Let $(X,I)$ be a matroid ($X$ is a finite set). We define $I subseteq 2^S$.

By definition $(X,I)$ is a matroid iff

a) $ Zin I$ and $Y subseteq Z implies Yin I$ (hereditary condition)

c) $Y,Z in I $ and $ |Y|lt |Z|$ then there exists an $z in Zsetminus Y$ such that $(Ycup z) in I$

Now, we consider finite sets of rational numbers $Q$, with the condition that the sum of the elements of any arbitrary $Q$ is less than or equal to 1.

That being said, if we define $Q$ so that $Q={-5,5}$, then it would mean that $Qin S$, however, according to part b) of our definition, a subset $Y$, say, in this case $Y={5}$, would also have to be in $S$, which is not the case. We conclude; $(X,I)$ can’t possibly be a matroid.

Is this proof permissible? I would really appreciate you guys’ thoughts đź™‚