Singleton Set

Singleton Set - Definition and Examples

A singleton set is a special type of set that contains only one element. It is the smallest possible non-empty set in set theory.

Definition of a Singleton Set

A set that has exactly one element is called a singleton set. It is also known as a unit set.

• In mathematical notation:

$A=\{a\}$ is a singleton set, because it contains only one element, a.

Characteristics of Singleton Sets

• Contains exactly one element.
• Is a finite set.
• The number of elements (cardinality) is:

$n(A)=1$

• Every singleton set is a subset of itself and other sets that include the same element.

Examples of Singleton Sets

Example 1:

$A=\{7\}$ This is a singleton set with one element, 7.

Example 2:

$B=\{\text{India}\}$ This set contains only one country name.

Example 3:

The set of natural numbers less than 2 and greater than 0: $C=\{1\}$ Only one natural number satisfies this condition, so the set is singleton.

Example 4:

Let D be the set of solutions to the equation $x^2=1$ in natural numbers: $D=\{1\}$ Since only 1 is a natural number solution, D is a singleton set.

Difference Between Singleton and Empty Set

Property	Singleton Set	Empty Set
Number of Elements	Exactly 1	0
Example	$\{5\}$	$\mathrm{\varnothing }$
Finite?	Yes	Yes
Subset of Other Sets?	Yes	Yes

Importance of Singleton Sets

• Useful in identifying specific outcomes in probability and statistics.
• Important in defining functions and relations.
• Forms the basis of understanding the concept of cardinality in set theory.

Conclusion

A singleton set is a set that contains just one element. It is an important type of finite set in mathematics and helps students grasp the idea of set size and structure. Knowing about singleton sets is essential for understanding more complex topics in mathematics.