# Questions tagged [partial-functions]

Questions on partial functions, history, usage, properties, significance for computability theory, connections to (inverse)-semi-groups and other algebraic theories.

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### How to distinguish partial functions from total functions?

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, ...
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### Can you define functions which are not primitive recursive, yet total, in Type Theory? [closed]

Ackermann's function is total but not primitive recursive. Can one define Ackermann's function in Type Theory, ie: Can you define functions which are not primitive recursive, yet total, in Type Theory?...
1 vote
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### A pathological partial function with pathological singularities

Motivation I was thinking about how computers should deal with partial functions. An example of such situation: I'm doing arbitrary real number computation and a division by zero must not fall into an ...
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### Quick question about integration of a partial function

I have the following "quick" question about the partial function integration: Say, I have to find $\int\limits_0^{1}f(x)\partial x$, with $f(x)$ being partially defined on $\mathbb{R}$ as ...
291 views

### Using Leibniz rule for multivariable integral.

How would I use the fact that integration is the inverse of differentiation, to find $∂F/∂x$ and $∂F/∂y$ for the case where : $F(x, y)$ = $\int_{0}^{xy^2} {e^{t^2}}\ dt$ I initially tried to use the ...
$u_t+cu_x=0$ show that $u(x,t)=f(ct-x)$. With conditions, it shows that $$u(x,0)=f(-x)=h(x),$$ $$u(0,t)=f(ct)=g(t)$$ I dont know how to find what $f(ct-x)$ is and what the purpose of condition: $c&... 2 votes 0 answers 41 views ### Analytic Continuation of Fractions Excuse my lack of expertise, I study natural sciences (physics) and not mathematics so I will be off with my terminology and mathematical vocabulary. Please feel free to poke and build at this idea ... 1 vote 0 answers 22 views ### Semilattice of the Left Inverse Hull This is a follow-up on this post, which is based upon this paper. First, let me set up some definitions, etc. A Semigroup$S$is said to be an inverse semigroup provided that for every$x \in X$, ... 1 vote 1 answer 81 views ### Inverse Semigroups, Partial Bijections, and Semilattice of Idempotents I have a question about a passage from this paper. First, some definitions A Semigroup$S$is said to be an inverse semigroup provided that for every$x \in X$, there exists a unique element$x^{-1}$... 1 vote 0 answers 46 views ### Notation for the scope of definition of a partial binary operation of a groupoid I have a groupoid${\displaystyle (G,\ast )}$with a partial binary operation${\displaystyle *:G\times G\rightharpoonup G}$. For every$(a,b)\displaystyle ∈G\times G$,$\displaystyle *$is defined if ... 0 votes 1 answer 111 views ### Function vs Partial Function Generally speaking, can we say that "any function is a partial function"? I know what is exactly a partial function, but is it correct to imply that arbitrary function can be seen as a ... 0 votes 1 answer 48 views ### Confusion regarding Computability and Recursion I'm reading about Computability and so far I have found it a bit difficult to understand, given that I have a narrow mathematical background, and that often times I would find different names for the ... 1 vote 1 answer 55 views ### Number of bijective partial functions between two finite sets? Suppose we have two finite sets$A, B.$If we are to create a partial function$A\to B,$for each$a\in A$we have$|B|+1$choices for the image of$a.$Hence there must be$(|B|+1)^{|A|}$partial ... 0 votes 0 answers 36 views ### Family of unary partial computable functions having total computable extension is computable. I have to prove: Family of unary partial computable functions having total computable extension is computable. But it is not so obvious for me. Here i provide some definitions: If the function$h$... 1 vote 1 answer 206 views ### How can I show that a computably enumerable set is the range of a partial computable function In recursion theory, by definition, a computably enumerable set (c.e.) is the range of a total computable function. However, I came across a textbook which asks to show how a c.e. set can also be the ... 0 votes 1 answer 34 views ### How to represent sections in math notation? E.g. (_ + 3) Functional programming languages have the concept of sections where they can partially apply a function that is represented by an operator. For instance, one can define a function that always adds ... 1 vote 2 answers 70 views ### Equivalence Relations and Partial Order - Symmetry and Anti-Symmetry I am confused on how to check symmetry and anti symmetry I came to a conclusion that '==' is symmetric, but can it be anti-symmetric? '>=' function is reflexive and transitive. But I cannot determine ... 1 vote 1 answer 745 views ### Inverse Laplace with a irreducible quadratic in the denominator and a 1 in the numerator Please help me find the Inverse Laplace transform of: $$F(s) = \frac{1}{s(s^2+8s+4)}$$ After completing the square, I obtained $$F(s) = \frac{1}{s((s+4)^2-12)}$$ Thank you. 1 vote 0 answers 41 views ### Equivalence of various notions of injectivity for partial functions Let$A$and$B$denote sets. Definition. Given a partial map$f : A \rightarrow B$, let us define that a partial inverse of$f$is any partial map$g : B \rightarrow A$satisfying $$fg \leq \mathrm{... 2 votes 3 answers 1k views ### The use and meaning of the "tilde-equal-symbol" for partial (recursive) functions in Girards monograph 'proof theory and logical complexity, volume 1' English is not my native language, so please forgive if I do not express myself properly. I am working with the book named above on proof theory, and I have a little problem with the authors use of ... 1 vote 1 answer 74 views ### Can I give undefined (\perp) as an argument to my function? Hopcroft & Ullman (1979) say that a function f(x) is undefined when f is not defined for x and they use (I think) the \perp symbol to denote that. My question is: since I can use \perp ... 2 votes 2 answers 60 views ### The logic behind adding this sentence to the second condition in this definition of isomorphism Below is the definition of isomorphism quoted from the textbook Introduction to Set Theory by Karel Hrbacek and Thomas Jech. First, we introduce relevant definitions: An n-ary relation R in A ... 2 votes 1 answer 113 views ### What equivalence relation is being used to define the category of partial maps? Here's what Awodey says in his Category Theory. For any category \mathbf{C} with pullbacks, define the category \mathbf{Par}(\mathbf{C}) of partial maps in \mathbf{C} as follows: the objects ... 0 votes 2 answers 40 views ### Directional Derivative with partial function I have been trying to do this question but I'm completely lost. Be f(x,y) a differentiable function, the maximum value of Duf(0,2) is equal to 2 and occurs when u = (\frac{\sqrt 2}{2},\frac{\... 4 votes 2 answers 175 views ### Definition of (lazy) conditional in partial recursive functions I'm currently working on formalizing the theory of partial recursive functions, and something seems peculiar about the standard definition. The definition of the primitive recursion clause of \mu-... 1 vote 0 answers 24 views ### Anything special on the ratio of partial over total functions? The number of functions that maps an element in A to another one in A is |A|^{|A|} for finite size of A, and the number of partial functions that maps an element in A to another one in A ... 0 votes 1 answer 358 views ### Prove that a language A is c.e. iff it is the range of a partially computable function I am having trouble in formally proving this statement. I looked online and most proofs just mention that it is the part of the definition of c.e (computably enumerable) language. 0 votes 1 answer 565 views ### The average values of a harmonic function over the ball and surface of the ball are the same? This is the proof I saw in my text. I can understand all the algebraic manipulations in the proof. But, what I am not sure about is the deriative of the function Φ(r). As far as I know, the average ... 1 vote 1 answer 77 views ### In what way is a partial function different from a partial transformation when both can always express the other's associated total function? 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 ... 1 vote 1 answer 96 views ### How do I show that this relation is a partial function? I don't quite get the definition yet. How do I show that a relation R : A \mapsto B is a partial function iff R \circ R^{-1} \subseteq id_B? I'm getting multiple definitions of partial ... 3 votes 1 answer 846 views ### Example function which is partial, injective, and surjective Can somebody give an example of a function f : \mathbb{N} → \mathbb{N} which is partial, injective, and surjective. I was thinking about f(x)=x-1, but I am not sure if it is surjective. 1 vote 1 answer 192 views ### Hasse Diagram and Partial Orders Consider a partial order on piles of 2 black stones and 2 white stones. Say that one configuration of pile of stones,A, is smaller than another,B, if you can join piles ... 1 vote 1 answer 158 views ### When is the \mu operator a partial function vs. total? In my computability theory course notes, I have written that, if f:\mathbb{N}^{n+1} \to \mathbb{N} is a partial function, then the function obtained from f by minimisation is the partial function ... 0 votes 0 answers 302 views ### Is this a total function, a partial function, or just a total, single-valued relation? Is the following a total function, a partial function or total, single-valued relation? h: \mathbb{N} \rightarrow \mathbb{N}, where h(n) = \Bigg\{0 if n is divisible by 2, 1 if n is ... 0 votes 1 answer 85 views ### Extension of definable functions in o-minimal structures The proof of the following theorem Let M be an o-minimal structure, and let a,b\in M,A\subseteq dom(M). If a\in acl(b,A) and a\notin acl(A), then b\in acl(A). starts like this: ... 1 vote 0 answers 57 views ### Partial sum of infinite series In a probability excercise I concluded that the expected value is the following infinite series: \sum_{i=1}^{\infty} \frac{2i}{2^i} I wrote down the first few terms and saw that it converges to 4, ... 2 votes 1 answer 96 views ### Showing \mathscr{P}_n=\langle\zeta, \tau, \pi, \xi\rangle. This is Exercise 1.9.13 of Howie's "Fundamentals of Semigroup Theory". The Details: Definition: The partial transformation semigroup \mathscr{P}_n is the set of all partial maps from \{1, 2, \... 1 vote 1 answer 140 views ### Soft question on notation regarding partial bijections I'm not sure if the following technicality should just be ignored, or if I should just learn more category theory (of which I know very little), but here it goes anyway. Let \mathcal{C} be the ... 2 votes 1 answer 187 views ### Union of Partial Functional relations A relation R is said to be partial functional if for all x, if y R x and y' R x, then y=y'. The union of two relations R and S is defined as follows:$$y (R \cup S) x \iff y R x \text{ ... 0 votes 0 answers 1k views ### Is this Hasse Diagram correct? I have the following question Let, A = {{1}, {2}, {3}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {1, 2, 4}, {1, 3, 4}, {1, 2, 3, 4}} and (A, R) is a POSET where R is the partial order a ⊆ b. I have drawn this ... 1 vote 1 answer 232 views ### Number of Partial Surjective Functions from X to Y The number of total surjective functions from$X$to$Y$is known to be$T = y!\left\{{x \atop y}\right\}$, with$|X|=x, |Y|=y$. However, I am interested in the number$P$of partial surjective ... 1 vote 3 answers 190 views ### Which properties of total functions are absent for partial functions? For a partial function$p:X\to Y$, we have$p^{-1}(A\cap B)=p^{-1}(A)\cap p^{-1}(B)p^{-1}(A\cup B)=p^{-1}(A)\cup p^{-1}(B)p^{-1}(A \setminus B)=p^{-1}(A) \setminus p^{-1}(B)p^{-1}(A \...
One of the ways of defining the set of recursive functions is to define first a language $L$ by induction in the following way: $\mathsf{Z}^1 \in L$; $\mathsf{S}^1 \in L$; $\mathsf{P}^n_k \in L$ for ...