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Can anyone explain to me why a cycle basis hones the properties of a matroid? Especially points 2 & 3.
How can a subset of I also be a member of I? Isn't a cycle basis supposed to be consisted of cycles and cycles only? If the cycle basis consisted a triangle, wouldn't a subset of the triangle become an edge? And by point 2 am I supposed to say that the edge is also a member of I?
I would really appreciate it if anyone can use a square or pentagon graph to explain, cuz my mind is really messed up...

Thanks in advance.

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    Are you referring to points 2 & 3 in the definition of the matroid *in terms of independent sets*? That's obviously not satisfied by the set of cycles in a graph. The independent sets in the "cycle matroid" of a graph $G$ are the forests in $G$, and the bases are the spanning trees. The cycles themselves are the *circuits* in that matroid, i.e., the minimal dependent sets. (If I've understood things correctly; I'm not an expert.)2011-03-28
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    youre confusing the various definitions i think... here is a survey paper www.math.lsu.edu/~oxley/survey4.pdf2011-03-28

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