So we got the following question in the lecture:
How many matroids with a single element exist?
Couldn't really think of an answer. Any assistance would be of help!
2026-03-25 09:22:45.1774430565
How many matroids with 1 element exist?
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I would interpret element to refer to the elements of the ground set. If $E=\{x\}$ is the ground set of a matroid $\langle E,\mathscr{I}\rangle$, how many different families $\mathscr{I}$ of independents sets are possible? Since $\varnothing$ must belong to $\mathscr{I}$, there are only two candidates, $\mathscr{I}=\{\varnothing\}$, and $\mathscr{I}=\{\varnothing, E\}$, and you need only check each to see whether it has the requisite properties.