partial function

a partial function $\text{[math]}$ from a set $\text{[math]}$ to a set $\text{[math]}$ is an assignment to each element $\text{[math]}$ in a subset of $\text{[math]}$ , called the domain of definition of $\text{[math]}$ , of a unique element $\text{[math]}$ in $\text{[math]}$ . the sets $\text{[math]}$ and $\text{[math]}$ are called the domain and codomain of $\text{[math]}$ , respectively. we say that $\text{[math]}$ is undefined for elements in $\text{[math]}$ that are not in the domain of definition of $\text{[math]}$ . when the domain of definition of $\text{[math]}$ equals $\text{[math]}$ . we say that $\text{[math]}$ is a total function.
[cite:;taken from @discrete_kenneth_2018 chapter 2.3.6 partial functions; definition 13]

some stuff from college

let $\text{[math]}$ be non-empty sets and let $\text{[math]}$ be a relation from $\text{[math]}$ to $\text{[math]}$
we say $\text{[math]}$ is a partial function from $\text{[math]}$ to $\text{[math]}$ if $\text{[math]}$ is single-valued meaning that for every $\text{[math]}$ there exists at most a single $\text{[math]}$ such that $\text{[math]}$

$\text{[math]}$

equality of partial functions

the functions $\text{[math]}$ are equal if:
$\text{[math]}$

of the following functions only 2 are equal
$\text{[math]}$