matroid independence axioms
The third axiom, (I3), is equivalent to the following alternative axiom:
(I3*) If and and and are both maximal subsets of with the property that they are in , then .
Suppose (I3) holds, and that and are maximal independent subsets of . Also assume, without loss of generality, that . Then there is some such that , but , contradicting maximality of .
Now suppose that (I3*) holds, and assume that with . Let . Then cannot be maximal in by (I3*), so there must be elements such that is maximal, and by construction these . So (I3) holds. ∎
|Title||matroid independence axioms|
|Date of creation||2013-03-22 14:44:05|
|Last modified on||2013-03-22 14:44:05|
|Last modified by||sgraves (6614)|
|Synonym||matroid independent sets|