Is a total function also a partial function?
Is there a consensus on whether a total function, i.e., a function defined for each element of the domain, is also a partial function?
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$\begingroup$When someone says "partial function", the usual interpretation is that the function may or may not be defined on the entire domain. The word is also sometimes used with the meaning "not total", but that meaning is relatively rare and will usually only be understood in contexts where the ordinary meaning would be clearly senseless.
The unambiguous way to say that a function is not total is "not total".
Note that in almost all mathematical subfields, the word "function" alone means "total function"; we only add the word "total" when there's a risk that the reader might otherwise think we were allowing non-total ones, too.
$\endgroup$ $\begingroup$Yes. A partial function from $A$ to $B$ is a total function $U \to B$ for some $U \subseteq A$; under this definition a total function is just a partial function with $U=A$. This is valid since $A \subseteq A$.
$\endgroup$ 1 $\begingroup$Yes, I think generally "total" anything is usually regarded or defined as a "partial" things that happens to be total. Usually in practice, one defines the more general partial concept first and then defined the total concept afterward by adding the additional totalness condition to the original definition.
For example, you must may have seen the concept of a partial ordering. A linear ordering or total ordering is just a partial ordering in which everything is comparable.
In terms of functions, in computability theory this convention is actually used, the partial computable functions are the function given by Turing Machines but may not be defined on all $n \in \omega$. (Intuitively, algorithms or computer programs do not necessarily halt on all inputs.) Afterward, the total computable functions (or computable functions for short) are the total partial computable functions, or less akward sounding the partial computable functions that are defined for all $n \in \omega$.
$\endgroup$ $\begingroup$In Set Theory (the basis of almost all mathematics) there is no such a thing as "partial function". Let see:
1) Given a set R is said to be relation if there are a couple of sets A, B such that R is a subset of AxB (the Cartesian product of A and B).
The domain of R (denoted dom(R)) is defined as dom(R) = {x $\in$ A / $\exists$ y $\in$ B / (x,y) $\in$ R}
The range of R (denoted ran(R)) is defined as ran(R) = {y $\in$ B / $\exists$ x $\in$ A / (x,y) $\in$ R}
Notice that domain and range are defined for relations not for functions but as we will see next, a function is a special kind of relation.
2) Given a relation F is said to be a function if and only for all (x1,y1) in F and for all (x2,y2) in F if x1 = x2, then y1 = y2.
Notice that A and B can be any set. For example, A = $\mathbb{R}$ $\cup$ {a1,a2} B = $\mathbb{N}$ $\cup$ {b1,b2} the set F={(a1,b1),(a2,b2)} is a subset of AxB, so it is a relation that is also a function (Why?).
Now some terminology (there are small variants):
A function f: A $\rightarrow$ B (that is: f $\subset$ AxB):
Is say to be in A if dom(f)=A
Other variant:
Is say to be from A to B if dom(f)=A
The origin of the mathematically wrong terminology is the idea in other disciplines (computer science of physics for instance) that a function is a "rule", so maybe natural to think that that "rule" works for some elements and for other does not works. In that case "partial rule" or "partial function" may have sense. But the term "partial function" is virtually inexistent in mathematics and should not be used.
A specific ordered pair (x,y) is in a function f or not, so there is no room for "partial".
$\endgroup$ 1 $\begingroup$My understanding is that a function 'from $A$ to $B$' is a relation $R$ (subset of $A \times B$) that is right-unique: for every $a \in A$, there cannot be $b_1$ and $b_2$ where $(a,b_1) \in R$ and $(a,b_2) \in R$
This means that the relation need not be left-total, i.e. it need not be the case that for all $a \in A$, there exists some $b \in B$ such that $(a,b) \in R$
When people talk about a total function, they clearly mean to talk about a function that is left-total.
When people talk about a partial function, things are a little less clear, as I think there are two different uses for it:
Sometimes, a 'partial function' is meant to be a function that is not total.
Other times, a 'partial function' is a function that may or may not be total.
The second use would be compatible with how you apparently saw the use of partial function. But I believe the first usage is more common, and that is for two reasons:
With the second usage, 'partial function' becomes synonymous with just 'function'. So .. what then is the point of adding 'partial'?
In many branches of mathematics, the functions we are dealing with are total. As such, it is often assumed that the use of 'function' implies that it is a total function. Or maybe better put: in the rare case where a function is not total, we say that the function is 'partial' (i.e. we use usage 1) ... and therefore if in those contexts we say 'function' we implicitly mean that it is a total function (because otherwise we would have said 'partial function').
But no, I don't think there are really any super-hard rules for this that universally apply across different contexts. Some will insist that a 'partial function' is one that is not total .. but others will say that a partial function is just a function.
In fact, some will insist that a 'function' is one that is a (left-)total function, whereas others will simply sick to right-uniqueness. So, we don't even seem to have such a super-hard rule for the definition of a 'function' that applies across different contexts. And indeed, one can regard the ambiguity of 'function' to be the soucse of the ambiguity of 'partial function'.
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