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Codomain and Range

Are you eager to learn about the codomain and the range and the difference between the two? If yes, then this guide will provide you with all the information related to the codomain and the range.

Often the term codomain and the range are used interchangeably and sometimes, they are considered as one. As a result, students get confused between the codomain and the range.  Students are confused between the two terms. But there is a minute difference between the two. This article will clear all the doubts related to the codomain and the range. Furthermore, it will highlight the key differences between the two.

What is a Set?

A set represents a model for the well-defined collection of different elements. The elements contained in the set may be anything like numbers, alphabets, symbols, geometrical shapes, and many more. 

Types of Sets

  • A set containing no elements is known as a null set
  • A set that contains only one element is known as the singleton set
  • A set that contains a finite number of elements is known as a finite set and the one that contains the infinite number of elements is known as an infinite set
  • The sets whose elements are the same are known as equal sets
The concept of Codomain and Range A set is formed by matching two functions. First, the two functions are given and a condition between the two is defined. Then, the condition is made to satisfy the elements of one function with the other. Finally, if one element of each function satisfies the given condition, then the elements are inserted into the set. This is how a set is formed.  A set consists of a domain, codomain, and range. For example, we can consider a function that gives the output as the square of the number or a function that gives the output only the positive numbers. To clearly differentiate between the input and output of a function, range, domain, and codomain are used. These terms are described below: Domain: Domain is referred to as the input of the function. Codomain: Codomain is the set of values that can come out as the function’s output. Range: Range is the set of values that comes out as the function’s output. Example Explaining Domain, Codomain, and Range Consider the square function defined as f(x) = x2. Let the domain and the range of the function be only the natural numbers.  So, we get,  Domain = {1, 2, 3, 4, 5, 6, ….} Range = {1, 2, 3, 4, 5, 6, ….} The required set is {(1,1), (2,4), (3,9), (4,16), ….} Hence, we can see that the required codomain is {1, 4, 9, 16, …}

Difference between the codomain and the range

Codomain Range 
It is the set of all values that can come as the function’s output The set of values comes out as the function’s output
In other words, the codomain can also be defined as the range plus the other set of values The range can be defined as the subset of the codomain
The cardinal number of the codomain is greater than the cardinal number of the range The cardinal number of the range is less than the cardinal number of the codomain
It is mainly related to the definition of the function It is mainly related to the image of the function

Conclusion

The difference between the domain and range is obvious, but the difference between the codomain and the range is very confusing. This article highlights the key differences between the codomain and the range with an example. The codomain is the set of all values that can come as the function’s output. The range is the set of values that actually comes out as the function’s output. We hope that this article was useful for the readers and now the difference between the codomain and the range has been clarified. 
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