 # Dependence relation

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

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

• if a S , then a S ;
• if a S , then there is a finite subset S 0 of S , such that a S 0 ;
• if T is a subset of X such that b S implies b T , then a S implies a T ;
• if a S but a S { b } for some b S , then b ( S { b } ) { a } .
• Given a dependence relation on X , a subset S of X is said to be independent if a S { a } for all a S . If S T , then S is said to span T if t S for every t T . 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 bases of X have the same cardinality.

## Examples

• Let V be a vector space over a field F . The relation , defined by υ S if υ is in the subspace spanned by S , is a dependence relation. This is equivalent to the definition of linear dependence.
• Let K be a field extension of F . Define by α S if α is algebraic over F ( S ) . Then is a dependence relation. This is equivalent to the definition of algebraic dependence.

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