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Using the definition given by wikipedia https://en.wikipedia.org/wiki/Partial_function a partial function $f:X\nrightarrow Y$ is just a function $f:X'\to Y$ for some subset $X'\subseteq X$ analgous to the way for a categorically defined function its image is some subset of its co-domain/target set. However they go on to say a "partial transformation" is a partial function from some set to itself. Yet by that description I could define a partial function $f:X\cup Y\nrightarrow X\cup Y$ analgous to our previous one by just throwing in extra values its not defined at or doesn't map to until I get one that maps from the same set back to itself and yet corresponds to an identical function $f:X'\to Y$ as it did previously.

So am I missing something here? In short it appears a partial function is just a functional relation equipped with two externally defined sets which just have to contain the domain/image. Though since we can always construct external sets equal to each other satisfying this whose associated functions are the same, it appears to me that the distinction between a partial function and a partial transformation is entirely arbitrary. Can anyone clear this up for me? Where did I go wrong?

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I don't think you went wrong anywhere, but at the same time, I don't think it's fair to say that a concept is completely arbitrary just because it can be interpreted as an instance of another concept.

Forgetting the 'partial' part for a moment, consider the difference between a function and a transformation (in the sense of a function from a set to itself). A transformation is a kind of function, and every function $f : X \to Y$ induces a transformation $f' : X \cup Y \to X \cup Y$ given by letting $f'(t)=f(t)$ if $t \in X$ and $f'(t) = t$ if $t \in Y \setminus X$. However, even though a function can be regarded as a transformation, and vice versa, each notion is useful for different reasons.

As another example, a function $f : X \to Y$ can be regarded as a relation $R$ on $X \sqcup Y$, where $t\,R\,t'$ if and only if $t \in X$, $t' \in Y$ and $t\,R\,t'$. Conversely, a relation $R$ on a set $X$ can be regarded as a function $f : X \times X \to \{ 0, 1 \}$ by letting $f(x,x')=1$ if $x\,R\,x'$ and $f(x,x')=0$ otherwise. This doesn't make the notions of function or relation redundant, it just allows us to translate from one context into another.

The same is true in the 'partial' case. Sometimes it might be convenient or important to view a partial function $f : A \rightharpoondown B$ as a partial transformation $X \nrightarrow X$ for some set $X$ containing both $A$ and $B$ as subsets, for example.

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