# 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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### 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
25 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 $A$ ...
540 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.
1 vote
812 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
103 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
138 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 ...
1k 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
221 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
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1 vote
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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 ...
221 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{ ...
2k 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 ...
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 ...