What does it mean to solve an equation?

Solution 1:

Interesting question. I'd say that solving $f(x) = 0$ amounts to

• exhibiting the set S = $\{ x \mid f(x) = 0 \}$, typically by enumerating its members, or giving a sequence whose elements are all the members of $S$

• demonstrating that the listed or enumerated items are exactly equal to $S$.

So if I say that the solutions of $\sin x = 0$ are $n\pi, n = 0, \pm 1, \pm 2, \ldots$, I've given a purported solution set $S'$. I now need to show that for each element $t$ of $S'$, we actually have $\sin t = 0$, and that no other values of $t$ satisfy $\sin t = 0$.

The method by which I arrive at the set $S$ is not really germane, despite the active verb "solve"; the solution might come from algebraic or geometric manipulations, or it might come to me in a dream. But the second part -- the demonstration that the purported solution set is the actual solution set -- that must follow the rules of logic and mathematics.

This is, however, mostly opinion about common mathematical speech, rather than a fact about mathematics.

PS: For infinite solution sets that are not countable, Christian Blatter's answer starts to get at a good description, although it doesn't take into account things like "the solution set is all irrationals," where a parametrization of the set may be very hard to come up with. Roughly speaking, as the solution sets get more complicated, exhibiting the set gets more and more complicated. No big surprise there...

Solution 2:

I agree with John Hughes; this is an interesting question. It is made more interesting by the observation that we often do answer questions like "Solve $f(x) = 0$" with something like $\{x \mid g(x) = 0\}$, which seems on the face of it like answering one question with another. Typically, of course, $g(x)$ is simpler in some sense than $f(x)$: For instance, we might answer

$$ \text{Solve for $x \in \mathbb{N}$ in }8x+1-y^2 = 0, y \in \mathbb{N} $$

with

$$ x \in \left\{\left.\frac{n(n+1)}{2} \,\,\right|\, n \in \mathbb{N}\right\} $$

where we might think of $x-n(n+1)/2$ as the equation $g(x)$ that characterizes the solutions to $f(x) = 8x+1-y^2 = 0$ for $y \in \mathbb{N}$. (There are better examples; this is just the one that comes to mind.)

What makes this characterization of $x$ simpler or better than the one that is posed as the problem? The explanation that I come up with is that we agree that—there is consensus that—the solution is a more immediately transparent description of the $x$ that solve the problem than the problem itself is. In other words, mathematics (like science in this regard) is a social activity, with (often unspoken) agreements about what constitutes progress toward a more primitive characterization of a mathematical object.

When we begin, as students, we get used to the idea of solutions being concrete, like the number $4$ for $3x-12 = 0$. Later, we apprehend that mathematics is a big web or network of relationships, and solutions often merely move from an expression that is less simple or transparent to one that is more simple or transparent. What it is that simplicity or transparency actually represent is not usually made explicit, and I'm not sure that it can be made explicit in any universal way.

Solution 3:

An equation, or a system of equations, defines a solution set $S$ as the set of all members $x$ belonging to some universe $X$ that satisfy certain conditions encoded in a formula, or a "story" ${\cal P}(x)$: $$S:=\{x\in X\>|\>{\cal P}(x)\}\ .\tag{1}$$ This is an implicit description of the set $S$. In most cases it is easy to check whether a proposed $x\in X$ actually belongs to $S$ or not.

Solving $(1)$ means producing an explicit description of $S$. Such an explicit description could consist in a proof that $S$ is in fact empty, it could consist in a finite list $S=\{x_1,\ldots, x_p\}$ of explicitly exhibited elements $x_k\in X$, or it could consist in a parametric representation $$f:\quad I\to X,\qquad \iota\mapsto x_\iota\in X\ ,\tag{2}$$ where $I$ is a certain "standard" set, e.g., $I={\mathbb N}$, $f(I)=S$, and $f$ is injective. In other words: Each element of $S$ is produced by $f$ exactly once in a well understood way.

Solution 4:

If someone says that the solutions of the equation are all the numbers in the set {y∈R|f(y)=0}{y∈R|f(y)=0} has he or she solved the equation? I don't think he/she has.

i think you have to understand 'solve' more like simplify in this context. From an arbitrary equation to the solution is like from abstract to specific. You specifie the abstract rules of your model. You have an equation, which represents something in abstract form, you solve it, you have the specific objects.