How to distinguish partial functions from total functions?

Question

How does one distinguish partial functions from a corresponding total function with a restricted domain? For example, I can consider the total function from non-negative reals to reals which gives, for any non-negative real number $x$, its non-negative square root. I want this function to be a different entity than the partial function from reals to reals which is undefined for negative real numbers, and gives the non-negative square root for non-negative real numbers. Is there a set-theoretic implementation that makes these two entities distinct entities?

Answer 1

Remember that there are two (broadly speaking) ways to implement a (total) function in set theory: as a mere set of ordered pairs, or as a set of ordered pairs together with an explicit choice of codomain. The former implementation doesn't let us talk about surjectivity, while the latter does.

Once we bring partiality into the picture, we have (again, broadly speaking) three ways of implementing a function, the new addition being as a set of ordered pairs together with an explicit choice of codomain and domain, the latter of which is no longer redundant. At this point partiality vs. totality is easy to express.

Ultimately you simply have to pick the right implementation choice for the type of object you want to consider. Not every method of implementation will be appropriately flexible. We "get" to be fairly blithe about this due to the ease of translation between different natural implementations.