Sets of Numbers - Essential Dictionary I

Mathematical Writing - Vivaldi Franco 2014

Sets of Numbers
Essential Dictionary I

The ’open face’ symbols $$\mathbb {N}$$, $$\mathbb {Z}$$, $$\mathbb {Q}$$ were introduced in Sect. 2.1.1 to represent the natural numbers, the integers, and the rationals, respectively. Likewise, we denote by $$\mathbb {R}$$ the set of real numbers (its symbolic definition is left as Exercise 2.13), while the set of complex numbers is denoted by $$\mathbb {C}$$. The set $$\mathbb {C}$$ may be written as

$$ \mathbb {C}\mathop {=}\limits ^\mathrm{def}\big \{x+\text{ i }y\,:\, \text{ i }^2=-1,\,\,x,y\in \mathbb {R}\big \}. $$

The symbol $$\text{ i }$$ is called the imaginary unit, while $$x$$ and $$y$$ are, respectively, the real part $$\mathrm {Re}(z)$$ and the imaginary part $$\mathrm {Im}(z)$$ of the complex number $$z=x+\text{ i }y$$. The sets $$\mathbb {R}$$ and $$\mathbb {C}$$ are represented geometrically as the real line and the complex plane (or Argand plane), respectively. A plot of complex numbers in the Argand plane is called an Argand diagram. We have the chain of proper inclusions

$$ \mathbb {N}\subset \mathbb {Z}\subset \mathbb {Q}\subset \mathbb {R}\subset \mathbb {C}. $$

We now construct new sets from the sets of numbers introduced above. An interval is a subset of $$\mathbb {R}$$ of the type

$$ [a,b]:=\{x\in \mathbb {R}\,:\, a\leqslant x\leqslant b\} $$

where $$a,b$$ are real numbers, with $$a<b$$. This interval is closed, that is, it contains its end points. (A point is sometimes regarded as a degenerate closed interval, by allowing $$a=b$$ in the definition.) We also have open intervals

$$ (a,b):=\{x\in \mathbb {R}\,:\, a<x< b\} $$

as well as half-open intervals

$$ [a,b)\qquad (a,b]. $$

The notational clash between an open interval $$(a,b)\subset \mathbb {R}$$ and an ordered pair $$(a,b)\in \mathbb {R}^2$$ is unfortunate but unavoidable, since both notations are firmly established. For (half) open intervals, there is the following alternative—and very logical—notation

$$ ]a,b[\qquad [a,b[\qquad ]a,b], $$

which, for some reason, is not so common.

The interval with end-points $$a=0$$ and $$b=1$$ is the (open, closed, half-open) unit interval. A semi-infinite interval

$$ \{x\in \mathbb {R}\,:\, a<x\} \qquad \{x\in \mathbb {R}\,:\, x\leqslant b\} $$

is called a ray. The rays consisting of all positive real or rational numbers are particularly important, and have a dedicated notation

$$\begin{aligned} \mathbb {R}^+:=\{x\in \mathbb {R},\,\,x>0\} \qquad \mathbb {Q}^+:=\{x\in \mathbb {Q},\,\,x>0\} \end{aligned}$$

(2.12)

whereas $$\mathbb {Z}^+$$ is just $$\mathbb {N}$$.

Some authors extend the meaning of interval to include also rays and lines, and use expressions such as

$$\begin{aligned} (-\infty ,\infty )\qquad [a,\infty )\qquad (-\infty ,b]. \end{aligned}$$

(2.13)

As infinity does not belong to the set of real numbers, the notation $$[1,\infty ]$$ is incorrect.

A variant of (2.12) is used to denote non-zero real and rational numbers:

$$\begin{aligned} \mathbb {R}^*:=\{x\in \mathbb {R},\,\,x\not =0\} \qquad \mathbb {Q}^*:=\{x\in \mathbb {Q},\,\,x\not =0\}. \end{aligned}$$

(2.14)

This notation is common but not universally recognised; before using these symbols, a clarifying comment may be appropriate (see Sect. 6.2).

The set $$\mathbb {R}^2$$ of all ordered pairs of real numbers is called the cartesian plane, which is the cartesian product of the real line with itself. If $$(x,y)\in \mathbb {R}^2$$, then the first component $$x$$ is called the abscissa and the second component $$y$$ the ordinate.

The set $$\mathbb {Q}^2\subset \mathbb {R}^2$$, the collection of points of the plane having rational coordinates, is called the set of rational points in $$\mathbb {R}^2$$. The set $$[0,1]^2\subset \mathbb {R}^2$$ is called the unit square. In $$\mathbb {R}^3$$ we have the unit cube $$[0,1]^3$$, and for $$n>3$$ we have the unit hypercube $$[0,1]^n\subset \mathbb {R}^n$$. The following subsets of the cartesian plane are related to the geometrical figure of the circle:

$$\begin{aligned}&\{(x,y)\in \mathbb {R}^2\,:\, x^2+y^2=1\} \qquad \mathbf{unit }\,\mathbf{circle } \nonumber \\&\{(x,y)\in \mathbb {R}^2\,:\, x^2+y^2\leqslant 1\} \qquad \mathbf{closed }\,\mathbf{unit }\,\mathbf{disc }\\&\{(x,y)\in \mathbb {R}^2\,:\, x^2+y^2< 1\} \qquad \mathbf{open }\,\mathbf{unit }\,\mathbf{disc }.\nonumber \end{aligned}$$

(2.15)

Thus the closed unit disc is the union of the open unit disc and the unit circle. The (unit) circle is denoted by the symbol $$\mathbb {S}^1$$.

For $$n\geqslant 0$$, the $$n$$-dimensional unit sphere $$\mathbb {S}^n$$ is defined as follows:

$$ \mathbb {S}^n= \{(x_0,\ldots ,x_n)\in \mathbb {R}^{n+1}\,:\, x_0^2+\cdots +x_n^2 =1\} $$

This Zermelo definition, to be compared with the Definition (2.15) of the unit circle $$\mathbb {S}^1$$, employs a combination of ordinary and raised ellipses. For $$n=0$$, we have $$\mathbb {S}^0=\{-1,1\}$$.