Matroids: Prove that for circuit $C$ and its cocircuit $C^*$: $|C \cap C^*| \neq 1$

Question

Prove that for circuit $C$ and its cocircuit $C^*$: $|C \cap C^*| \neq 1$

Any hints and assistance would be very nice! Thank you.

Answer 1

I'm not sure what you mean for "its cocircuit". I'm not aware of any correspondence between circuits and cocircuits. But the statement is true for any circuits and cocircuits in general.

Assume by way for contradiction $C \cap C^* = {x}$. The complement of $C^*$ is a hyperplane $H$, meaning $cl(H) = H$. Note that $cl(C - x) \supseteq C$ because a linear combination of the other elements of $C$ can be taken to form $x$ by the definition of circuit.

We have $C - x \subseteq H$, and since hyperplanes are closed under closure, we get $C \subseteq H$. So $x$ is in $H$ and its complement, a contradiction.

Answer 2

Here's a shorter answer: if $C \cap C^* = x$ in a matroid $M$, then in $M \backslash (C^* - x)$, the set $C$ is a circuit and $x$ is a coloop that belongs to a circuit; a contradiction.