Infinite Set: Definition, Examples, Properties, Countable & Uncountable 

1.0Introduction to Sets and Their Types

In mathematics, a set is a collection of distinct objects. Sets are foundational to almost every area of mathematics, from algebra to probability. They can be classified into two main categories based on the number of elements they contain:

• Finite Sets: These are sets with a countable number of elements. The process of counting the elements comes to an end. Examples include the set of all students in a classroom or the set of vowels in the English alphabet.
• Infinite Sets: These are sets with an unlimited, or uncountable, number of elements. The process of counting would never end.

This guide will focus on the fascinating world of infinite sets, a concept that goes beyond simple counting and delves into the deeper nature of numbers.

2.0What is an Infinite Set?

An infinite set is a set that contains an endless number of elements. You cannot list all the elements of an infinite set, as the list would never be complete.

Formally, a set A is considered infinite if there exists a proper subset B of A such that a one-to-one correspondence (a bijection) can be established between the elements of A and B.

This formal definition is what distinguishes infinite sets from finite ones. For instance, the set of natural numbers N={1,2,3,4,...} is an infinite set. Consider its proper subset of even numbers E={2,4,6,8,...}. We can pair each element of N with an element of E using the mapping n→2n. Every natural number has a unique even number counterpart, and vice versa. This shows that the set of natural numbers is indeed infinite.

3.0Examples of Infinite Sets

Infinite sets are more common in higher mathematics than you might think. Here are some key examples:

• The set of all Natural Numbers (N): {1,2,3,4,...}
• The set of all Integers (Z): {...,−3,−2,−1,0,1,2,3,...}
• The set of all Rational Numbers (Q): All numbers that can be expressed as a fraction p/q, where p,q are integers and q=0. Examples include 1/2,−5,10.75.
• The set of all Real Numbers (R): All numbers on the number line, including rationals and irrationals $(like\sqrt{2},\pi ,e)$
• The set of all points on a line segment: Even a short line segment contains an infinite number of points.
• The set of all circles in a plane.

These examples highlight that infinite sets can be quite different from one another.

4.0Cardinality of Infinite Sets

For finite sets, we use the term cardinality to mean the number of elements. For infinite sets, the concept of cardinality gets more complex. We cannot assign a whole number to it. Instead, we use a special kind of number called a transfinite number to describe the size of an infinite set.

The smallest transfinite number is called Aleph-null (ℵ0​). This number represents the cardinality of the set of natural numbers (N). Any set that can be put into a one-to-one correspondence with the set of natural numbers has a cardinality of ℵ0​. These are a special type of infinite set called countably infinite sets.

Interestingly, it has been proven that not all infinite sets have the same size. For instance, the set of real numbers is "larger" than the set of natural numbers.

5.0Countable vs. Uncountable Infinite Sets

This is a crucial distinction for JEE-level understanding.

Countably Infinite Sets

A set is countably infinite if its elements can be put into a one-to-one correspondence with the set of natural numbers. In simple terms, you can "count" them, even if the counting process never ends. This means you can create a list where every element of the set has a unique position in the list.

Examples:

• Natural Numbers (N): {1,2,3,4,...}
• Integers (Z): We can list them as {0,1,−1,2,−2,3,−3,...}. Every integer will appear exactly once on this list.
• Rational Numbers (Q): While seemingly more complex, it can be proven that the set of all rational numbers is also countably infinite. This is a famous proof by Georg Cantor.

The cardinality of all countably infinite sets is ℵ0​.

Uncountably Infinite Sets

A set is uncountably infinite if it is an infinite set whose elements cannot be put into a one-to-one correspondence with the set of natural numbers. You cannot create a list that includes every element of the set.

Examples:

• Real Numbers (R): This is the most famous example. Georg Cantor's Diagonal Argument proves that the set of real numbers is uncountable. No matter how you try to list them, there will always be a real number that you missed.
• The set of points on a line segment or an interval: For example, the set of all numbers in the interval [0,1].
• The set of irrational numbers.

The cardinality of the set of real numbers is denoted by c (the continuum). It has been proven that ℵ0​<c. This means there are different "sizes" of infinity.

6.0Properties of Infinite Sets

Infinite sets have some surprising and counter-intuitive properties.

• An infinite set can have a proper subset that is equivalent to the original set (i.e., they have the same cardinality). For example, the set of natural numbers N and the set of even numbers E are equivalent, even though E is a proper subset of N.
• The union of two infinite sets is an infinite set.
• The intersection of two infinite sets can be either finite or infinite. For example, the intersection of the set of all integers and the set of all real numbers is the set of integers, which is infinite. The intersection of the set of all positive integers and the set of all negative integers is an empty set (a finite set).

7.0Operations on Infinite Sets

Operations like union, intersection, and Cartesian products apply to infinite sets as well.

• Union (A∪B): The union of two infinite sets A and B is always an infinite set. For example, N∪Z=Z, which is infinite.
• Intersection (A∩B): The intersection of two infinite sets can be either infinite or finite. For example, the intersection of the set of all positive integers and the set of all natural numbers is the set of positive integers, which is infinite. The intersection of the set of even integers and the set of odd integers is the empty set, which is finite.
• Cartesian Product (A×B): The Cartesian product of two infinite sets is always an infinite set. The cardinality of the product of two countably infinite sets is also countably infinite.

8.0JEE-Level Concepts and Problems on Infinite Sets

While you won't be asked to perform complex proofs like Cantor's Diagonal Argument, JEE-level questions often test your fundamental understanding of these concepts.

• Identifying Countable vs. Uncountable Sets: Be able to classify sets like N,Z,Q,R as countable or uncountable.
• Cardinality Comparisons: Understand that the cardinality of N is the same as Z and Q, but smaller than that of R.
• Set Operations on Infinite Sets: Solve problems involving the union, intersection, and difference of infinite sets, especially in the context of intervals on the real number line (e.g., (−∞,5]∩[2,∞)).
• Countability of Subsets: A subset of a countably infinite set is either finite or countably infinite.

Example Problem:

Which of the following sets is uncountable?

(A) The set of all integers.

(B) The set of all rational numbers.

(C) The set of all real numbers.

(D) The set of all integers divisible by 5.

Solution:

Sets (A), (B), and (D) are all countably infinite. Set (C), the set of all real numbers, is uncountably infinite. Therefore, the correct answer is (C).