Equations

Why

We name a statement which involves an identity.¹

Definition

An equation is a statement (see Statements) relating two terms by the relation of identity (see Identities). Some authors also call an equation an equality. The symbol “$=$” is called the (or an) equals sign or equals symbol.

Variables

Let $X$ and $Y$ be sets and let $f,g: X \to Y$. In the statement

\[ (\forall x)(f(x) = g(x)), \]

$f$ and $g$ are free names and $x$ is a bound name (see Quantified Statements).

For convenience, we often refer to the equation $f(x) = g(x)$ without the quantifier $\forall x$. In this case, $x$ appears free, but is not. In this context, the statement $f(x) = g(x)$ has $x$ implicitly bound. There are two senses here, though. The first is that $x$ is bound because it is “subordinate” to the quantifier $\forall$. The particular symbol is irrelevant; the symbol $y$ works just as well. In a second sense, though, the name is “free,” as it is a placeholder and the choice of the symbol $x$ does not matter.

We use different terminology for this common case. In discussing $f(x) = g(x)$, we call the placeholder name $x$ a variable and we call the names $f$ and $g$ constants. The language is meant to convey $f$ and $g$ are fixed in the present discussion, as indicated by the usual language “Let $f$ and $g$ ...”.

Solutions

We are often interested in finding objects in some set to satisfy an equation. For example, we are interested in finding an object $\xi \in X$ to satisfy $f(\xi ) = g(\xi )$. In this setting we call the variable $\xi $ in the equation an unknown.

We call an element $\xi \in X$ a solution of the equation if $f(\xi ) = g(\xi )$. We call the set

\[ \Set*{\xi \in X}{f(\xi ) = g(\xi )} \]

the solution set. If the solution set is non-empty, we say that a solution exists. If the solution set is a singleton, we say that the solution is unique.

We are often interested in solutions which satisfy several equations at once. For example, we have the equations $f_1(x) = g_1(x)$ and $h(x) = i(x)$ and so on. We want $x$ to satisfy these. Here it is set of equations, simultaneous equations, or a system of equations.

Finding solutions

We often talk about finding or searching for solutions or solving equations. We say: “We want to find $x \in X$ to satisfy $f(x) = g(x)$.” In addition to $f(x) = g(x)$, we may include other statements about $x$. The language is meant to convey that we are searching for an object which we will name, as a variable, $x$, and we want this object to satisfy the statements.

Future editions will modify this statement and sheet. ↩︎