Involuție (matematică)

O involuție este o funcție $f:X\to X$ care, aplicată de două ori, se întoarce la punctul de pornire

În matematică, o involuție, sau o funcție involutivă, este o funcție $f$ care este inversa ei înseși

f (f (x)) = x

pentru orice $x$ din domeniul de definiție al $f$ .^[1] Echivalent, rezultatul obținut prin aplicarea funcției $f$ de două ori este valoarea inițială.

Proprietăți generale

Orice involuție este bijectivă.

Funcția identitate ( $f(x)=x$ ) este exemplul banal de involuție. Exemple simple de involuții sunt înmulțirea cu −1 în aritmetică, elementul invers, conjugatul complex, anumite permutări^[2] și complementul din teoria mulțimilor. Alte exemple sunt rotația cu o jumătate de tură, transpoziția de tip ROT13, matrice transpusă, cifrarea simetrică, cifrul Beaufort sau cifrul polialfabetic.

Numărul involuțiilor, inclusiv involuția identică, pe o mulțime cu elementele n = 0, 1, 2, ... este dat de o relație de recurență găsită de Heinrich August Rothe în 1800:

a_{0}=a_{1}=1

și

a_{n}=a_{n-1}+(n-1)a_{n-2}

pentru

n>1.

Primii câțiva termeni ai acestui șir sunt 1, 1, 2, 4, 10, 26, 76, 232 (șirul OEIS A000085^[3]); aceste numere se numesc numere de telefon^⁠(d) și indică și numărul de tablouri Young^[2] cu un număr de celule dat.^[4] Compunerea g ∘ f a două involuții f și g este o involuție dacă și numai dacă operația este comutativă: g ∘ f = f ∘ g.^[5]

Orice involuție pe un număr impar de elemente are cel puțin un punct fix. Mai general, pentru o involuție pe o mulțime finită de elemente, numărul de elemente și numărul de puncte fixe au aceeași paritate.^[6]

Note

^ en Russell, Bertrand (1903), Principles of mathematics (ed. 2nd), W. W. Norton & Company, Inc, p. 426, ISBN 9781440054167
^ ^a ^b Knuth, Donald E. (1976), Tratat de programarea calculatoarelor, vol 3: Sortare și căutare, Ed. Tehnică, p. 47 .
^ Șirul A000085 la Enciclopedia electronică a șirurilor de numere întregi (OEIS)
^ en Knuth, Donald E. (1973), The Art of Computer Programming, Volume 3: Sorting and Searching, Reading, Mass.: Addison-Wesley, pp. 48, 65, MR 0445948 .
^ en Kubrusly, Carlos S. (2011), The Elements of Operator Theory, Springer Science & Business Media, Problem 1.11(a), p. 27, ISBN 9780817649982 .
^ en Zagier, D. (1990), „A one-sentence proof that every prime p≡ 1 (mod 4) is a sum of two squares”, American Mathematical Monthly, 97 (2): 144, doi:10.2307/2323918, JSTOR 2323918, MR 1041893 .

Lectură suplimentară

en Ell, Todd A.; Sangwine, Stephen J. (2007). „Quaternion involutions and anti-involutions”. Computers & Mathematics with Applications. 53 (1): 137–143. arXiv:math/0506034 . doi:10.1016/j.camwa.2006.10.029.
en Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998), The book of involutions, Colloquium Publications, 44, With a preface by J. Tits, Providence, RI: American Mathematical Society, ISBN 0-8218-0904-0, Zbl 0955.16001
en Involution, Encyclopedia of Mathematics, EMS Press (Springer-Verlag), 2001

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