Power set

Consider a small garden with three types of plants: It includes roses, tulips, and daisies. The power set incorporates everything that has been noted above, including the empty set, every plant as an individual member, combinations of two plants, and all three plants as the power set members. The set of all subsets of a set is called the power set.

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Power sets determine the links among various subsets of elements. This means that the power set contains all the possible subsets of a particular set, depicting all the possibilities in a certain range of the set. Sets are very fundamental concepts in mathematics, which have applications across various domains like statistics, calculus, computer science, etc.

In this article, we will cover the concept of power sets. This concept falls under the broader category of sets, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of three questions have been asked on this concept, including one in 2014, one in 2022, and one in 2023.

What is a Power Set?

The set of all subsets of a set $S$ is called the power set of ' $S$ '. It is denoted by $P(S)$. It also includes the empty set and the set itself. The power set always contains $2^n$ elements, where $n$ is the number of elements in the original set.

Power Set Definition: The collection of all subsets of a set $A$ is called the power set of $A$. It is denoted by $P(A)$.

Power Set Example: Let $\mathrm{A}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}\}$, then
$P(A)=\{\varphi,\{a\},\{b\},\{c\},\{a, b\},\{b, c\},\{c, a\},\{a, b, c\}\}$

Cardinality of Power Set

The number of distinct elements in a finite set $A$ is called the Cardinal number or cardinality of set $A$ and it is denoted by $n(A)$.

The power set formula to find the cardinality(number of elements) of the power set of any set $A$ is $2^n$ where $n$ is the number of elements in $A$.

Eg. Let $\mathrm{A}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}\}$. The cardinality of the set $A$ is $3$. The cardinality of the power set of $A$, $P(A)$ is $2^n = 2^3 = 8$.

Properties of Power Set

The properties of the power set are,

• The power set of any set is non-empty.
• Each element of a Power set is a set.
• Number of elements in $P(A)=$ Number of subsets of $\operatorname{set} A
  =2 ^{n(A)}$ where $n(A)$ is the number of elements in set $A$
• It is much larger than the original set.
• The power set of a finite set has finite number of elements.
• For a set of natural numbers, we can do one-to-one mapping of the resulting set, $P(S)$ with the real numbers.
• $P(S)$ of set $S$, if operated with the union of sets, the intersection of sets and complement of sets, denotes the example of Boolean Algebra.

Recommended Video on Power Set

Solved Examples Based on Power Set

Example 1: Power set of $\{1,2,3\}$ is

$
\begin{aligned}
& \text { 1) } P(A)=\{\phi,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\}\} \\
& \text { 2) } P(A)=\{\{1\},\{2\},\{3\},\{1,2\},\{1,3\}\{2,3\},\{1,2,3\}\} \\
& \text { 3) } P(A)=\{\phi,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}
\end{aligned}
$
4) None of the above

Solution

As we learned
In this Question,
Option (3) shows all the subsets of $\{1,2,3\}$
Hence, the answer is the option 3.

Example 2: Find the number of elements in the power set of $\{1,2,3,4\}$.
1) $8$
2) $16$
3) $32$
4) $64$

Solution
As we learned
In this Question,
The power set of $\{1,2,3,4\}$ will have $16$ elements.
Hence, the answer is the option 2.

Example 3: Let $\mathrm{A}=\{1,2,3,4,5,6,7\}$ and $\mathrm{B}=\{3,6,7,9\}$. Then the number of elements in the set $\{\mathrm{C} \subseteq \mathrm{A}: \mathrm{C} \cap \mathrm{B} \neq \phi\}$ is. $\qquad$ .

Solution

Total subsets of $\mathrm{A}=2^7=128$
Subsets of $A$ such that $C \cap B=\phi$

$
\begin{aligned}
& =\text { Subsets of }\{1,2,4,5\}=2^4=16 \\
& \text { Required Subsets }=128-16=112
\end{aligned}
$

Hence, the answer is $112$.

Example 4: The number of elements in the power set of $P(P(P(\phi))$ is
1) $1$
2) $2$
3) $3$
4) $4$

Solution

Number of elements in $\mathrm{P}(\varphi)$ is $2^0=1$
Number of elements in $\mathrm{P}(\mathrm{P}(\varphi))$ is $2^1=2$
Number of elements in $\mathrm{P}(\mathrm{P}(\mathrm{P}(\varphi)))$ is $2^2=4$
Hence, the answer is the option 4.

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