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.
Re:Partial functions and total functions
by eximius on Tuesday December 14, 2021 @01:36AM
(#29548887)
It does not imply one to one or onto.
Unique pair (a, b) only implies for any a, the corresponding b is unambiguous.
There could easily be (a1, b) and (a2, b), thus not one to one, and there could easily be a b1 such that there is no corresponding A, thus not onto.
"You can make partial functions rigorous by defining them to be relations rather than functions. A relation between sets A and B is a subset of their Cartesian product A × B. A function is a relation such that for every a in A there exists a unique pair (a, b) in the relation. A partial function is a relation in which for every a in A there is at most one pair (a, b) in the relation."
Does this equate with a partial function being one-to-one and onto? [0]
[0] See https://www.math.auckland.ac.nz/class255/08s1/04s2/week5.pdf