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Topic:
[HM] Hauber [was: lexicology of mathematics]
Replies:
3
Last Post:
May 16, 1999 9:15 AM




[HM] Hauber [was: lexicology of mathematics]
Posted:
May 14, 1999 3:39 PM


Yes, "Hauber's theorem", as stated by Mr. Felscher (as a tautology), is exactly the result I referred to. In Brazil, this result used to be taught in introductory courses in plane geometry. Here is a typical formulation (not a very rigorous one):
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 settheoretic formulation involving three (3) sets. This version can be easily generalized to any number of sets as follows. Let f and g be two setvalued 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



