Table of Contents
Last modified on July 12th, 2024
A singleton set is a set that contains only one element. It is thus also called a unit set.
For example, a set of even prime numbers is a singleton set since only one prime number is even = {2}; all others are odd.
Here are a few more examples of singleton sets:
If ‘A’ is a singleton set containing the element ‘a,’ it is written as:
A = {a}
Since there is always only one element in a singleton set, its cardinality is 1
Thus, if A is a singleton set, its cardinality is |A| = n(A) = 1.
Since the complement of any singleton set is an open set, we conclude that the singleton sets are always closed.
Since a singleton set has exactly one element, it is a finite set.
Empty sets are subsets of every set, as well as of the singleton sets. Thus, the number of subsets of a singleton set is two: the empty set (ɸ) and the set itself.
Also, since singleton sets have only one element, they are subsets of all sets containing that element.
The power set of a set consists of all subsets of the given set. Thus, the power set of any singleton set always contains 2 elements.
If A = {a} is a singleton set, its power set is P(A) = {ɸ, {a}}
Thus, any singleton set has a powerset with a cardinality of 2.
For example,
If B = {12}, the power set of B is P(B) = {ɸ, {12}} and its cardinality |P(B)| = 2
The Venn diagram of a singleton set is a circle with only one element, as shown.
| Singleton Set | Empty Set |
|---|---|
| Contains only one element. | Does not contain any elements |
| The singleton set A is represented by A = {a} | An empty set is represented by ɸ = { } |
| Its cardinality is 1, means n(A) or |A| = 1 | Its cardinality is 0, means n(ɸ) = |ɸ| = 0 |
| Also known as a unit set. | Also known as a null or void set. |
| It has two subsets: the empty set and the set itself. | It has only one subset: the empty set. |
The zero set {0} is a set with ‘0’ as its only element. Thus, it is a singleton set.
However, the zero set {0} is not an empty set.
Define the set of vowels in the color BROWN. Is it a singleton set?
Let X be a set containing vowels in the color BROWN
Then,
X = {O} has 1 element. Thus, it is a singleton set.
Find the cardinality of the power set of X = {x | 78 < x < 80 and x is an integer}
Given, X = {x | 78 < x < 80 and x is an integer} = {79}
Now, its power set is {ɸ, {79}}
Thus, the cardinality of the given power set is 2
Which of the following is a singleton set?
a) A = {the prime numbers less than 10}
b) B = {x | x2 = 16 and x is a natural number}
a) The prime numbers less than 10 are 2, 3, 5, and 7
Here, A = {2, 3, 5, 7} has 4 elements.
Thus, A is not a singleton set.
b) x2 = 16
⇒ x = ± 4
Since x is a natural number, B = {4}
Thus, B is a singleton set.
Last modified on July 12th, 2024