Please help prove the following statement:

The indexed family of all unary partial computable functions that have total computable extension is computable.

Definitions:

1. If the function $h$ is obtained from the computable functions $f$ and $g$ and they are defined everywhere, then $h$ is a total function.

2. A function $g$ is called an extension of the partial function $f$ if $\operatorname{Dom}(f) ⊆ \operatorname{Dom}(g)$ and $g(x) = f(x)$ for any $x \in \operatorname{Dom}(f)$, where $\operatorname{Dom}(f)$ is the domain of the function.

3. Let $S$ be nonempty countable set (possibly finite). Any surjective map $\nu\colon \omega \twoheadrightarrow S$ from the set $\omega$ of natural numbers onto the set $S$ is called an enumeration .

4. An indexed family of functions is called computable if it has at least one computable enumeration.

Additional information:

• The indexed family of all unary computable functions is not computable.

Please help prove the following statement:

The indexed family of all unary partial computable functions that have total computable extension is computable.

Definitions:

1. Total function - a function which is defined for all inputs of the right type, that is, for all of a domain.

2. A function $g$ is called an extension of the partial function $f$ if $\operatorname{Dom}(f) ⊆ \operatorname{Dom}(g)$ and $g(x) = f(x)$ for any $x \in \operatorname{Dom}(f)$, where $\operatorname{Dom}(f)$ is the domain of the function.

3. Let $S$ be nonempty countable set (possibly finite). Any surjective map $\nu\colon \omega \twoheadrightarrow S$ from the set $\omega$ of natural numbers onto the set $S$ is called an enumeration .

4. An indexed family of functions is called computable if it has at least one computable enumeration.

Additional information:

• The indexed family of all unary computable functions is not computable.

Please help prove the following statement:

The indexed family of all unary partial computable functions that have total computable extension is computable.

Definitions:

1. If the function $h$ is obtained from the computable functions $f$ and $g$ and they are defined everywhere, then $h$ is a total function.

2. A function $g$ is called an extension of the partial function $f$ if $\operatorname{Dom}(f) ⊆ \operatorname{Dom}(g)$ and $g(x) = f(x)$ for any $x \in \operatorname{Dom}(f)$, where $\operatorname{Dom}(f)$ is the domain of the function.

3. Let $S$ be nonempty countable set (possibly finite). Any surjective map $\nu\colon \omega \twoheadrightarrow S$ from the set $\omega$ of natural numbers onto the set $S$ is called a numbering.

4. An indexed family of functions is called computable if it has at least one computable numbering.

Additional information:

• The indexed family of all unary computable functions is not computable.

Please help prove the following statement:

The indexed family of all unary partial computable functions that have total computable extension is computable.

Definitions:

1. If the function $h$ is obtained from the computable functions $f$ and $g$ and they are defined everywhere, then $h$ is a total function.

2. A function $g$ is called an extension of the partial function $f$ if $\operatorname{Dom}(f) ⊆ \operatorname{Dom}(g)$ and $g(x) = f(x)$ for any $x \in \operatorname{Dom}(f)$, where $\operatorname{Dom}(f)$ is the domain of the function.

3. Let $S$ be nonempty countable set (possibly finite). Any surjective map $\nu\colon \omega \twoheadrightarrow S$ from the set $\omega$ of natural numbers onto the set $S$ is called an enumeration .

4. An indexed family of functions is called computable if it has at least one computable enumeration.

Additional information:

• The indexed family of all unary computable functions is not computable.

Please help prove this statement.

"The indexed family of all unary partial computable functions that have total computable extension is computable."

Definitions.

1. If the function h is obtained from the computable functions f and g and they are defined everywhere, then h is a total function.

2. A function g is called an extension of the function f (partial function) if Dom(f) ⊆ Dom(g) and g(x) = f(x) for any x ∈ Dom(f). Where Dom(f) is the domain of the function.

3. Let S be nonempty no more than a countable set. Any map ν:ω->>S (surjection) from the set ω of natural numbers onto the set S is called numbering

4. An indexed family of functions is called computable if it has at least one computable numbering.

Additional Information:

The indexed family of all unary computable functions is not computable

Thanks in advance!

Please help prove the following statement:

The indexed family of all unary partial computable functions that have total computable extension is computable.

Definitions:

1. If the function $h$ is obtained from the computable functions $f$ and $g$ and they are defined everywhere, then $h$ is a total function.

2. A function $g$ is called an extension of the partial function $f$ if $\operatorname{Dom}(f) ⊆ \operatorname{Dom}(g)$ and $g(x) = f(x)$ for any $x \in \operatorname{Dom}(f)$, where $\operatorname{Dom}(f)$ is the domain of the function.

3. Let $S$ be nonempty countable set (possibly finite). Any surjective map $\nu\colon \omega \twoheadrightarrow S$ from the set $\omega$ of natural numbers onto the set $S$ is called a numbering.

4. An indexed family of functions is called computable if it has at least one computable numbering.

Additional information:

• The indexed family of all unary computable functions is not computable.