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I'm a math student and I'm studying matroids. I tried to prove it myself, but I just couldn’t do it.

Just note that the book I'm following, called Coxeter Matroids by A. V. Borovik, I. M. Gelfand and N. White, introduced Matroids as a collection of subsets of a certain set E that satisfies the Exchange Property, which states that if $A$ and $B$ are elements of the matroid, and $a’ \in A \setminus B$, there exists $b’ \in B \setminus A$ such that $A \setminus \{a’\} \cup \{b’\}$ is an element of the matroid.

To that book, independent sets are just subsets of E that are contained it at least one base of the matroid, so a matroid contains only bases. I tried to search the proof of the theorem cited in the title in several books but they all use the rank function. I’m writing a thesis and I didn’t define the rank function, and I was able to prove all the theorems but this one. It would be silly to make a whole bunch of definitions just to prove this theorem that isn’t essential at all in the thesis.

Could someone help me? Thanks in advance.

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  • $\begingroup$ The proof you want depends on which cryptomorphic definition of matroid quotient you are using. Have you characterized matroid quotients in terms of circuits/cocircuits? $\endgroup$ Commented Sep 13, 2020 at 16:37
  • $\begingroup$ As I introduced them, $M’$ Is a matroid quotient of $M$ if it is a matroid itself and each circuit C of $M$ can be expressed as an union of circuits in $M’$. We defined the rank of a matroid $M$ as the cardinality of one of its elements, thus the cardinality of all of its elements. The properties I was able to show so far are: if $M’$ is a quotient of $M$ then rank($M’$) $\leq$ rank($M$). If rank($M’$) = rank($M$) then $M’ = M$; each base in $M’$ is contained in a base of $M’$; each base in $M$ contains an indipendent set in $M’$. I haven’t defined cocircuits. Thanks a lot in advance! $\endgroup$ Commented Sep 13, 2020 at 18:56

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You can prove this using the relationship between independent sets and circuits. A set $I$ is independent in $M'$ if and only if every circuit $C'$ of $M'$ is disjoint from $I$. Now you can apply your definition of matroid quotient--do you see how to proceed?

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