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?
Is a total function also a partial function?
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2I would guess so. In programming language semantics, a partial function $A\to B$ is understood exactly as a function $A\to B\cup \{\bot\}$, where $\bot$ is a special value that indicates failure or nontermination. So programming language researchers all think of total functions as a special case of partial functions; they are functions for which $\forall x.f(x)\ne\bot$. – 2012-08-01
4 Answers
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.
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 $n \in \omega$.
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 $A \subseteq A$.
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".