Formal definition of a graph? $ \{ (x,f(x)) \mid x \in A \} $

Question

working my way through the first few pages of James Stewart's' Essential Calculus and came across some notation I'm trying to decode in plain English. In this section he talks about ordered pairs and graphs:

“ If $f$ is a function with domain $A$, then its graph is the set of ordered pairs $$ \{ (x,f(x)) \mid x \in A \} ” $$

So I'm trying to decode that into plain English.

I know the { } indicate a set and that $ \{ x\mid \ldots \} \to$ The set of all $x$ such that...

My take is that the original line is:

If $f$ is a function with domain $A$, then its graph is, The set of all $x$, $f(x)$ pairs such that $x$ is an element of $A$?

Is that correct? Does anyone have a better way of saying this?

Answer 1

Definitions:

A relation is a set of ordered pairs $(x, y)$ and is usually defined by some property or rule. For example, the general equation $y = mx + c$ is a rule that defines the set of ordered pairs $(x, y)$ to be linear, because the rule is the linear relationship described by multiplying the $x$-value by the gradient and adding a constant $c$ to that result.

The domain of a relation is the set of all first elements of the ordered pairs, namely $x$. For example, let $X = \{(x, f(x))\mid x\in A\}$ then since $X$ is a relation and $A$ is the set of all elements $x$, we define $A$ as the domain of $X$.

The range of a relation is the set of all second elements of the ordered pairs, namely $y$. In the set of ordered pairs $(x, f(x))$, we define $y$ as $f(x)$.

The Cartesian Product is the set of all possible pairs of two sets $X$ and $Y$. In symbols, it is denoted as $X\times Y$. $$\begin{align} X\times Y &= \{(x, y) \mid x\in X, y\in Y\} \\ \Leftrightarrow X &= \text{the domain.} \\ \& \ \ Y &= \text{the range.}\end{align}$$ Any subset of a Cartesian product is a relation (since every set is a subset of itself) and in most cases the relation is defined by some rule, i.e. $y$ is related to $x$ in some way and is expressed thus: $$y \ R \ x.$$ A relation that is defined by the rule $y > x$ is sometimes referred to as a many-many correspon-dence because this relation can be represented graphically by a diagram and not just by plotting points on a plane.

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Questions:

If $y$ can be defined as $f(x)$, can $x$ be defined as $f(y)$?

No. In the set of ordered pairs $(x, y)$, if we consider such a relation where no two pairs have the same first elements, this is known as a function. Since the first elements $x_1, x_2,\ldots$ cannot be the same, our input of this function is $x$ and our output is $y$. In symbols, $$f(x) = y$$ $***$

What happens when the domain of a relation is not mentioned?

When a domain is not mentioned, it is assumed that the domain is the set of all real numbers $\mathbb{R}$ which is the set of every number that is not expressed as a multiple of $\sqrt{-1} = i$. $$\mathbb{R} = \left\{\mathbb{Q}\cup \mathbb{K}\mid \mathbb{Q} = \text{the rationals}, \mathbb{K} =\text{the irrationals}\right\}.$$ Occasionally, you might see $\mathbb{K} = \mathbb{I}$ but generally $\mathbb{I} = [0, 1]$ and also $\mathbb{K} = \mathbb{P}$ but generally $\mathbb{P}$ is the set of all prime numbers. (Some people also write $\mathbb{K}=\mathbb{Q}'$.)

$***$

How can a function $f$ have a domain, for example $A$?

Because like mentioned in the foregoing, every function is a relation with the constraint that if $\{(x, y)\}\subset f(x)$ then no two subsets of ordered pairs can have the same first element. And since every relation can have a domain (and a range), then every function has a domain.

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Direct Answer:

$$\{(x, f(x))\mid x\in A\} = \text{the set of ordered pairs $(x, f(x))$ such that the domain of this set is $A$.}$$

Since $y = f(x)$ then this entire set (or relation in particular) is a function of $x$. This is simply joining solutions $(x, y)$ of $y = f(x)$ as pairs into a set to construct a relation. With the domain defined as $A$, then $x\in A$, as asserted. We cannot state, however, that this function is the set of all ordered pairs $(x, f(x))$ because we do not know whether or not the domain $A$ is finite or infinite.

Answer 2

I will use informal languages (since I am not good at set theory to accurately describe something).

$\{(x,f(x))|x\in A\}$ is a set (a bag), which contains something (elements) in the form of $(x,f(x))$. All elements are in form of $2$-tuple (2-component thing, such as on coordinate plane, $(1,2)$ is a $2$-tuple).

There are something to note in this set. First of all, I assume the word function has been defined - it maps all elements of $A$, to a unique element in codomain (say $B$). Any mapping satisfying the former property is called function.

Hence, the set $G:=\{(x,f(x))|x\in A\}$ has two properties too:

1. For all elements $a\in A$, there exists an element $(x,y)\in G$ such that $x=a$.
   (All elements in $A$ is mapped)
   
2. If $(x,y),(x,z)\in G$, then we have $y=z$.
   (Uniqueness of the mapping)

Answer 3

A point on a graph is an ordered pair $(x,y)$ and the graph is the collection of all of these points.

We plot the $(x,y)$ pairs, where $x$ is the input to your function and $y$ is the output.

It is the same thing you have been doing since elementary school, you just haven't seen it phrased so formally.

Answer 4

Yes, it's read almost that way, or better still as: 'The graph of $f$ is the set of all points $(x,f(x))$ in the cartesian plane, where $x$ is in the domain of $f$.'

Basically, the idea is just that the graph of a function represents the function. Many people simply identity a function with its graph, although it's loose language, but it doesn't usually cause confusion -- just like we identify a line and its pictorial representation, or a number and its symbolic representation, even though they're quite different.