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} .