Principio de Hauber [Hauber's principle]
"Se num teorema fizermos todas as hipoteses possiveis e elas nos
conduzirem a teses distintas e mutuamente exclusivas, entao podemos
afirmar que o teorema reciproco e tambem verdadeiro."
[If in a theorem we make all possible hypotheses and they lead us to
distinct and mutually exclusive theses, then we can affirm that the
converse theorem is also true.]
Thus, after being told that
(a) If two sides of a triangle are equal, then the opposite angles are
equal.
(b) If two sides are unequal, then the angle opposite the greater side
is the greater.
the student would conclude IMMEDIATELY, by invoking Hauber's principle
(and trichotomy), the converses of (a) and (b).
In my Spanish version of Ackermann's Grundzu"ge der Theoretischen
Logik (4th ed. 1958) (Chapter II, Exercise 2) one can see (under the
name "Hauberian theorem") a set-theoretic formulation involving three
(3) sets.
This version can be easily generalized to any number of sets as
follows. Let f and g be two set-valued functions defined on the same
index set I and such that:
(H1) Ui f(i) = Ui g(i);
(H2) AiAj i <> j ==> g(i) I g(j) = O;
(H3) Ai ( f(i) < g(i) ),
where U, I, O and < stand for union, intersection, empty set and
inclusion, respectively. (That is, f and g are families of pairwise
disjoint sets such that Uf = Ug and f < g.) Then
Ai ( g(i) < f(i) ).
(Equivalently: f = g.)
In fact, in virtue of (H2) and (H3) we see that f(i) I g(j) is O or
f(i) according to if i <>j or i = j. It follows from this and (H1)
that
f(j) = g(j) I (Ui f(i)) = g(j) I (Ui g(i)) = g(j).
(I want to thank Mr. Felscher for his response to my question on who
was Hauber. It would be interesting to know how Hauber himself stated
his "principle".)
Carlos Cesar Araujo