Infinite Set - Definition, Examples and Comparison

An infinite set is a set that contains an unlimited or uncountable number of elements. Unlike finite sets, infinite sets cannot be completely listed because they go on forever.

Definition of an Infinite Set

A set is called an infinite set if the number of its elements is not countable. In other words, it is impossible to list all the elements of the set completely, as they continue indefinitely.

If the number of elements in a set is not a finite number, then the set is infinite.

Characteristics of Infinite Sets

Contains endless elements.
Elements cannot be counted one-by-one in a finite amount of time.
Infinite sets are often described using patterns or rules.
They are commonly used in advanced mathematics, including calculus, sequences, and series.

Examples of Infinite Sets

Example 1:

The set of all natural numbers: $N = {1, 2, 3, 4, 5, \dots}$ This set never ends, so it is infinite.

Example 2:

The set of all integers: $Z = {\dots, - 3, - 2, - 1, 0, 1, 2, 3, \dots}$ This set extends infinitely in both positive and negative directions.

Example 3:

The set of all real numbers between 0 and 1: $R = {x ∣ 0 < x < 1, x \in ℝ}$ There are infinitely many real numbers between any two real numbers.

Example 4:

The set of square numbers: $S = {1, 4, 9, 16, 25, 36, \dots}$ This set continues endlessly, making it infinite.

Finite vs Infinite Sets

Property	Finite Set	Infinite Set
Number of Elements	Countable and limited	Uncountable or unlimited
Example	${1, 2, 3}$	${1, 2, 3, 4, \dots}$
Can Be Listed Completely?	Yes	No
Cardinality	A natural number (0 or more)	Not a specific number

How to Identify an Infinite Set

A set is likely infinite if:

It uses a pattern like $\dots$ to show continuation.
It includes a range like " $x \in ℕ$ " or " $x > 0$ " without an upper limit.
It cannot be described using a fixed list of items.

Uses of Infinite Sets

Essential in understanding concepts in:

 * Calculus
 * Limits
 * Sequences and Series
 * Real and Complex Numbers

Also found in computer science (e.g., infinite loops, recursion), logic, and physics.

Conclusion

An infinite set contains elements that go on endlessly, unlike finite sets which stop at a certain point. Infinite sets are foundational in many areas of higher mathematics and understanding them prepares students for advanced topics.