What exactly is an equation?

Question

It seems to me an equation, in an abstract sense, must always involve some varying quantities where the varying quantities belong in some space (set, algebraic structure, what have you). In order to make precise the phrase, "vary quantities", it seems to me that one must have a mechanism for evaluation of each side of the equation. Ultimately, I think solving an equation must always be, in essence, finding the pre-image of a mapping. If the pre-image is empty then there are no solutions.

Consider the equation over $\mathbb{C}$ $$x^2-3x+2=0$$

We have that $x^2-3x+2$ is a polynomial and this polynomial induces a natural map from to $\mathbb{C}$ to $\mathbb{C}$ called evaluation. The equation is really asking for the pre-image of $0$ of this map.

Consider the functional equation $$f(x+y) + f(x-y) = 0$$ where we are looking for solutions that are functions from $\mathbb{R}$ to $\mathbb{R}$. I think ultimately that this equation can be thought of in terms of the pre-image of the map $G$ that takes functions like $f$ and maps them to the function from $\mathbb{R^2}$ to $\mathbb{R}$ by sending $f$ to the function of two variables $f(x+y)+f(x-y)$. And the equation is really asking for the pre-image of the zero function under this map.

Is it correct to view all equations in this manner? That is finding the solutions must always be equivalent to finding the pre-image of some element of some mapping?

Think of basic equations one finds in college algebra books. I tell my students that we take the given equation and apply solution preserving operations to it to transform the equation into a simpler one. The goal is to ultimately end up with a simpler equation whose solutions we can find by inspection. For instance,

$$3x - 2 = 5 \implies 3x = 7 \implies x = \frac{7}{3}$$

At each step we transform the equation to a simpler one whose solution set is the same. We can solve the last equation by inspection. Ultimately, isn't this how all equations are solved? We transform the equation to simpler equation(s) and end up with an equation that can be solved by inspection.

Answer 1

One approach: An equation is a predicate, $P(x)$, of the form $s(x) = t(x)$ where $x$ is a free variable (or vector of free variables) and $s(x), t(x)$ are terms -- expressions which evaluate to elements of the universe (e.g. real numbers if you are doing mathematics over the reals) when values are substituted for variables. This means that $P(x)$ is something which evaluates to either $true$ or $false$ when $x$ is replaced by members of the universe. In the 1-variable case it can be thought of as a function of the form

$$P(x): U \mapsto \{true,false\}$$

where $U$ is the domain of discourse.

To solve an equation is to determine $P^{-1}(true)$, the set of all values which make the predicate true.

Answer 2

Short answer touching on foundations and notation.

Mathematicians use equations to tell their readers that two expressions (the things on either side of the $=$ sign) are actually two (different) names for the same underlying object. But there are contexts in which that simple meaning can be lost or forgotten.

In elementary school kids (and teachers) are uncomfortable writing $$ 3 = 1 + 2 $$ because they want to think of the $+$ and $=$ as operations, analogous to the buttons on a calculator, so want to read them only from left to right.

In beginning algebra the equation $$ 3x + 2 = 8 $$ is meant to be "solved". That is, you are to find the values of the variable $x$ that make the two sides of that equation represent the same number, namely $8$. The rules that say you can "do the same thing to both sides" essentially preserve the fact that the two sides continue to name the same object.

If all the $x$'s are on the same side of the equation you can (but need not) think of this as asking for the preimages of the other side under the given map.

Later on when you encounter $$ f(x) = x^2 $$ you can be confused because there's nothing to "solve". This equation tells us that we will use "$f$" to name the squaring function.

When you encounter a "functional equation" like $$ f(x+y) = f(x) + f(y) $$ you may try to solve it: find all the "values" for the function $f$ that make the equation true (for every $x$ and $y$). In this case (assuming continuity or some other weak regularity) the answer is $$ f(x) = cx \quad \text{for some constant } c $$ but you can't get to that conclusion by transforming

the equation to simpler equation(s) and end up with an equation that can be solved by inspection.

In all these cases the equality tells you two things are really the same. What you do with that information depends on the context.

Answer 3

The universe of equations does not have uniform semantics.

Equations can be tautologies. $$ 1=1, x+x=2x, \tan x = \frac{\sin x}{\cos x}, \frac{\mathrm{d}}{\mathrm{d}x}\mathrm{e}^x = \mathrm{e}^x, \dots$$

Equations can be explicit definitions. $$ f(x) = f(f(x-1)), \\\frac{\mathrm{d}}{\mathrm{d}x}g(x) = \lim_{h \rightarrow 0} \frac{g(x+h) - g(x)}{h}, \dots $$

Equations can be implicit specifications. This seems to be the only kind of equations you are discussing. $$ 8x+7 = 15, x^5+x-1 = 0, 0 = \int_{\Omega}f \cdot g^* \,\mathrm{d}\mu, \dots $$

These semantics can be nested. $$ f''(x) = - f(x), \mathrm{Tor}_n^R(A,B) = (L_nT)(A), \dots $$

Frequently, these equations are (implicitly) augmented by specifying the sets from which the variables may be drawn. For instance, \begin{align*} g &\in \{\mathbb{R} \rightarrow \mathbb{R}\}, \\ U &\subset \mathrm{dom}(g), \\ m(U) &> 0, \\ H_0(U; \mathbb{Z}) &= \mathbb{Z}, \\ x &\in \mathrm{int}(U), \\ h &\in \mathbb{R}, \text{[footnote]} \\ x+h &\in \mathrm{dom}(g), \\ \frac{\mathrm{d}}{\mathrm{d}x}g(x) &= \lim_{h \rightarrow 0} \frac{g(x+h) - g(x)}{h} \end{align*} Changing these specifications may alter what the final equation denotes, if it denotes anything.

[footnote]: There's notational abuse here. "$h$" only appears as a dummy variable in the limit -- it is expanded to infinite sequences. We really want to say that every element of such a sequence is real and when you add $x$ to any of them, you still land in the domain of $g$. This is (in some sense) really a defect of our usual notation for limits that we do not get to specify the set from which the sequences are drawn. Some would write these two conditions with the "$h \rightarrow 0$", but this doesn't change that these constraints are really to be applied to the various $h_i$.

Answer 4

I think most of these answers are too complicated. The simple answer to the simple question:

An equation is the statement that two things are equal

"equal", "equate". "equality" and "equation" are different forms of the same word.

In mathematics the "=" sign is put between two things that are being said to be. or required to be, equal.

A formula is a special type of equation that we remember for working out a problem without having to always start from scratch.

A simple example:

Distance traveled at uniform speed = the speed multiplied by the time taken

Using symbols, we might write this as d=st