Beweise, dass für alle Mengen ´A, B, C, D sube M´ folgende Aussage gilt:

´(A xx B) nn (C xx D) = (A nn C) xx (B nn D)´

Zeige zum Training einzeln ´sube´ und ´supe´.

Solution

  • Zu beweisen: ´(A xx B) nn (C xx D) sube (A nn C) xx (B nn D)´

    ´(s, t) in (A xx B) nn (C xx D)´ (Definition Schnittmenge) ´=> (s, t) in A xx B ^^ (s, t) in C xx D´ (Definition Kreuzprodukt) ´=> s in A ^^ t in B ^^ s in C ^^ t in D´ (Kommutativität) ´=> s in A ^^ s in C ^^ t in B ^^ t in D´ (Definition Schnittmenge) ´=> s in (A nn C) ^^ t in (B nn D)´ (Definition Kreuzprodukt) ´=> (s, t) in (A nn C) xx (B nn D)´ ´=> (A xx B) nn (C xx D) sube (A nn C) xx (B nn D)´ ´q.e.d.´

    Zu beweisen: ´(A xx B) nn (C xx D) supe (A nn C) xx (B nn D)´

    ´(s, t) in (A nn C) xx (B nn D)´ ´=> s in (A nn C) ^^ t in (B nn D)´ (Definition Kreuzprodukts) ´=> s in A ^^ s in C ^^ t in B ^^ t in D´ (Definition Schnittmenge) ´=> s in A ^^ t in B ^^ s in C ^^ t in D´ (Kommutativität) ´=> (s, t) in (A xx B) ^^ (s, t) in (C xx D)´ (Definition Kreuzprodukt) ´=> (s, t) in (A xx B) nn (C xx D)´ (Definition Schnittmenge) ´=> (A xx B) nn (C xx D) supe (A nn C) xx (B nn D)´ ´q.e.d.´

    ´(A xx B) nn (C xx D) sube (A nn C) xx (B nn D) ^^ (A xx B) nn (C xx D) supe (A nn C) xx (B nn D)´´=> (A xx B) nn (C xx D) = (A nn C) xx (B nn D)´ ´q.e.d.´

  • URL:
  • Language:
  • Subjects: math
  • Type: Proof
  • Duration: 40min
  • Credits: 2
  • Difficulty: 0.5
  • Tags: proof set
  • Note:
    HPI, Mathematik I - Diskrete Strukturen und Logik, Wintersemester 2012/2013
  • Created By: adius
  • Created At:
    2013-04-12 16:49:14 UTC
  • Last Modified:
    2014-07-20 18:22:30 UTC