In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence.

Let be a set. A (binary) relation between an element of and a subset of is called a dependence relation, written , if it satisfies the following properties:

  1. if , then ;
  2. if , then there is a finite subset of , such that ;
  3. if is a subset of such that implies , then implies ;
  4. if but for some , then .

Given a dependence relation on , a subset of is said to be independent if for all If , then is said to span if for every is said to be a basis of if is independent and spans

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

If and , then , using property 3. and 1.

Examples

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  • Let   be a vector space over a field   The relation  , defined by   if   is in the subspace spanned by  , is a dependence relation. This is equivalent to the definition of linear dependence.
  • Let   be a field extension of   Define   by   if   is algebraic over   Then   is a dependence relation. This is equivalent to the definition of algebraic dependence.

See also

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This article incorporates material from Dependence relation on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.