Countable Set

Last Updated : 01 May, 2024

A countable set is one that has the same cardinality (size) as the set of natural numbers, which is denoted by N (or often expressed as 0, 1, 2, 3,… in set theory). In other words, a set is countable if its elements have a one-to-one mapping with natural numbers. In this article, we are going to look at various methods to prove if a given set is countable or not. In simple terms, if you can build a list of the members of a set, it is said to be countable. A list is one in which you can discover a first member, a second one, and so on, eventually assigning an integer to each member, possibly indefinitely.

What is Countable Set?

In the theory of computation and mathematics, a given set S is called countable if and only if either of the following two conditions holds:

S is finite. or,

There exists a bijection or one-to-one correspondence between the elements of set S and the set of all natural numbers N. In other words, the cardinality of set S is the same as the cardinality of N.

Method 1: Using Formal Definition

This is the most formal method to prove that a given set is countable or not. Follow the steps mentioned below:

If the given set is finite, that means it is already countable by definition.

If the given set is infinite, then try to create a one-to-one function between the given set and the set of all natural numbers, N.

Example

For example lets take the set of binary strings of finite length. The set of binary strings of finite length simply means all the finite length strings over the alphabet Σ = {0, 1} or in other words Σ*. In this article we are going to prove that Σ* is countable.

The given set is infinite. So according to the definition we have to create a one-to-one function between Σ* and N. Let’s define a function between Σ* and N.

w is a finite length binary string such that, w ∈ Σ*
length of w = n

f(w) = 1             n=0
f(w) = 2ⁿ + decimal_representation(w)

f(w) is a one-to-one function from Σ* → N, because,

For every two distinct w1,w2 ∈ Σ* we have a different value of f(w). That means f(w) is injective.
For every natural number x ∈ N, we have some w ∈ Σ* for which f(w) = x. That means f(w) is surjective.

Let us consider the following diagrammatic representation:

f(w) : Σ* → N

Now as we have successfully created a one-to-one function f(w) : Σ* → N, it is proven that Σ* is a countable set.

Method 2: Using Alternate Definition

Now as you can see from the first proof it is not always easy to come up with a one to one function from a given set S to N. So now we are going to use a more informal and easy, alternate definition to prove a given set is countable or not.

The Alternate Definition

A given set S is countable if and only if for all n ∈ N we can assign a empty or non empty finite subset of S such that every element of S is covered at least once in the entire mapping.

Example : Let’s take the same example of the set of binary strings of finite length as before and see how much easier our lives have become this time.

All we have to do is assign a finite subset of Σ* to each n ∈ N, such that all the elements of Σ* is covered at least once.

To each n ∈ N we are going to assign the subset of Σ* that consists of the strings of length n. There are exactly 2ⁿ strings of length n. Which is finite.

1 → {λ, 0, 1} empty string and all the strings of length 1, We are taking the empty string with 1 as it was being left out.

2 → {00, 01, 10, 11} All the strings of length 2

3 → {000, 001, 010, 011, 100, 101, 110, 111} All the strings of length 3

n → All the strings of length n

In this way we will be able to cover all the elements of Σ* at least once. Therefore Σ* is finite.

Method 2: Using Alternate Definition

Conclusion

In conclusion, understanding the concept of countable sets is essential in the realm of mathematics and computer science. A countable set is one that either has a finite number of elements or can be mapped one-to-one with the set of natural numbers, denoted as N. In this article, we explored two methods for proving whether a given set is countable or not.

The first method, using the formal definition, involves establishing a one-to-one function between the given set and N. This method can be rigorous and challenging, requiring careful construction of such a function. However, as demonstrated with the example of Σ* (the set of binary strings of finite length), it is a powerful approach when applied correctly.

The second method, based on an alternate definition, offers a more intuitive and straightforward way to determine countability. According to this definition, a set is countable if, for each natural number n, we can assign an empty or non-empty finite subset of the set that covers all its elements. This method simplifies the process of proving countability, as shown in the example of Σ*.

In summary, these methods provide valuable tools for classifying sets as countable or not, offering flexibility in choosing the most suitable approach for a given problem. Whether through formal definitions or alternate strategies, understanding countable sets is fundamental to various fields within mathematics and computer science.

FAQs on Prove that a given set is countable

1. Can an infinite set be countable?

Answer:

Yes, an infinite set can be countable if there exists a one-to-one correspondence between its elements and the natural numbers. For example, the set of all integers (positive, negative, and zero) is countable because it can be mapped to the natural numbers.

2. Is the set of real numbers countable?

Answer:

No, the set of real numbers is not countable. It is uncountably infinite, which means there is no one-to-one correspondence between real numbers and natural numbers. This result is known as the “uncountability of the real numbers.”

UGC NET CS 2015 Dec – II | Question 3

3. What is the difference between a countable set and a finite set?

Answer:

A finite set contains a specific, finite number of elements and is, by definition, countable. In contrast, a countable set can be either finite or infinite, as long as there exists a one-to-one correspondence with the natural numbers. Countable sets may or may not be finite.

4. How do countable sets relate to computer science and algorithms?

Answer:

Countable sets have practical implications in computer science and algorithms, particularly in analyzing the complexity of algorithms and data structures. They help determine the efficiency of algorithms when dealing with finite or countably infinite data sets, which is essential for designing efficient computational solutions.