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Mathematics-Online course: Basic Mathematics - Sets

Properties of Relations

A (binary) relation on a set is called
• reflexive, if each element is related to itself:
• symmetric, if the order of the elements is irrelevant:
• asymmetric, if symmetry implies the identity of the respective elements:
• transitive, if the middle element of a chain can be removed:
• complete, if any two distinct elements are related to each other in at least one direction:

A reflexive, symmetric and transitive relation is called an equivalence relation, usually symbolized by instead of . An equivalence relation divides a set in disjoint subsets (equivalence classes), with any two elements of a particular subset being related (equivalent) to each other, while two elements of distinct subsets are not related to one another.

A reflexive, asymmetric and transitive relation is called a partial order, symbolized as instead of . If a partial order is complete, it is called a (total) order; is then ordered by .

(Authors: Hörner/Abele)

The inclusion of sets is a partial order in the power set of a set since it is

reflexive ( ),

asymmetric ( ),

and

transitive ( ).

However, if contains more than one element, then the inclusion is not an order:

i.e. it is not complete.

The relation ,,has an equal number of elements  is an equivalence relation in the power set of a finite set since it is

reflexive (),

symmetric ( ),

and

transitive ( ).

(Authors: Hörner/Abele)