Logical Data Modeling - Reflexive relationship property

About

reflexive is a relationship property that indicates:

  • that the relationship relates every element to itself.
  • in other word, that the relation set representing the relationship contains every tuple:
    • in a binary relationship - the relation set contains all <a,b> such as a=b
    • in a N relationship: <a,b,c…, n> such as a=b=c…=n

Mathematical Definition

A binary relation Rel is called reflexive:

  • on a a dimensional set A, if the relation set contains every tuple <a, b> for every element a of A.

<MATH> \{(a)\in\Bbb A\mid a = a\} \in Rel </MATH>

Example

Less than or equal

The relation less than or equal to (<=) on the set of integers {1, 2, 3} is the following set of tuple

<1, 1>, 
<1, 2>, 
<1, 3>, 
<2, 2>, 
<2, 3>, 
<3, 3>

It is reflexive because the tuples <1, 1>, <2, 2>, <3, 3> are in this relation.

As a matter of fact, this relation is reflexive on any set of numbers (not only integer but also real numbers, …).

Against the <math>R</math> , the relation less than or equal to is the below gray area in <math>R^2</math>

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Equivalence relationship

Similarly and because every number is equal to itself, the relation greater than or equal and is equal to on any set of numbers are reflexive.

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Plot

A few relations:

  • between x and y
  • on subsets of <math>R</math> (real number)
  • represented as subsets of <math>R^2</math>

The gray area is the relation.

In the set of real number, a relation is reflexive if it contains the dotted line y=x

Example:

  • Because the line y=x is in the relation, this is then a reflexive relation (but it's not symmetric, nor antisymmetric)

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  • Because the line y=x is in the relation, this is a reflexive relation (and it's symmetric)

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  • Because the line y=x is partially in relation, this is NOT a reflexive relation (nor symmetric or antisymmetric )

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  • Because the line y=x is not in the relation, this is then NOT a reflexive relation (but it's an antisymmetric)

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Documentation / Reference


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