How can expressions like $x^2+y^2 = 4$ be defined?

Question

I'm wondering how to define the expression(?) $x^2+y^2 = 4$, because I realised it's not a function because it cannot be expressed in terms of $x$ or $y$ alone. Is it even called an expression?

Of course I then thought that you could split it into two functions $f(x) = \pm\sqrt{4-x^2}$, but then I realized that $x^2+y^2 = 4$ doesn't have a domain nor a range, so this doesn't really work, as a function needs these. So how do you define a range and domain for things like $x^2+y^2=4$?

In short: What's the technical way of defining $x^2+y^2=4$ in terms of domain/range of the variables, and what is the name for a thing like this?

Edit:

A function can be defined as:

\begin{align}f:\quad &[-2,2]\to[0,2]\\ &x \mapsto \sqrt{x^2-4}\end{align}

Is there a similar way to describe $x^2+y^2=4$ ?

Answer 1

$f(x)=\sqrt{4-x^2}$ does have a domain and range.

$$\text{Dom}\,f=[-2,2]\quad\text{and}\quad\text{Range}\,f=[0,1]$$

As for the equation, $x^2+y^2=4$, is a subset of $\Bbb R^2$, and is called a $\textbf{relation}$.

Answer 2

A relation is any set of ordered pairs. For instance, \begin{equation} R=\{(x,y)|x^2+y^2=4\} \end{equation} Tersely, the domain is the set of all first coordinates while the range is the set of all second coordinates. That is,

$Dom=\{x| \exists y$ so that $(x,y) \in R\}$

with the range being defined similarly. A relation in which for each $x$ in the domain there is only one $y$ in the range so that $(x,y) \in R$ is called a function. Your relation is clearly not a function, as $0$ is in the domain, but both $y=2$ and $y=-2$ satisfy $(0,y) \in R$.

Answer 3

Hint: $\theta\rightarrow (2\cos\theta,2\sin\theta)$ where $\theta\in[0,2\pi).$ This represents a circle.

Answer 4

You're right in noticing that the equation for a circle cannot be written as a function of just the $x$ coordinate, because for any $x$ value (except $2$ and $-2$) you have two distinct $y$ values on the circle, and similarly for $y$. But you can generalise a bit and think of $x^2+y^2=4$ as the set of zeroes of the two-variable function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ which sends a pair $(x,y)$ to $x^2+y^2-4$, and this one is actually a function.

Or alternatively as noted in other comments, you can view it as a relation on $\mathbb{R}$.

Answer 5

The most generic term is equation. Mathematicians tend to use this word for many things. Your $x^2+y^2=4$ could be thought in terms of algebraic curves, but it can also be considered as a shorthand for $$\left\{ (x,y) \in \mathbb{R}^2 \mid x^2+y^2=4 \right\}.$$

If an equation if any statement of the form $$F(x)=0,$$ then yours is an equation. We are not thinking of the solution set of this equation, and can be taken as a primitive concept like sets and elements of sets.

Therefore, to summarize: when you write $x^2+y^2=4$, you write an equation. But then you must ask yourself a pregnant question: what do I mean by $x^2+y^2=4$? Do I mean the circle in the plane, do I mean the two implicit functions $y=y(x)$ that this equation defines, what do I mean?

Answer 6

As a start, $$\tag1 x^2+y^2=4$$ is an equation.

You may be interested in the set of solutions $$\tag2\{\,(x,y)\in A^2\mid x^2+y^2=4\,\}$$ over a certain set $A$ (for example, $A=\mathbb R$ or $A=\mathbb C$ or $A=\mathbb Z$).

For $A=\mathbb R$, the set in $(2)$ may be described as the union of the graphs of the two functions $$\begin{align}\tag3f_{1,2}\colon [-2,2]&\to\mathbb R\\ x&\mapsto\pm\sqrt{4-x^2}\end{align}$$

However, since $(1)$ may be considered in the context at least of any ring with unit, there is little sense in assuming any specific $A$ in $(2)$ or $(3)$ if no such ring is either specifically mentioned or "understood". As such we should simply leave $(1)$ alone, as an equation.

There is also a notion of algebraic set, which loosely speaking consists of talking about $(2)$ without specifiýing $A$ (except that certain minimal conditions on $A$ be required - but not in out example).

Answer 7

Addressing your comment about, "Why can't $x, y\in\mathbb{C}$?":

The set $S$ of $(x, y)\in\mathbb{R}^2$ which form the circle w/center the origin, and radius 2, has a really interesting property: it's definable, in this case as the zeroes of a polynomial expression. I'm deliberately being vague about what I mean by "definable" - there are a lot of different notions of definability, and they're all interesting. Certainly we shouldn't expect every set of points in the plane to be definable - in fact, if we make things more precise, we can prove that there are undefinable sets of points! EDIT: And, more concretely, we shouldn't even expect every "nice" set of points to be definable in a specific way - say, as the graph of a function. As you correctly say, there is no function whose graph is that circle.

Why is this relevant? Well, let $\varphi$ be the expression "$x^2+y^2=4$." Clearly $\varphi$ defines the set $S$ in $\mathbb{R}^2$ - that is, $S$ is the set of points $(x, y)$ satisfying $\varphi$. However - as you point out correctly - the expression "$x^2+y^2=4$" makes sense in contexts other than $\mathbb{R}^2$. $\varphi$ defines a subset of $\mathbb{C}^2$, and of $\mathbb{Z}^2$, and of $\mathbb{Q}_2^2$ ($\mathbb{Q}_2$ being the really interesting structure called the $2$-adic numbers).

Note that this isn't really all that surprising - the expression "the people with green hair" can be used to define a subset of

• the people in this room,

• the people in Canada,

• the people at this totally awesome Phish concert, or

• the rocks which Steve has touched over the last ten years (in this case, not a very interesting subset, but still).

(NOTE: there are some restrictions on where definitions can even be applied. For example, your expression "$x^2+y^2=4$" only makes sense in a context where "$\times$," "$+$," and "$4$" are all meaningful (so it makes sense in an arbitrary ring, but not in an arbitrary group, for example).)

There's an implicit question here - which do I care more about, the set $S$ or the definition (maybe description is a more intuitive word) $\varphi$? One of the most surprising facts, to my mind, of mathematics is:

If you care about some thing $S$, you should also care about the ways $S$ can be defined - and often the best way to understand $S$ is to understand how one of its definitions behaves in other contexts.

The study of definitions as interesting objects in and of themselves is really one of the main driving forces of mathematical logic, and in particular the subfield model theory (http://en.wikipedia.org/wiki/Model_theory). (For me it's the entire motivation, but I don't think that's universal.)

A really striking example of this is something incredibly basic: studying the integer solutions to polynomials with integer coefficients. It turns out that even such basic objects as "natural numbers satisfying some polynomial" hide incredible structure, which is best understood by - among other things - looking at what they would be if "natural numbers" meant something completely different.

And, just to end on a ridiculous note, a really really extreme version of "understanding $S$ by looking at its analogues in other contexts" is this utterly weird and opaque, but brilliant, thing called "interuniversal Teichmuller theory" which was recently discovered/invented by a Japanese mathematician Shinichi Mochizuki. (BTW I don't actually know anything about ITT - my sentence about it is based purely on what I took away from my friend's excited ramblings on the subject.)

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We need to be careful, of course: a thing may have different definitions which agree in the context where that thing lives, but disagree in other contexts. For example, the circle $S$ you mention in your question can be described - in the context of $\mathbb{R}$ - as the set of points $(x, y)$ which:

• EITHER satisfy "$x^2+y^2=1$,"

• OR satisfy "$x^2+y^2=1$, or $x=7, y=-\pi$, and -1 has a square root."

Obviously these expressions define the same subset of $\mathbb{R}^2$, but different subsets of $\mathbb{C}$. This example might strike you as stupid, but this is an important phenomenon which can cause a lot of problems.

Answer 8

The expression $x^2+y^2=4$ can very well be seen as a Boolean function of two real variables.

$$f:\mathbb R^2\to \{False,True\}:(x,y)\to f(x,y)= (x^2+y^2=4).$$

The domain of $f$ is the whole of $\mathbb R^2$, as the expression is everywhere defined, and the range is $\{False,True\}$, as both truth values occur.

More interesting is the subset of the domain that maps to $True$, i.e. a circle.