also, set Y
includes 4 subsets that do not include element 3 and 4 other subsets that have
element 3. So, in all, for set Y there are 4 subsets without the element '3' and
4 subsets with the element '3'.


POWER SET OF EMPTY SET

Power set of an empty set also has an element. We know that if the number of
elements in a set is 'n', then there will be 2n elements in the power set. The
empty set is a set with no elements. It is denoted by { } or the symbol Ø. This
implies, { } is a subset of every set. An empty set does not contain any
element. Therefore, the power set of the empty set is an empty set only.

We just read that an empty set does not contain any elements, this means the
power set of empty set will contain 20 elements. Therefore, the power set of the
empty set is an empty set with one element, i.e., 20 = 1. So, P(E) = {}.

☛Articles on Power Set

Given below is the list of topics that are closely connected to the Power Set.
These topics will also give you a glimpse of how such concepts are covered in
Cuemath

 * Set Builder Notation
 * Roster Notation
 * Operations on Sets
 * Intersection of Sets
 * Subset
 * Universal Set

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POWER SET EXAMPLES

 1. Example 1: Find the number of elements in the power sets of the following.
    a) An empty set, set A = {}
    b) A set with 'k + 1' elements.
    
    Solution:
    a) If the number of elements in a set is 'n', then there will be 2n elements
    in the power set. Since an empty set does not contain any elements, the
    power set will contain 20 elements or 1 element. Therefore, the power set of
    the empty set is an empty set, P(E) = {}.
    
    b) Power set of a set with 'n' elements is given by P(A) = 2n. It is given
    that a set has 'k+1' elements, therefore, the power set of the set will
    contain 2k+1 elements.

 2. Example 2: Find the power set of the following set. Set B = {1,2,3,4}
    
    Solution:
    
    The power set for a set with 'n' elements is given by 2n.
    
    The number of elements in set B = 4. Therefore, there will be 24 elements in
    the power set of B.
    Subsets of B = {}, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2,
    4}, {3, 4},{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1,2,3,4}.
    Power set of B, P(B) = { {}, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2,
    3}, {2, 4}, {3, 4},{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1,2,3,4} }.
    Therefore, there are 16 elements in the power set of B.

 3. Example 3: For the set A = {p, q, r, s, t, u, v, w, x, y}, how many members
    will the power set have?
    
    Solution:
    
    Set A has total 10 members A = {p, q, r, s, t, u, v, w, x, y}
    So, |P(A)| = 2n = 210 = 1,024.
    Therefore, there are 1024 members in the power set of A.

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PRACTICE QUESTIONS ON POWER SET

 1. Q1.The power set is a set that includes:
    Responses
     * Empty Set
       Empty Set
     * All the elements of the set
       All the elements of the set
     * All the elements of the set including the empty set.
       All the elements of the set including the empty set.
    
    
    Check Answer
 2. Q2.The cardinality of a power set for a set consisting of 'n' elements is. 
    Responses
     * 2n 
       2, n,  
     * 2n+1 
       2, n+1,  
     * 2n-1 
       2, n-1 
    
    
    Check Answer

Check Answer >
Submit Answers




FAQS ON POWER SET


WHAT IS A POWER SET?

A set that contains all the subsets of a given set along with the empty set is
called a power set. For example, if set A = {a,b}, then the power set of A is {
{}, {a}, {b}, {a,b}}.


WHAT IS THE NOTATION OF A POWER SET?

A power set is denoted by the letter P(Set Name). For example, if set B =
{1,2,3}, then power set of B is denoted as P(B).


WHAT IS THE CARDINALITY OF A POWER SET?

Cardinality denotes the total number of elements in the power set. It is denoted
by |P(X)|. The cardinality of a power set for a set of 'n' elements is given by
'2n'. For example, if set X = {a,b,c}, then the cardinality of the power set is
|P(X)| = 23 or 8. This means there will be 8 subsets present in the power set: {
{}, {a}, {b}, {c}, {a,b}, {a, c}, {b, c}, {a, b, c} }.


WHAT IS THE POWER SET OF AN EMPTY SET?

A set that has no elements is said to be an empty set. A power set always has
the empty set as an element. Therefore, the power set of an empty set is an
empty set only. It just has one element. P(ϕ) = {ϕ}.


HOW DO YOU FIND THE POWER SET OF A SET?

To find the power set of a set, write down all the subsets of the given set
along with the empty set. For example, if set S = {p,q,r}, then the power set of
S, P(S) = { {}, {p},{q},{r},{p,q},{q,r},{p,r},{p,q,r} }


WHAT ARE THE PROPERTIES OF A POWER SET?

A set is a collection of distinct elements and a power set is a combination of
an empty set and all the subsets of a given set. The properties of a power set
are as follows.

 * A power set has an empty set as an element for sure.
 * The cardinality of a power set for a set of 'n' elements is given by 2n.
 * The power set of an empty set has only one element which is the empty set or
   the null set.
 * The power set of a finite set of elements is countable. For example, a set
   with 2 elements has 22 or 4 elements in the power set.


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