Some Advanced Terms - Sets and Sequences - Essential Dictionary II

Mathematical Writing - Vivaldi Franco 2014

Some Advanced Terms - Sets and Sequences
Essential Dictionary II

We introduce some advanced words and symbols concerning sets, sequences, and equations.

3.5.1 Sets and Sequences

Let $$(A_k)$$ be a sequence of sets, which may be finite or infinite. The binary set operations of union and intersection generalise to an arbitrary number of operands:

$$ \bigcup _k A_k=A_1\cup A_2\cup \cdots \qquad \bigcap _k A_k=A_1\cap A_2\cap \cdots . $$

These expressions denote the set of elements belonging to at least one of the sets $$A_k$$, and to every one of the sets $$A_k$$, respectively. The integer index conforms to the notational rules for sums and products (Sect. 3.2):

$$ \bigcap _{k=2}^\infty \varTheta _k \qquad \qquad \bigcup _{k\in 2\mathbb {N}\atop k \geqslant K} (A_k{\backslash }B_k). $$

However, infinite unions and intersections may also be controlled by a real (as opposed to integer) index, which adds flexibility to this notation:

$$ \varOmega =\bigcup _{\alpha >0} \varOmega _\alpha \qquad \qquad \varOmega _\alpha =\{(x,y)\in \mathbb {R}^2\,:\,y=x^2+\alpha x\}. $$

(Sketch the set $$\varOmega $$.)

A sequence of sets is descending (or nested) if

$$ A_1\supset A_2\supset A_3\supset \cdots $$

and ascending if

$$ A_1\subset A_2\subset A_3\subset \cdots . $$

EXAMPLE. The following symbols

$$ m\mathbb {Z}\supset m^2\mathbb {Z}\supset \cdots \supset m^k\mathbb {Z}\supset \cdots \qquad \qquad \qquad \bigcap _{k\geqslant 1} m^k\mathbb {Z}= {\left\{ \begin{array}{ll}\{0\}&{} \mathrm{{if}}\,|m|\not =1\\ \mathbb {Z}&{} \mathrm{{if}}\,|m|=1\\ \end{array}\right. } $$

tell a story. Say it with words. [ $$\not \varepsilon \,$$ ]

The set of multiples of a power of an integer contains the set of multiples of any higher power of the same integer. By considering increasing powers we obtain an infinite nested sequence of sets. Apart from a trivial case, the only element common to all these sets is the origin.

EXAMPLE. The Farey sequence 4 $$({\fancyscript{F}}_n)$$ is an ascending sequence of sets. Its general term $$\mathcal{F}_n$$ is the set of all reduced fractions in the unit interval whose denominator does not exceed $$n$$:

$$\begin{aligned} {\fancyscript{F}}_1&= \left\{ 0,1\right\} \\ {\fancyscript{F}}_2&= \left\{ 0,\frac{1}{2},1\right\} \\ {\fancyscript{F}}_3&= \left\{ 0,\frac{1}{3},\frac{1}{2},\frac{2}{3},1\right\} \\ {\fancyscript{F}}_4&= \left\{ 0,\frac{1}{4},\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{3}{4},1\right\} \\&\vdots&\end{aligned}$$

Let us now consider the representation of sets of sequences. Let $$A$$ be any set. We form the set of all finite sequences of elements of $$A$$, with $$n$$ terms. This set is the cartesian product $$A^n$$ of $$n$$ copies of the set $$A$$: the first element $$a_1$$ of a sequence is chosen from the first copy of $$A$$, the second element from the second copy of $$A$$, and so on. For instance, the set $$\{0,1\}^n$$ is the set of all binary sequences with $$n$$-digits, while $$\mathbb {Q}^n$$ is the set of all $$n$$-uples of rational numbers. It follows that the infinite union

$$ \bigcup _{n\geqslant 1}\mathbb {Z}^n $$

represents the set of all finite integer sequences.

The set of all infinite sequences of elements of $$A$$ has the structure of a cartesian product with infinitely many terms. It would seem natural to denote it by $$A^\infty $$. However, the idiomatic notation $$A^\mathbb {N}$$ is more common, due to its greater flexibility. Thus $$A^\mathbb {Z}$$ denotes the set of doubly-infinite sequences of elements of $$A$$, and one even finds $$A^{\mathbb {Z}^2}$$ for the set of two-dimensional arrays: $$(a_{i,j})\in A^{\mathbb {Z}^2}$$.