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Linear Algebra II

Linear Algebra II

Last update 

Basics for Binary Relations (Part of linear Algebra - Applied Computer Science)

Items (28)

  • A binary relation between sets A and B is a subset of

    A x B

  • R ⊆

    A x B

  • A{1,2,3,4} R={(a, b) | a < b}, R=?

    {(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)}

  • A{1,2,3,4} R={(a, b) | a > b}, R=?

    {(2,1),(3,1),(3,2),(4,1),(4,2),(4,3)}

  • A{1,2,3} R={(a,b) | a = b}, R=?

    {(1,1),(2,2),(3,3)}

  • A{1,2,3} R={(a,b) | a < b}, R=?

    {(1,2),(1,3),(2,3)}

  • A{1,2,3,4} R={(a,b) | 2a < b}, R=?

    {(1,4)}

  • How many Elements exists in A x A?

  • How many relations exists on A?

    2^n²

  • A relation is reflexive if...

    ∀a (a,a) ∈ R

  • A relation is symmetric if...

    Whenever (a,b)∈R, (b,a)∈R

  • A relation is antisymmetric if...

    Whenever (a,b)∧(b,a)∈R, a = b

  • A relation is asymmetric if...

    Whenever (a,b)∈R, (b,a)∉R

  • A relation is transitive if...

    Whenever (a,b)∧(b,c)∈R, (a,c)∈R

  • {1,2,3,4} R={(1,1),(1,2),(2,1),(3,3),(4,4)}

    symmetric, transitive

  • {1,2,3,4} R={(1,1),(2,2),(2,1),(3,3),(4,4)}

    reflexive

  • {1,2,3,4} R={(1,1),(1,2),(2,2),(2,1),(3,3),(4,4)}

    sym, reflexive

  • {1,2,3,4} R={(1,1),(2,2),(3,3)}

    sym, antisym

  • {1,2,3,4} R={(1,3),(3,2),(2,1)}

    antisym, asym

  • {1,2,3,4} R={(4,4),(3,3),(1,4)}

    antisymmetric

  • {1,2,3,4} R={(1,3),(3,2),(2,1)}

    asymmetric

  • {1,2,4] R={(1,4),(4,2),(2,1),(4,1),(1,2)}

    transitive

  • {1,2,4] R={(1,4),(4,2),(2,1),(4,1),(1,1),(1,2)}

    transitive

  • R1{(1,2),(1,3),(1,1)}R2{(1,1),(2,1)}, P(A∩B)?

    sym, antisym

  • R1{(1,2),(1,3),(1,1)}R2{(1,1)}, P(A∪B)?

    antisymmetric

  • A relation is irreflexive if...

    (a,a)∉R

  • A = Ø

    reflexive, irreflexive

  • Equivalence Relation

    reflexive, sym, transitive