Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies
- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who
Difference Between Codomain and Range
In mathematics, functions play a vital role in describing and modeling various phenomena. A function is a rule that assigns a unique output value to each input value. The set of input values is called the domain, and the set of output values is called the range. However, in some cases, the range is not the same as the codomain, which can lead to confusion. In this essay, we will explore the difference between codomain and range.
Codomain of a Function
The “codomain” of a function or relation is a set of values that might possibly come out of it. It’s actually part of the definition of the function, but it restricts the output of the function. For instance, let’s take the function notation f: R -> R. It means that f is a function from the real numbers to the real numbers. Here, codomain is the set of real numbers R or the set of possible outputs that come out of it.
Domain is also the set of real numbers R. Here, you can also specify the function or relation to restrict any negative values that output produces. In simple terms, codomain is a set within which the values of a function fall.
Let N be the set of natural numbers and the relation is defined as
R = {(x, y): y = 2x, x, y $ \epsilon $ N}
Here, x and y both are always natural numbers. So,
Domain = N, and
Codomain = N that is the set of natural numbers.
Range of a Function
The “range” of a function is referred to as the set of values that it produces or simply as the output set of its values. The term range is often used as codomain, however, in a broader sense, the term is reserved for the subset of the codomain. In simple terms, range is the set of all output values of a function and function is the correspondence between the domain and the range.
In native set theory, range refers to the image of the function or codomain of the function. In modern mathematics, range is often used to refer to image of a function. Older books referred range to what presently known as codomain and modern books generally use the term range to refer to what is currently known as the image. Most books don’t use the word range at all to avoid confusions altogether.
For instance, let A = {1, 2, 3, 4} and B = {1, 4, 9, 25, 64}. The function f: A -> B is defined by f (x) = x ^2. So here, set A is the domain and set B is the codomain, and Range = {1, 4, 9}. The range is the square of A as defined by the function, but the square of 4, which is 16, is not present in either the codomain or the range.
Differences: Codomain and Range
To understand the difference between range and codomain, let's consider an example. Suppose we have a function "f" that maps each real number "x" to its square, f(x) = x^2. The domain of f is the set of all real numbers, and the range is the set of all non-negative real numbers. However, the codomain of "f" is also the set of all real numbers, because for any real number x, f(x) can take any real value, whether or not it is actually produced by "f".
Another example to consider is a function "g" that maps each real number "x" to its square root, g(x) = sqrt(x). The domain of "g" is the set of all non-negative real numbers, and the range is the set of all non-negative real numbers. However, the codomain of g is the set of all non-negative real numbers, because for any non-negative real number "x", g(x) can take any non-negative real value, whether or not it is actually produced by g.
From these examples, we can see that the range is always a subset of the codomain. That is, the range is a set of values that are actually produced by the function, while the codomain is a set of values that could be produced by the function. The range is determined by the function itself, while the codomain is a property of the function that is independent of the function's actual output values.
One important consequence of this difference between range and codomain is that a function can have multiple codomains, but it can have only one range. For example, consider the function f(x) = x^2 again. We have already seen that the range of f is the set of all non-negative real numbers. However, we could also define a new function g(x) = x^2 with a different codomain, such as the set of all complex numbers. In this case, the range of g would still be the set of all non-negative real numbers, but the codomain would be different.
Characteristics |
Codomain |
Range |
|---|---|---|
| Definition | Both the terms are related to output of a function, but the difference is subtle. While codomain of a function is set of values that might possibly come out of it, it’s actually part of the definition of the function, but it restricts the output of the function. |
Range of a function, on the other hand, refers to the set of values that it actually produces. |
| Purpose | Codomain of a function is a set of values that includes the range but may include some additional values. The purpose of codomain is to restrict the output of a function. |
The range can be difficult to specify sometimes, but larger set of values that include the entire range can be specified. The codomain of a function sometimes serves the same purpose as the range. |
| Example | If A = {1, 2, 3, 4} and B = {1, 2, 3, 4, 5, 6, 7, 8, 9} and the relation f: A -> B is defined by f (x) = x ^2, then codomain = Set B = {1, 2, 3, 4, 5, 6, 7, 8, 9} and Range = {1, 4, 9}. The range is the square of set A but the square of 4 (that is 16) is not present in either set B (codomain) or the range. |
If A = {1, 2, 3, 4} and B = {1, 2, 3, 4, 5, 6, 7, 8, 9} and the relation f: A -> B is defined by f (x) = x ^2, then codomain = Set B = {1, 2, 3, 4, 5, 6, 7, 8, 9} and Range = {1, 4, 9}. The range is the square of set A but the square of 4 (that is 16) is not present in either set B (codomain) or the range. |
Conclusion
In conclusion, the difference between codomain and range is an important concept in mathematics. The range is the set of all actual output values produced by a function, while the codomain is the set of all possible output values that the function could produce. The range is a subset of the codomain, and a function can have multiple codomains but only one range. Understanding this difference is important for correctly interpreting and using mathematical functions.
1K+ Views
To Continue Learning Please Login