Definitions - Existence and Definitions

Mathematical Writing - Vivaldi Franco 2014

Definitions
Existence and Definitions

Definitions are closely related to unique existence. When we define a symbol or a name, we must ensure that this quantity actually exists, and that it has the stated properties. A definition may identify a unique quantity (Let $$a$$ be the length of a diagonal of a regular pentagon with unit area), or specify membership to a unique non-empty set (Let $$p$$ be a prime such that $$p+2$$ is also prime). If these conditions are met, then our object is well-defined; otherwise it is ill-defined.

Let us begin with definitions of sets. Here there is a distinctive safety net: if the conditions imposed on a set are too restrictive, then this set will be empty rather than ill-defined. Nonetheless, we should be alert to this possibility, because the consequences of an empty definition could be more serious than formally correct nonsense. For example, consider the following definition:

Let $$\fancyscript{M}$$ be the set of 2 by 2 integral matrices with odd entries and unit determinant.

The set $$\fancyscript{M}$$ is empty because the determinant of a matrix with odd entries is an even integer, and so it cannot be 1. It’s easy to run into trouble now: Let $$M\in \fancyscript{M}$$.

Sets defined by several conditions are commonplace; for instance, the set of solutions of a system of simultaneous equations is the intersection of the solution sets of the individual equations. A considerate definition should flag the possibility of an empty intersection:

Let $$A_1,\ldots ,A_n$$ be sets, and let $$A$$ denote their (possibly empty) common intersection

$$\begin{aligned} A=A_1\cap A_2\cap \cdots \cap A_n. \end{aligned}$$

(9.2)

For $$n>2$$, the set $$A$$ is well-defined, because the intersection operator is associative.

For $$n=1$$ the Formula (9.2) is, strictly speaking, unspecified, but the interpretation $$A=A_1$$ is quite natural, so a separate treatment is unnecessary. One could even stretch this definition to mean $$A=\emptyset $$ if $$n=0$$, although a clarifying remark would be needed in this case.

Next we consider an innocent-looking integer sequence whose existence is more delicate than expected.

Let $$q_n$$ be the smallest prime number with $$n$$ decimal digits.

Does such a prime exist for all $$n$$? Let’s look at the first few terms of the sequence:

$$\begin{aligned} (q_n)_1^\infty =(2,\, 11,\, 101,\,1009,\,10007,\ldots ). \end{aligned}$$

(9.3)

In all cases, the prime $$q_n$$ lies just above $$10^{n-1}$$, and it seems unavoidable that there is at least one prime between $$10^{n-1}$$ and $$10^n$$.

However, given that there are arbitrarily large gaps between consecutive primes,3 an argument is needed to rule out the possibility that a large gap could include all integers with $$n$$ digits, for some $$n$$. Such an argument is not elementary, and we shouldn’t keep the reader pondering on this. So the definition of the sequence $$(q_n)$$ should incorporate a remark or a footnote of the type:

This sequence is well-defined due to Bertrand’s postulate 4: for all $$n\geqslant 1$$ there is at least one prime $$p$$ such that $$n < p \leqslant 2n$$ [15, p. 343].

A function definition $$f{:}\,A\rightarrow B$$ requires specifying two sets, the domain $$A$$ and the co-domain $$B$$, as well as a rule that associates to every point $$x\in A$$ a unique point of $$f(x)\in B$$. For the function $$f$$ to be well-defined, we must ensure that the specification of $$A$$ and of the rule $$x\mapsto f(x)$$ do not contain any ambiguity.

By contrast, there is flexibility in the specification of the co-domain $$B$$, in the sense that any set containing $$f(A)$$ may serve as a co-domain. Formally, different choices of $$B$$ correspond to different functions, although such distinctions are often unimportant. But then why don’t we always choose $$f(A)$$ as co-domain? This would have the advantage of making every function surjective. The problem is that we may not know what $$f(A)$$ is, or the description of $$f(A)$$ may be exceedingly complicated.

For instance, let us return to the sequence $$(q_n)$$ given in (9.3). Writing

$$\begin{aligned} q_n=10^{n-1}+a_n \end{aligned}$$

(9.4)

we find

$$ (a_n)_1^\infty =(1,\, 1,\, 1,\,9,\,7,\ldots ). $$

Let us now define the function

$$ f{:}\,\mathbb {N}\rightarrow \mathbb {N}\qquad f(n)=a_n. $$

This function is well-defined but we have limited knowledge of the image $$f(\mathbb {N})$$. It can be shown that $$f(\mathbb {N})$$ cannot intersect any of the sets $$2\mathbb {N}$$, $$2+3\mathbb {N}$$, and $$5\mathbb {N}$$, but we don’t know if there are other constraints—see Exercise 10.4.3.