Power Set (Set of All Subsets)
What Is a Power Set?
The power set of a set A, denoted by P(A), is the set containing every possible subset of A, including the empty set and the set A itself. $$ P(A) = \{ S \mid S \subseteq A \} $$
In other words, if a set can be formed using only elements from A, then it belongs to the power set of A.
If S is a subset of A, we write:
$$ S \subseteq A $$
Since every subset of A belongs to the power set, S is also an element of P(A):
$$ S \in P(A) $$
Note. It is important to distinguish between the symbols \( \subseteq \) and \( \in \). A subset S is contained in A, so we write \( S \subseteq A \). The same subset becomes an element when it is viewed as a member of the power set, so we write \( S \in P(A) \).
An Example of a Power Set
Consider the set A containing two elements:
$$ A = \{ a,b \} $$
The elements a and b belong to A.
$$ a,b \in A $$
The following Venn diagram represents the set A.
To find the power set, we first list all subsets of A:
$$ \emptyset \\ \{ a \} \\ \{ b \} \\ \{ a,b \} $$
These subsets become the elements of the power set.
Therefore:
$$ P(A) = \{ \emptyset,\ \{ a \},\ \{ b \},\ \{ a,b \} \} $$
Since \(\{a,b\}=A\), the power set can also be written as:
$$ P(A) = \{ \emptyset,\ \{ a \},\ \{ b \},\ A \} $$
The subsets \(\{a\}\) and \(\{b\}\) are the proper subsets of A.
The subsets \(\emptyset\) and A are the improper subsets of A.
Note. Because these sets are elements of the power set, we can write: $$ \emptyset,\ \{a\},\ \{b\},\ A \in P(A) $$
The following diagram shows the power set P(A).
Notice that A itself appears in the power set. This is because every set is a subset of itself.
Why Does the Empty Set Belong to the Power Set?
In set theory, the empty set \(\emptyset\) is a subset of every set.
As a result, it is also a subset of A.
Since the power set contains all subsets of A, the empty set must be included in P(A).
The Power Set of the Empty Set
If A is the empty set, that is:
$$ A = \emptyset $$
then its power set is:
$$ P(A) = \{ \emptyset \} $$
Do not confuse \(\emptyset\) with \(\{\emptyset\}\).
The symbol \(\emptyset\) represents a set with no elements.
The symbol \(\{\emptyset\}\) represents a set containing one element, namely the empty set.
Why Is It Called a Power Set?
The term power set comes from a remarkable property.
If a finite set A contains n elements, then its power set P(A) contains \(2^n\) elements.
Example
The set:
$$ A = \{a,b\} $$
contains two elements, so \(n=2\).
Therefore, the number of elements in its power set is:
$$ 2^n = 2^2 = 4 $$
This matches the four subsets listed in the previous example.
How Many Subsets Does a Set Have?
A set with n elements has exactly \(2^n\) subsets.
Why does this formula work?
To build a subset of a set A, we examine each element of A and decide whether to include it or exclude it.
For every element, there are only two possible choices:
- include the element in the subset;
- leave the element out.
If the set contains n elements, this decision must be made n times.
Since each choice is independent, the total number of possible subsets is:
\[ 2 \cdot 2 \cdot \ldots \cdot 2 = 2^n \]
where the factor 2 appears once for each element of the set.
Note. The count includes both extreme cases: the empty set, which contains no elements, and the set itself, which contains all its elements.
Example 1
Consider the set:
\[ A = \{a, b, c\} \]
Since A contains three elements, the number of subsets is:
\[ 2^3 = 8 \]
The eight subsets are:
\[ \emptyset,\ \{a\},\ \{b\},\ \{c\},\ \{a,b\},\ \{a,c\},\ \{b,c\},\ \{a,b,c\} \]
Notice that the list includes both the empty set and the set A itself.
Example 2
Now consider the set:
\[ B = \{1,2,3,4\} \]
Since B contains four elements:
\[ 2^4 = 16 \]
Therefore, B has sixteen distinct subsets.
These include the empty set, all one-element subsets, all possible pairs, all possible three-element subsets, and the set B itself.
In general, whenever a finite set contains n elements, the number of possible subsets is always \(2^n\).
