dependence relation

Let X be a set. A (binary) relationMathworldPlanetmath between an element a of X and a subset S of X is called a dependence relation, written aS, when the following conditions are satisfied:

  1. 1.

    if aS, then aS;

  2. 2.

    if aS, then there is a finite subset S0 of S, such that aS0;

  3. 3.

    if T is a subset of X such that bS implies bT, then aS implies aT;

  4. 4.

    if aS but aS-{b} for some bS, then b(S-{b}){a}.

Given a dependence relation on X, a subset S of X is said to be independent if aS-{a} for all aS. If ST, then S is said to span T if tS for every tT. S is said to be a basis of X if S is independent and S spans X.

Remark. If X is a non-empty set with a dependence relation , then X always has a basis with respect to . Furthermore, any two of X have the same cardinality.


  • Let V be a vector spaceMathworldPlanetmath over a field F. The relation , defined by υS if υ is in the subspaceMathworldPlanetmathPlanetmathPlanetmath S, is a dependence relatoin. This is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the definition of linear dependence (

  • Let K be a field extension of F. Define by αS if α is algebraic over F(S). Then is a dependence relation. This is equivalent to the definition of algebraic dependence.

Title dependence relation
Canonical name DependenceRelation
Date of creation 2013-03-22 14:19:25
Last modified on 2013-03-22 14:19:25
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 03E20
Classification msc 05B35
Related topic LinearIndependence
Related topic AlgebraicallyDependent
Related topic MatroidMathworldPlanetmath
Related topic AxiomatizationOfDependence