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:
- if, then ;
- if, then there is a finite subset of, such that ;
- if is a subset of such that implies, then implies ;
- 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
Remark. 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.
Read more about Dependence Relation: Examples
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—Georg Wilhelm Friedrich Hegel (1770–1831)