Equations and Identities - Essential Dictionary II

Mathematical Writing - Vivaldi Franco 2014

Equations and Identities
Essential Dictionary II

Let $$f$$ and $$g$$ be functions with the same domain $$X$$ and co-domain $$Y$$. An equation (on or over $$X$$) is an expression of the type

$$\begin{aligned} f(x)=g(x). \end{aligned}$$

(3.15)

The quantity $$x$$ is the equation’s unknown.

$$\begin{aligned} \begin{array}{l@{\qquad \quad \qquad }l} x^2-2x-4=0 &{} {\textit{An algebraic equation.}} \\ \cos (x)=\sin (x) &{} {\textit{A trigonometric equation.}}\\ \log (1+x)=-x &{} {\textit{A transcendental equation.}} \end{array} \end{aligned}$$

(3.16)

These are equations over (a subset of) $$\mathbb {R}$$, and each equation is described in broad terms by an attribute (algebraic, trigonometric, transcendental). We shall learn more about these attributes in Sect. 3.4.2.

The expression (3.15) defines a property that each point $$x\in X$$ either has or doesn’t have.3 This prompts the definition of the solution set of Eq. (3.15), given by

$$ \{x\in X\,:\, f(x)=g(x)\}. $$

For instance, over the real numbers the solution set of the first equation in (3.16) is $$\{1+\sqrt{5},1-\sqrt{5}\}$$, while over $$\mathbb {Q}$$ the solution set is empty. We see that the solution set of an equation depends on the ambient set.

If the co-domain $$Y$$ of $$f$$ and $$g$$ is a set of numbers (e.g., $$Y=\mathbb {Z},\mathbb {Q},\mathbb {R},\mathbb {C}$$), then, by replacing $$f$$ by $$f-g$$, we can reduce Eq. (3.15) to the simpler form

$$\begin{aligned} f(x)=0. \end{aligned}$$

(3.17)

An element $$x$$ of the solution set of this equation is called a zero of $$f$$, but if $$f(x)$$ is a polynomial, we speak of a root of $$f$$. We also say that $$f$$ vanishes at $$x$$. A function $$f$$ vanishes identically on a set if it vanishes at every point of this set, in which case we use the emphatic notation $$f(x)\equiv 0$$. For example, the real function $$x\mapsto \sin (\pi x)$$ vanishes identically on $$\mathbb {Z}$$.

More generally, an equation may be reduced to the form (3.17) if the co-domain of $$f$$ and $$g$$ is a group with respect to addition. In this case the zero on the right-hand side is the zero element of the group. This is not necessarily the number 0, but it could be a zero matrix, a zero polynomial, a zero function, etc.

An example is given by the differential equations, which relate a function to its derivatives:

$$\begin{aligned} \frac{\text {d}x}{\text {d}t}-x=0\qquad \qquad \qquad t^2\frac{\text {d}x^2}{\text {d}t^2}+t\frac{\text {d}x}{\text {d}t}+(t^2-1)x=0. \end{aligned}$$

(3.18)

Here the unknown is $$x$$ (not $$t$$), where $$x=x(t)$$ is a function, and the underlying ambient set is a set of differentiable functions. The symbol 0 represents the zero function, namely the constant function that assumes the value zero everywhere. We’ll consider other types of equations in Sect. 3.5.

An identity (or indeterminate equation) is an equation whose solution set is equal to the ambient set:

$$\begin{aligned} (x-1)^3=x^3-3x^2+3x-1. \end{aligned}$$

After simplification, every identity reduces to the standard form $$0=0$$. This doesn’t mean that identities are trivial, far from it; identities express equivalence of functions. However, they are ephemeral quantities, which disappear if they are simplified.

EXAMPLE. The identity

$$ x^{2^n}-y^{2^n}=(x-y)\prod _{k=0}^{n-1}\left( x^{2^{k}}+y^{2^{k}}\right) $$

gives the full factorisation of the difference of two monomials whose degree is a power of 2, into the product of polynomials with integer coefficients.

EXAMPLE. Over the set $$\mathbb {R}^2$$, the expression $$x+y=y+x$$ is an identity, representing the commutativity of the addition of real numbers. The similar expression $$x+y=1-y$$ is an equation, whose solution set is a line in $$\mathbb {R}^2$$.

By restricting the ambient set to the solution set, every equation becomes an identity. Clearly, $$\sin (\pi x)=0$$ is an equation over $$\mathbb {R}$$ and an identity over $$\mathbb {Z}$$. For a more subtle example, consider the equation $$f(x)=x^5-x=0$$. The factorisation $$x^5-x=x(x-1)(x+1)(x^2+1)$$ shows that the solution set over $$\mathbb {C}$$ is $$\{0,\pm 1,\pm \sqrt{-1}\}$$, a set with five elements. Consider now the set

$$ X=\mathbb {Z}/5\mathbb {Z}=\{0+5\mathbb {Z},\,1+5\mathbb {Z},\,2+5\mathbb {Z},\,3+5\mathbb {Z},\,4+5\mathbb {Z}\} $$

of congruence classes modulo 5. Let us evaluate our function $$f$$ at all points of $$\mathbb {Z}/5\mathbb {Z}$$, writing $$k$$ for $$k+5\mathbb {Z}$$:

$$\begin{aligned}&f(0)=0\equiv 0\,(\mathrm{mod \, }5)\\&f(1)=0\equiv 0\,(\mathrm{mod \, }5)\\&f(2)=30\equiv 0 \,(\mathrm{mod \, }5)\\&f(3)=240\equiv 0 \,(\mathrm{mod \, }5)\\&f(4)=1020\equiv 0\,(\mathrm{mod \, }5). \end{aligned}$$

The function $$f$$ vanishes identically over $$\mathbb {Z}/5\mathbb {Z}$$, and hence the expression $$x^5=x$$ is an identity!

An expression of the type

$$\begin{aligned} {\left\{ \begin{array}{ll} f_{1}(x)=0&{}\\ f_{2}(x)=0&{}\\ \quad \vdots &{}\\ f_{n}(x)=0&{}\\ \end{array}\right. } \qquad x=(x_{1},\ldots ,x_m) \end{aligned}$$

(3.19)

where all functions have the same domain and co-domain, is called a system of $$n$$ simultaneous equations in $$m$$ unknowns. The solution set of a system of equations is the intersection of the solution sets of the individual equations.

EXAMPLE. Explain what is an equation, and its solutions. [ $$\not \varepsilon \,$$ ]

BAD:

An equation is when we equate two functions. The solution is when the functions are the same.

The inappropriate use of ’when’ is easily spotted (see Sect. 1.1), but there is a more serious flaw. The expression ’equating two functions’ means that we seek conditions under which the two functions become the same function. That’s not what we had in mind. The operands of the equal sign in expression (3.15) are not functions, but rather values of functions.

GOOD:

An equation is an expression that identifies the value of two functions at a generic point of their common domain. The solutions of an equation are the points at which the two functions assume the same value.

(The expression ’equating two functions’ may be appropriate for functional equations, see (3.21).)