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18 CHAPTER 1. INTRODUCTION
(viewed as a function R → R). It might be tempting to consider F to be zero,
but in
{
fact F (−1) = −2. In fact, F can also be written as x ↦→ x − |x| or
0 x ≥ 0
x ↦→
. Is (1.1) therefore candid? It depends on the definition of
2x x ≤ 0
“simpler”.
Since practically everyone would agree that {0} is a simple class, candid
expressions are automatically normal, but need not be canonical (see (1.1)), so
this definition provides a useul intermediate ground, which often coïncides with
naïve concepts of “simpler”. We refer the reader to [Sto11] for further, very
illustrative, examples, and to section 2.2.1 for a discussion of candidness for
rational functions.
1.2.1 A Digression on “Functions”
In colloquial mathematics, the word “function” has many uses and meanings,
and one of the distinctive features of computer algebra is that it plays on these
various uses.
In principle, (pure) mathematics is clear on the meaning of “function”.
On dit qu’un graphe F est un graphe fonctionnel si, pour tout x, il
existe au plus un objet correspondant à x par F (I, p. 40). On dit
qu’une correspondance f = (F, A, B) est une fonction si son graphe
F est un graphe fonctionnel, et si son ensemble de départ A est égal
à son ensemble de définition pr 1 F [pr 1 is “projection on the first
component”].
[Bou70, p. E.II.13]
The present author’s loose translation is as follows.
We say that a graph (i.e. a subset of A × B) F is a functional graph
if, for every x, there is at most one pair (x, y) ∈ F [alternatively
(x, y 1 ), (x, y 2 ) ∈ F implies y 1 = y 2 ]. We then say that a correspondance
f = (F, A, B) is a function if its graph F is a functional graph
and the starting set A is equal to pr 1 F = {x | ∃y(x, y) ∈ F }.
So for Bourbaki a function includes the definition of the domain and codomain,
and is total and single-valued.
Notation 3 We will write (F, A, B) B
for such a function definition. If C ⊂ A,
we will write (F, A, B) B
| C , “(F, A, B) B
restricted to C”, for (G, C, B) B
where
G = {(x, y) ∈ F |x ∈ C}.
Hence the function sin is, formally, ({(x, sin x) : x ∈ R}, R, R) B
, or, equally
possibly, ({(x, sin x) : x ∈ C}, C, C) B
. But what about x + 1? We might be
thinking of it as a function, e.g. ({(x, x + 1) : x ∈ R}, R, R) B
, but equally we
might just be thinking of it as an abstract polynomial, a member of Q[x]. In
fact, the abstract view can be pursued with sin as well, as seen in Chapter 7,
especially Observation 13, where sin is viewed as 1 2i (θ − 1/θ) ∈ Q(i, x, θ|θ′ =
iθ). The relationship between the abstract view and the Bourbakist view is
pursued further in Chapter 8.