Beweise für alle Relationen ´T,R,S sube M^2´ folgende Aussage:

´T @ (R uu S) = (T @ R) uu (T @ S)´

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

Solution

  • Zu beweisen: ´T @ (R uu S) sube (T @ R) uu (T @ S)´

    ´(x, y) in T @ (R uu S)´ (Definition Komposition) ´=> EE z in M : (x, z) in T ^^ (z, y) in R uu S´ (Definition Vereinigung) ´=> EE z in M : (x, z) in T ^^ ((z, y) in R vv (z, y) in S)´ (Distributivität) ´=> EE z in M : ((x, z) in T ^^ (z, y) in R) vv ((x, z) in T ^^ (z, y) in S)´ (Definition Komposition) ´=> ((x, y) in T @ R) vv ((x, y) in T @ S)´ (Definition Vereinigung) ´=> (x, y) in (T @ R) uu (T @ S)´ ´=> T @ (R uu S) sube (T @ R) uu (T @ S)´ ´q.e.d.´

    Zu beweisen: ´T @ (R uu S) supe (T @ R) uu (T @ S)´

    ´(x, y) in (T @ R) uu (T @ S)´ (Definition Vereinigung) ´=> (x, y) in (T @ R) vv (x, y) in (T @ S)´ (Definition Komposition) ´=> EE z_1, z_2 in M : ((x, z_1) in T ^^ (z_1, y) in R) vv ((x, z_2) in T ^^ (z_2, y) in S)´ (´z = (z_1 vv z_2)´) ´=> EE z in M : ((x, z) in T ^^ (z, y) in R) vv ((x, z) in T ^^ (z, y) in S)´ (Distributivität) ´=> EE z in M : (x, z) in T ^^ ((z, y) in R vv (z, y) in S)´ (Definition Vereinigung) ´=> EE z in M : (x, z) in T ^^ ((z, y) in R uu S)´ (Definition Komposition) ´=> (x, y) in T @ (R uu S)´ ´=> T @ (R uu S) supe (T @ R) uu (T @ S)´ ´q.e.d.´

    ´T @ (R uu S) sube (T @ R) uu (T @ S) ^^ T @ (R uu S) supe (T @ R) uu (T @ S)´´=> T @ (R uu S) = (T @ R) uu (T @ S)´ ´q.e.d.´

  • URL:
  • Language:
  • Subjects: math
  • Type: Proof
  • Duration: 40min
  • Credits: 2
  • Difficulty: 0.6
  • 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 18:25:35 UTC