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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Why can't a base be negative in an exponential function?
The function $f(x) = a^x$ is generally taught to only allow $a > 0$. This is usually justified by giving a few examples of complex points in cases where $a < 0$. For example:
$f(x) = (-2)^x$, if ...
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1
answer
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Topological perspective of Rice-Shapiro theorem
Given a Gödel numbering $\phi_n$ of the partial recursive unary functions, the Rice-Shapiro theorem states
Let $\mathcal A$ be a family of partial-recursive unary functions such that its index set $...
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Partial functions over a finite set and the relationship between fixpoints of composed functions and individual ones.
As the title suggests, I am looking at partial functions over a finite set $\mathcal{A}$.
Lets denote by $Y_f$ the set of fixpoints of $f$, $C_f$ the set of changed values of $f$, and $B_f$ the points ...
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Define the partial function where if n is the Godel number of a statement with proof in formal arithmetic, then $f(n) = 1$; otherwise $f(n) = 0$.
I'm a bit confused with accurately defining partial functions. I'm trying to do the following exercise.
Define the partial function $f$ from $\mathbb{N} \to \mathbb{N} $ where if n is the Godel number ...
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Elliptic equations and variational formulation
My question: We now consider the following equation:
$$
\left\{\begin{array}{l}
-\Delta_x u+u(x)=f(x), \quad x \in \Omega \\
\left\langle\nabla_x u(x), \vec{n}(x)\right\rangle=g(x), \quad x \in \...
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1
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Find P if Integrating factor of $x(1-x^2)dy+(2x^2y-y-ax^3)dx=0$ is $e^{\int Pdx}$.
I have seen one method which correctly evaluates $P=\frac{2x^2-1}{x(1-x^2)}$.
But I have seen a method that says if $\frac{\frac{\partial M}{\partial y}-\frac{\partial N}{\partial x}}{N}$ is a ...
0
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1
answer
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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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1
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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?...
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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 piecewise defined function
I have the following "quick" question about the piecewise function integration:
Say, I have to find $\int\limits_0^{1}f(x)d x$, with $f(x)$ being piecewisely defined on $\mathbb{R}$ as ...
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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 ...
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Solve $u_t+cu_x=0$ with $c>0$ when $x,t >0$ and contions $u(x,0)=h(x)$ with $x>0$ and $u(0,t)=g(t)$ with $t>0$.
$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&...
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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 ...
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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$, ...
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1
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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}$ ...
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vote
0
answers
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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 ...
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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 ...
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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
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1
answer
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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 ...
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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
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1
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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 ...
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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 ...
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2
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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 ...
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1
answer
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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)}$$
Thank you.
- 13
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0
answers
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Equivalence of various notions of injectivity for partial functions
Let $A$ and $B$ denote sets.
Definition. Given a partial map $f : A \rightarrow B$, let us define that a partial inverse of $f$ is any partial map $g : B \rightarrow A$ satisfying $$fg \leq \mathrm{...
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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 ...
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1
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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 $\perp$ ...
user592630
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2
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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 $A$ ...
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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 ...
- 3,202
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2
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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{\...
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2
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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 $\mu$-...
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0
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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 $A$ ...
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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.
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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 ...
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1
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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 ...
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1
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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 ...
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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.
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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 ...
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1
answer
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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 $...
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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 ...
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1
answer
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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:
...
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0
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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, ...
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answer
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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, \...
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1
vote
1
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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 ...
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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{ ...
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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 ...
3
votes
1
answer
322
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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 ...
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3
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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 \...
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votes
1
answer
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Eager vs. lazy interpretation of recursive functions
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 ...
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