Ambiguities in the meaning of the term "function"

Question

In my mathematics class, i have learnt that a function from a set $A$ to a set $B$ is a subset of the cartesian product of $A$ and $B$, that is, $A$ X $B$ with the condition that the first sub-element of each element of the subset are distinct and that all the elements of A are present.

In some books i have read that it is a rule which which connects or maps the values of two variable quantities. Recently i saw that in a book it is written y is a function of x.

What exactly is a function and is it related to variables or sets?

Answer 1

To be very informal, functions are a way of representing any relation between two quantities, on the condition that they are uniquely related. Why is it that function are defined in such a way that the relation is unique? It is because otherwise functions would not be that useful as they are. Consider any expression in some variable $x$: $2x-5$, $x\sin(e^x)$, $\ln(\sin x)$, anything. And, try to calculate the values of each of these expressions for some particular $x$. Do the expressions produce multiple values, or just a single value?

Rigorous approach: Let there be two sets $A$ and $B$. Then, a function $f$ from $A$ to $B$, written as $f:A\to B$, is a subset of the cartesian product $A\times B$, such that all $(x,y)\in f$ have one and only one $y$ for each $x$. Here, $A$ is called the domain of $f$, and $B$ is called the co-domain of $f$. For all $y$ in $(x,y)\in f$, the notation $y=f(x)$ is used. The domain of $f$ can also be thought of all the arguments(inputs) $f$ can take to produce well defined results. And finally, the set of all $f(x),x\in A$ is called the Range of $f$, and intuitively, is the set of all outputs the function produces.

Answer 2

Intuitively and conceptually and woefully informally a function is a mapping between groups so that each input item in the Input Group goes to an item in the Output group.

But that in informal and not rigorous.

1) So that is why we have the first definition. It's rigorous and tangible:"a subset of the cartesian product of two sets(A, B;A X B) with the condition that the elements (ordered pair) of the subset(a,b) do not have more than 2 same first element(a) and that all the elements of the original first set(domain)(A) are present "

The simple act of being in a pair serves as a "mapping" and the restriction that every element of the left Set (A) of the Cartesian product is present in exactly one pair and one pair only, assures that this is a mapping of individual items in the Input Group (which is the left Set(A)).

The only problem with this definition is that it is abstract and it is not intuitively obvious (at least not to me) that it really is the same thing as the intuitive idea of a "mapping". It IS the same but.... one needs to wrap one's mind around it and practice before it becomes intuitive (at least I did; but I did and it does become intuitive eventually).

2) is just an intuitive description. There is a relation between variables. One variable can be a range of values. The other variable can be a associate values following some "rule". Thus there is set of all value inputs and outputs which basically a subset of the cartesian coordinate, isn't it?

This is, IMO, way too informal and incomplete. And it doesn't explain the essential requirement that each input has exactly one output. According to this $x^2 + y^2 = 1$ would be a function between $x$ and $y$.

Also "rule" implies only describable, calculable relations are allowed. Not so. The mapping and the subset of pairs, can be utterly arbitrary as long as each element of the domain is mapped/in a pair exactly once.

3) " y is a function of x"

That's obviously not a definition of a function. It's just a statement of the the relation between two specific variables. It assumes the concept of "function" is intuitively understood. It is a single instance of a single example in one book. As a phrase to define or describe a function it's.... meaningless.

OKAY, So here is fleablood interpretation of a function:

A function $f:A \rightarrow B$ is a mapping between two sets: the domain $A$ and the codomain $B$ so that each element, $x$ of the domain is mapped to one (not necessarily unique) element $f(x)$ of the codomain.

The function can be thought of either as the idea of the "mapping" itself or as the complete set of all paired input/output values. i.e. as the set $\{(x, f(x))| x\in A\} \subset A\times B$.

Both ways of thinking are legitimate but as a subset of Cartesian Coordinates is direct and tangible while a "mapping" relies on undefined abstractions

Answer 3

It's all the same. Formally, language of mathematics based on set theory, so function defined in set theory rigorously as a specific kind of relation (see this answer, for example). Any other way of defining a function is just a informal way for the same definition. Words like "rule", "connects"... can be translated to rigorous set theory.