Finite Set

A finite set is a set with a countable number of elements. In other words, a finite set has a specific number of elements, which can be listed. The cardinality (number of elements) of a finite set is a non-negative integer.

Example:
Consider the set of vowels in the English alphabet:
[ V = {a, e, i, o, u} ]
This set is finite because it contains exactly 5 elements.

Infinite Set

An infinite set is a set with an uncountable number of elements. This means that the set has no finite limit and its elements cannot be completely listed.

Example:
Consider the set of all natural numbers:
[ \mathbb{N} = {1, 2, 3, 4, 5, \ldots} ]
This set is infinite because there is no end to the number of natural numbers.

Null Set (Empty Set)

A null set (or empty set) is a set that contains no elements. It is denoted by ( \emptyset ) or ({}).

Example:
Consider the set of all real numbers ( x ) such that ( x^2 + 1 = 0 ):
[ A = { x \in \mathbb{R} \mid x^2 + 1 = 0 } ]
There are no real numbers that satisfy this equation, so ( A ) is a null set:
[ A = \emptyset ]

Singleton Set

A singleton set is a set containing exactly one element. The cardinality of a singleton set is 1.

Example:
Consider the set containing only the number 7:
[ S = {7} ]
This set is a singleton set because it contains exactly one element.

Summary

• Finite Set: A set with a limited number of elements.
• Example: ( V = {a, e, i, o, u} )
• Infinite Set: A set with an unlimited number of elements.
• Example: ( \mathbb{N} = {1, 2, 3, 4, 5, \ldots} )
• Null Set: A set with no elements.
• Example: ( \emptyset )
• Singleton Set: A set with exactly one element.
• Example: ( S = {7} )