Set Theory
6.0 Power set
6.0 Power set
The collection or family of all possible subsets of a given set is called the power set.
If $A$ is a set, then the collection of all possible subsets is called the power set of $A$ and is denoted as $P(A)$.
From the theorems of the subsets, we can conclude that,
i. Set $A$ is always an element of $P(A)$
ii. $\emptyset $ is always an element of $P(A)$.
iii. The number of elements in $P(A)$ is same as total number of possible subsets that can be formed from set $A$.
Hence, the total number of elements is in $P(A)$ is ${2^n}$, where $n$ is the total number of elements in set $A$.
Illustration 15. Write down the power set of each set. Also write the number of elements in each power sets.
a. Set of vowels in the English alphabet
b. $A = \{ x:x = 1,x \in N\} $
c. $\emptyset $
d. Set of letters used in the word GRATITUDE
e. Set of natural numbers less than $3$
f. Set of vowels not used in the word EDUCATOR
Solution:
| SET | POWER SET AND NUMBER OF ELEMENTS |
| a. Set of vowels in the first ten letters of the English alphabet | There are $3$ vowels in the first ten letters of the English alphabet. Thus, number of elements of the power set is ${2^3} = 8$ $P(A) = \{ \emptyset ,\{ a\} ,\{ e\} ,\{ i\} ,\{ a,e\} ,\{ a,i\} ,\{ e,i\} ,\{ a,e,i\} \} $ |
| b. $A = \{ x:x = 1 \} $ | There is exactly one element. Thus, the number of elements is ${2^1} = 2$ $P(A) = \{ \emptyset ,\{ 1\} \} $ |
| c. $\emptyset $ | The number of elements in an empty set is $0$. Thus, the number of elements is ${2^0} = 1$ $P(A) = \{ \emptyset \} $ |
| d. Set of letters used in the word GRATITUDE | The elements of the set are, $A,D,E,G,I,R,T,U$. The number of elements in the set is $8$. Hence, the number of elements in the power set is ${2^8} = 256$. |
| e. Set of natural numbers less than $3$ | The numbers are, $1,2$. Hence, number of elements in power set is ${2^2} = 4$. $P(A) = \{ \emptyset ,\{ 1\} ,\{ 2\} ,\{ 1,2\} \} $ |
| f. Set of vowels not used in the word EDUCATOR | The vowels used are $A,E,O,U$. The vowel not used is, $I$. $P(A) = \{ \emptyset ,\{ i\} \} $ |
