More on Equations - Essential Dictionary II

Mathematical Writing - Vivaldi Franco 2014

More on Equations
Essential Dictionary II

An equation whose unknown is a function is called a functional equation.

$$\begin{aligned} f(x)=f(x+1) f(f(x))=x. \end{aligned}$$

(3.21)

Here the unknown is $$f$$, not $$x$$. To make this plain, let $$1\!\!1$$ be the identity function in the appropriate set of functions, and let $$\tau $$ be the function that increases its argument by 1: $$\tau (x)=x+1$$. Equations (3.21) may now be rewritten without reference to the function’s argument, using the composition operator—see (2.19).

$$\begin{aligned} f=f\circ \tau \qquad \qquad f\circ f=1\!\!1. \end{aligned}$$

(3.22)

Solutions of these equations are periodic functions and involutions, respectively.

The class of functional equations includes the differential equations (which we’ve seen in Sect. 3.3) and the integral equations. The latter are equations in which the unknown function appears under an integral sign, e.g.,

$$ \sqrt{\pi } f(x)=\int \limits _{0}^\infty \text {e}^{-x y} f(y) dy. $$

An equation whose unknown is a set is called a set equation:

$$\begin{aligned} (X+X)\cap X=\emptyset . \end{aligned}$$

(3.23)

Here the symbol $$X$$ necessarily represents a set, and hence $$X+X$$ is an algebraic sum of sets [defined in (2.21)]. This set equation may be defined over any family of sets of numbers, such as $$\mathbf {P}(\mathbb {Z})$$, the power set of the integers. A solution of this equation is called a sum-free set; thus the set of odd integers is sum-free.

It should be clear that equations can be written having any type of mathematical object as unknown.

Exercise 3.5

Same as in Exercise 3.2.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Exercise 3.6

Write an equation whose unknown is (i) a matrix; (ii) a polynomial; (iii) a sequence.

Exercise 3.7

Ask some questions about the Farey sequence.

Exercise 3.8

Let $$X,Y$$ be sets, and let $$\fancyscript{F},\fancyscript{G}:X \rightarrow \mathbf {P}(Y)$$. Prove that the set equation $$\fancyscript{F}(x)=\fancyscript{G}(x)$$ is equivalent to the equation

$$ \fancyscript{F}(x)\,\varDelta \, \fancyscript{G}(x)=\emptyset $$

in the sense that the two equations have the same solution set. Thus every set equation may be reduced to the form $$\fancyscript{F}(x)=\emptyset $$, which resembles (3.17).

Footnotes

1

Jean Baptiste Joseph Fourier (French: 1768—1830).

2

Leonhard Euler (Swiss: 1707—1783). An accessible account of Euler’s mathematics is found in [11].

3

In Chap. 4 we shall see that an equation is a special type of predicate, which in turn is a special type of function.

4

This sequence, named after John Farey, Sr. (English: 1766—1826), was actually discovered by the Frenchman Charles Haros.