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Equivalence Relations

  • Iain T. Adamson

Abstract

Let X be a class and R a relation on X. Recall that the diagonal D X = (x,x): xX. The relation R is said to be
  1. (1)

    reflexive if D X + R, i.e. if for every x in X we have (x, x) ∈ R;

     
  2. (2)

    irreflexive ifD X R = Ø, i.e. if for every x; in X we have (x,x) ∉ R;

     
  3. (3)

    symmetric if R = R-1, i.e. if for every x, y in X such that (x, y) ∈ R we have (y, x) ∈ R;

     
  4. (4)

    antisymmetric if RR-1 + D x , i.e. if for all x, y in X such that (x,y) ∈ R and (y, x) ∈ R we have x = y.

     
  5. (5)

    transitive if R o R + R, i.e. if for all x, y, z in X such that (x, y) ∈ R and (y, z) ∈ R we have (x, z) ∈ R.

     

Keywords

Equivalence Relation Identity Mapping Reverse Inclusion Straightforward Application Usual Device 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Iain T. Adamson
    • 1
  1. 1.Department of MathematicsThe University of DundeeDundeeScotland

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