Singleton Set

A singleton, also known as a unit set in mathematics, is a set containing precisely one element. The set null, for example, is a singleton that contains the element null. Further we will discuss the properties of singleton sets and also take some examples for more clarity about the topic. A singleton set has one element, which is represented by the number 1. In the singleton set, there is just a single element. 

Category Theory

• The sentence above demonstrates that singleton sets are exactly the terminal objects in the category of sets, as seen in the following diagram. There are no additional sets that are terminal. 
• Any singleton admits a topological space structure that is distinct from the others (both subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces and continuous functions, and they may be used to construct other topological spaces. There are no other spots in that category that are terminal. 
• Singletons are capable of admitting a single group structure (with the unique element functioning as the identity element). These singleton groups are zero objects in the category of groups and group homomorphisms, and they are represented by the symbol. There are no additional groupings in the category that are terminal. 

Definition of Singleton Set

A singleton set is one that only has one element. The singleton set has the form A = a, where A stands for the set and the little alphabet ‘a’ stands for the singleton set’s element. A singleton set is sometimes known as a unit set because it only has one element. A singleton set’s cardinal number is one. A singleton set has two subsets: the empty set and the set itself with a single element.

The number of singleton sets in a given set that are subsets is equal to the number of elements in the given set. There are 5 items in the set A = {a, e, i, o, u}. As a result, the set contains five singleton sets, {a}, {e}, {i}, {o}, and {u}, which are subsets of the given set.

Properties of Singleton Set

When applied to Zermelo–Fraenkel set theory, the regularity axiom ensures that no set is an element of itself, which is a crucial property for many applications. This indicates that a singleton is always different from the element it contains; for example, 1 and 1 are not the same thing, and the empty set is distinct from the set containing just the empty set, among other implications. A singleton is a set that only includes a single element, as opposed to a collection (which itself is a set, however, not a singleton). 

A set is a singleton if and only if the number of elements in it is one. Von Neumann’s set-theoretic definition of the natural numbers defines the number 1 as the singleton, which means that it is the only possible value. 

The existence of singletons is a consequence of the axiom of pairing in axiomatic set theory: for any set A, the axiom applied to A and A affirms the existence of which is the same as the existence of the singleton (since it includes just A and no other set as an element). 

If A is any set and S is any singleton, then there exists exactly one function from A to S, the function that sends every element of A to the single element of S. If A is any set and S is any singleton, then there exists exactly one function from A to S. As a result, every singleton belongs to the category of sets and is a terminal object. 

Some of the most important properties of a singleton set are listed below.

• There is only one element in the singleton set.
• A singleton set has one cardinality.
• There are two subsets of the singleton set.
• Every singleton set has a null set as a subset.
• The null set and the singleton set are the two subsets of a singleton set.
• A singleton set’s power set has a cardinal number of two.

Examples of Singleton Set

1 Find the singleton sets that are subsets of the set A = {1, 3, 5, 7, 11}.

Solution: A = {1, 3, 5, 7, 11} is the provided set.

The given set has 5 elements and 5 singleton subsets, each of which can only have one element.

As a result, the five singleton sets that are subsets of the given set A are {1}, {3}, {5}, {7}, and {11}.

2. Determine the singleton set 5’s powerset.

Solution:

A = 5 is the supplied singleton set and 5 are two possible subsets of this singleton set.

These subsets can be used to create the power set’s elements.

A = 5 is a powerset of A.

As a result, the powerset of the singleton set A is 5.

Conclusion:-

So, till now we understood that a singleton set is one that only has one element. The singleton set, often known as a unit set, has the form A = a. Due to the fact that the singleton set has just one element, the singleton set is often referred to as the unit set. The null set and the set itself are two subsets of the singleton set. In this article, we have also covered properties of singleton sets and have taken some examples too for better understanding of the topic.