Codomain
Since you may be dealing with functions and relations unto their depths, knowing better about domain and range sets would be considerable.
Codomain of a function
The codomain of a function is known to be its set of possible outputs. In other words, codomain is a set of elements that may possibly and logically be produced by the function and the inputs that may be entered.
For instance, consider the use of function notation f: R→R, It would certainly mean that f is a function from real number to real number. In other words, it could be said that the codomain of f is a set of real numbers R (also the set of its possible inputs or domains is also supposed to be a set of real numbers RR).
Now, just because an object exists as the codomain of a function, it does not imply that it would come out as an outcome for the inputs entered for the function. Consider an example,
Suppose that we have defined a function f: R→R, $f(x)=x^{2}$. Here, since f(x) would always be non-negative, the number -3 despite being a codomain of the function cannot come out as an outcome since there are no inputs that could actually result in a negative outcome. The set of all outputs that would be received from putting in all inputs into the function is called range. While range is a set of non-negative real numbers, codomain is a set of all considerable real numbers.
It may now be clear to you that a “codomain of a relation or a function is a set of values which include the range as explained earlier and can include additional values apart from the ones in the range.
Codomains are of importance in the following cases:
- When you are required to restrict the outputs of a considered function. For instance, by specifying a codomain as a “set of positive real numbers”, you may be instructing the ones who’re ignoring any negative values while using the function.
- It might be difficult to specify the range exactly. However, a larger set of numbers that includes a few that could possibly be the part of the range can surely be specified. For instance, a codomain could define a set of entirely positive real numbers even though; a function does not generate all positive real numbers.
Since the range is quite difficult to be specified, thinking about the codomain could help in attaining the range.