How to Check If a Set Is Finite or Infinite?
To check whether a set is finite or infinite, we use the following methods:
For Finite Sets
To verify that a set is finite, follow these steps:
- If you can list or count all the elements of the set, then the given set is a finite set.
- If the set is described using set-builder notation, check if the condition results in a finite set.
- For example, {x | x is an even number between 1 and 10} is finite because you can list all the even numbers between 1 and 10 i.e., {2, 4, 6, 8}
For Infinite Sets
Here are a few common methods to identify infinite sets:
- Observe Patterns: If any set follows a specific pattern for its element without any end, it should be an infinite set.
- For example, the set of natural numbers {1, 2, 3, 4, …} is infinite as you can see the pattern of adding 1 to each further element without any end.
- Bijection with a known infinite set: If you can establish a one-to-one and onto correspondence (bijection) between the given set and a known infinite set, then the given set is also infinite.
- For example, we can establish the bijection between a set of integers and natural numbers and we know that the set of natural numbers is the infinite set. Thus, a set of integers is also an infinite set.
- Using Mathematical Properties: If the set has a property it can help us establish that this set has an infinite number of elements.
- For example, we know that between any two rational or real numbers there are infinitely many rational or real numbers respectively. Thus the set of rational numbers between 1 and 2 and real numbers between 0 and 1 both are infinite sets.
Infinite Set
Infinte set is one of the types of Sets based upon the cardinality in Set Theory and sets are one of the important topics in mathematics. In simple words, an infinite set is a set with infinite elements i.e., the number of elements in an infinite set never depletes. This concept of infinite sets seems to be complicated at first sight but we’ll try our best to make it as comprehensive and understanding as possible.
This article deals with this concept and tries its best to explain the concept in detail. Other than that, this article covers definition, notation, types, cardinality, examples, and properties of Infinite Sets. So, let’s start learning about Infinite Sets.
Table of Content
- What are Infinite Sets?
- Infinite Sets Definition
- Infinite Set Notation
- Infinite Set Examples
- Types of Infinite Sets
- Properties of Infinite Sets
- Venn Diagram for Infinite Sets
- Difference Between Finite Sets and Infinite Sets