# 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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24 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 ...
15 views

### Partial Functions and Relations

Is it possible to infer that "Any function is a partial function" is true? If not, what are some counterexamples for that (for example, $f(x) = \frac{1}{x}$)? I know that any partial ...
29 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 ...
31 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 ...
25 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 $h$ ...
34 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 ...
27 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 ...
39 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 ...
271 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.
39 views

825 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 ...