This definition of a partial means that not it may be the case that not every element of A gets mapped to an element of B. This is not related to being one-to-one or being onto. In math a function from A to B means each element of A gets mapped to an element of B. With a partial function we allow the possibility that some a in A is not mapped to an element of B. In mathematical language we’d say that a partial function from A to B is a function to B with domain a subset of A. This can be made more rigorous and precise but I don’t feel like writing out the nuance involved.