A singleton set is defined as a set that contains exactly one element.

In mathematics, a set is a collection of distinct objects considered as a single entity. A singleton set, also referred to as a unit set, is a specific type of set that includes only one element. For instance, the set $\{5\}$ is a singleton set because it contains solely the number $5$. Similarly, the set $\{"apple"\}$ is also a singleton set as it consists of only the word “apple”.

Grasping the concept of singleton sets is crucial because they represent the simplest form of sets and act as the foundational building blocks for more intricate concepts in set theory. When discussing sets, we typically use curly braces $\{\}$ to enclose the elements. If there is only one element within the braces, we identify it as a singleton set.

In formal notation, if $a$ is an element, then the set containing $a$ is represented as $\{a\}$. This notation clearly indicates that the set has just one member. Singleton sets are particularly useful in various branches of mathematics, including functions, sequences, and probability, as they help simplify and elucidate fundamental concepts.

For example, in the context of functions, a singleton set can represent the domain or range of a function that maps a single input to a single output. In probability, a singleton set can denote an event that has only one possible outcome. Mastering these basic principles will greatly assist you as you advance to more complex topics in mathematics.