Lecture 9
نویسندگان
چکیده
(Note that for circuits we implicitly assume that ∅ / ∈ C(M), just as we assume that for matroids ∅ ∈ I.) Proof: 1. follows from the definition that a circuit is a minimally dependent set, and therefore a circuit cannot contain another circuit. 2. Let X, Y ∈ C(M) where X 6= Y , and e ∈ X ∩ Y . From 1, it follows that X \ Y is non-empty; let f ∈ X \ Y . Assume on the contrary that (X ∪ Y ) − e is independent. Since X is a circuit, therefore X − f ∈ I. Extend X − f to a maximal independent set in X ∪ Y , call it Z. Then Z ⊆ X∪Y , and Z does not contain Y (otherwise Y would be an independent set as well). Therefore |Z| < |(X ∪ Y )− e|, which is a contradiction to the maximality of Z. One can give an alternative definition of matroids in terms of circuits as follows; this is given without proof.
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