Let E = {1, 2, 3}. Show that there are exactly eight non-isomorphic matroids on E.
So far I listed 7 non-isomorphic matroids but I have no idea what the eighth one should be.
$\{ \emptyset \}$
$\{ \emptyset, \{1\} \}, \{ \emptyset, \{2\} \}, \{\emptyset, \{3\}\}$
$\{ \emptyset, \{1\}, \{2\}\}, \{ \emptyset, \{1\}, \{3\}\}, \{ \emptyset, \{2\}, \{3\}\}$
$\{ \emptyset, \{1\}, \{2\}, \{1, 2\}\}, \{ \emptyset, \{1\}, \{3\}, \{1, 3\}\}, \{ \emptyset, \{2\}, \{3\}, \{2, 3\}\}$
$\{ \emptyset, \{1\}, \{2\}, \{3\}\}$
$\{ \emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}\}$
$\{ \emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\}\}$
Can one of these matroids that I listed be non-isomorphic on their own right (instead of being isomorphic with others as I've shown) or did I just miss something?