My question is aroused by this article;
"By definition of a function, the image of an element x of the domain is always a single element y of the codomain. Conversely, though, the preimage of a singleton set (a set with exactly one element) may in general contain any number of elements. For example, if f(x) = 7 (the constant function taking value 7), then the preimage of {5} is the empty set but the preimage of {7} is the entire domain."
Here,I can understand that the preimage of the singleton set is the entire domain.But if so,then how does the inverse of a singleton function be a "function" according to the function definition as there exists more than one element map singleton set to preimage.
Best Answer
For a function to have an inverse, it has to be injective--that is, distinct domain elements must be taken to distinct codomain elements. In such a case, preimages of singletons will either be singletons or the empty set. However, preimages are defined even when a function doesn't have an inverse.