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 be the length of a diagonal of a regular pentagon with unit area), or specify membership to a unique non-empty set (Let be a prime such that 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 be the set of 2 by 2 integral matrices with odd entries and unit determinant.
The set 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 .
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 be sets, and let denote their (possibly empty) common intersection
(9.2)
For , the set is well-defined, because the intersection operator is associative.
For the Formula (9.2) is, strictly speaking, unspecified, but the interpretation is quite natural, so a separate treatment is unnecessary. One could even stretch this definition to mean if , 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 be the smallest prime number with decimal digits.
Does such a prime exist for all ? Let’s look at the first few terms of the sequence:
(9.3)
In all cases, the prime lies just above , and it seems unavoidable that there is at least one prime between and .
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 digits, for some . Such an argument is not elementary, and we shouldn’t keep the reader pondering on this. So the definition of the sequence should incorporate a remark or a footnote of the type:
This sequence is well-defined due to Bertrand’s postulate 4: for all there is at least one prime such that [15, p. 343].
A function definition requires specifying two sets, the domain and the co-domain , as well as a rule that associates to every point a unique point of . For the function to be well-defined, we must ensure that the specification of and of the rule do not contain any ambiguity.
By contrast, there is flexibility in the specification of the co-domain , in the sense that any set containing may serve as a co-domain. Formally, different choices of correspond to different functions, although such distinctions are often unimportant. But then why don’t we always choose as co-domain? This would have the advantage of making every function surjective. The problem is that we may not know what is, or the description of may be exceedingly complicated.
For instance, let us return to the sequence given in (9.3). Writing
(9.4)
we find
Let us now define the function
This function is well-defined but we have limited knowledge of the image . It can be shown that cannot intersect any of the sets , , and , but we don’t know if there are other constraints—see Exercise 10.4.3.