Dear sir, I have this question about the domain, range and co domain of a function.
What is the correct definition for the co domain and what is the importance of it ?
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Codomain Y is a set stated by the definition of a function such that all outputs of the function need to be within it , in other words it is a set of all possible outputs. Range R is the set of all outputs of a function and which can be determined by the relation and domain of the function. Therefore by definition R should be a subset of Y. If R = Y function maps it’s domain onto codomain , if R ⊂ Y function maps it’s domain into codomain.
To get a better insight of the issue we can consider probability function because it has practical importance and we can clearly see the difference of the two sets . Here the domain is Event space and codomain is the set [0,1] but range is only particular values within the set [0,1]. In the case of tossing a coin once range is the set {0,1/2 , 1} .
Thank you very much sir.
Actually it should be tossing a fair coin not just a coin.
Of course 😊
For a function f : A→B, the set A is called the domain
of f . (Think of the domain as the set of possible “input values” for f .)
The set B is called the codomain of f .(Think of the codomain as a sort of “target” for the outputs.)
The range of f is the set { f(a) : a ∈ A} ( Think of the range as the set of all possible “output values” for f .)
I use the sport of archery to understand this concept. You can think of an arrow as an input value (an arrow can hit only one place on the target board which is also similar to a function having only one relationship for each input value. Finally satisfying one condition to be a function) and the bow as the function. The target board is the codomain and the places where the arrows hit on the target board are called range. Here the target board cannot be set up by the competitor because it is already set by the organizers.
Consider the function f : Z→N, where f (n)= |n|+5.
The domain is Z and the codomain is N.
The range of this function is the set { f(a) : a ∈ Z}={ |n|+5 : a ∈ Z}={5,6,7,…}
Notice that the range is a subset of the codomain N, but it does not (in this case) equal the codomain.
In general, the range of a function is a subset of the codomain. In this sense, the codomain could have been any set that contains the range.
We might just as well have said that this f is a function f : Z→Z, or even f : Z→R.
♣ This illustrates an important point: the codomain of a function is not an intrinsic feature of the function; it is more a matter of choice or context.
In general, the codomain of a function can be any set that contains the function’s range as a subset.
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