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.
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?