Before my reading on Linear Algebra by Hoffman/Kunze, I was under the impression that a function was "a rule (or mathematical object) that maps/assigns each $x \in X$ (domain) to an element $y \in Y$ (codomain)". However, stumbling across Hoffman/Kunze's LA book, in the appendix, it defines a functions as a mathematical object which consists of:
- Domain $\longrightarrow$ set of possible inputs
- Co-domain $\longrightarrow$ set of possible values outputted
- Rule $\longrightarrow$ associates each element in the domain to a single element in the codomain
This got me confused as I previously interpreted the rule being the whole function, not being only a part of the function. This changed my understanding because functions were no longer just "a rule that maps..." but instead an object that consisted of a rule and other parts. I did some researching and stumbled across a few "more precise" (but less common) definitions where it also stated that function as three parts. For example, Joe's answer here states that a function is...
Definition. A function is an (ordered) triple ($X$, $Y$, $f$), where $X$ and $Y$ are sets, and $f$ is a subset of $X \times Y$ satisfying the following properties:
- For every $x \in X$, there is a $y \in Y$ such that ($x$, $y$) $\in f$.
- For every $x \in X$, and for all $z$, $z' \in Y$, if ($x$, $z$) $\in f$ and ($x$, $z'$) $\in f$ then $z=z'$.
Additionally, a few other sources (that I could find online) define it nearly the same way (with a few saying $f$ is a graph instead of a rule), such as Asaf Karangila's answer, Reed College's Math 111 Lecture Note, Topoi by Robert Goldblatt, etc.
Eventually, I concluded my search and problems by accepting the ordered triple definition of the function. I told myself that the reason why the less precise definition of the function defined it as a rule was because of something that Joe and my LA book mentioned in common. Taken from the same answer by Joe, it says:
Commentary. If our definition is to be taken seriously, then $f$ is not the function: rather it is the graph of the function. Nevertheless, it is conventional to abuse notation and refer to the triple ($X$, $Y$, $f$) as $f$ for short.
Similarly, in Hoffman/Kunze's LA book, it comments:
If ($X$, $Y$, $f$) is a function, we shall also say $f$ is a function from $X$ into $Y$. This is a bit sloppy, since it is not $f$ which is the function; $f$ is the rule of the function. However, this use of the same symbol for the function and its rule provides one with a much more tractable way of speaking about functions.
To clarify, I told/convinced myself that functions are commonly defined as rules because people would "sloppily" abuse the notation $f$ and refer it to both the function and its rule, which therefore caused an adaptation to the definition of a function being a rule.
However, I still feel that my reasoning isn't perfect and in some way, still don't understand whether the definition of a function as a rule is correct or not and why people define it like that. Is it saying something different from the precise definition of a function, or am I not understanding something correctly? Are there any flaws to the rule definition of the function?