Sets - Types of Sets: Finite, Infinite, Empty, Singleton

A Set is a well-defined collection of distinct objects. These objects are called elements or members of the set. Visually, a set is often represented by a closed loop or circle (Venn diagram) containing points that represent each element.

A Finite Set is a set that contains a countable number of elements. The process of counting the elements comes to an end. For example, $P = \{x : x \text{ is a factor of } 10\}$ results in $P = \{1, 2, 5, 10\}$. In a visual diagram, all elements are contained within a fixed boundary.

An Infinite Set is a set where the number of elements is unlimited and the counting process never ends. It is represented by listing a few elements followed by three dots (ellipses) like $N = \{1, 2, 3, \dots\}$. Visually, this can be imagined as a collection that extends infinitely in a specific direction.

An Empty Set (or Null Set/Void Set) is a set that contains no elements at all. It is denoted by the symbol $\phi$ or empty curly brackets $\{ \}$. Visually, an empty set is represented by a circle or region with nothing inside it. Note that $\{0\}$ is NOT an empty set; it is a set containing the number zero.

A Singleton Set (or Unit Set) is a set that contains exactly one element. For example, the set of even prime numbers is $A = \{2\}$. Visually, this is represented by a circle containing a single point or object.

The Cardinal Number of a finite set is the number of distinct elements present in that set, denoted by $n(A)$. For example, if $A = \{a, e, i, o, u\}$, then $n(A) = 5$. For an empty set, $n(\phi) = 0$, and for a singleton set, the cardinal number is always $1$.