Given a matroid $(S, F)$, $\forall x,y,z \in S$, if $\{x\}, \{y\}, \{z\} \in F$, $\{x,y\} \notin F, \{y,z\} \notin F$, will $\{x, z\} \notin F$? I can't figure this out by definition of matroid.
Motivation from an example from Corollary 3.1.7 in Lovasz's Matching Theory:
Given a graph $G$, the class of sets of vertices, where each set can be missed by some maximum matching in $G$, forms a matroid. Let $D \subseteq V(G)$ be the set of those points of $G$, where each point can be missed by some maximum matching. For all $u,v \in D$, define $u \sim v$ if and only if $u=v$ or no maximum matching misses both $u$ and $v$. Then $\sim$ is an equivalence relation on $D$.
Thanks!