if the union of three subspaces is a subspace, then one of the subspaces is contained in the union of the other two [duplicate]
Let $K=\Bbb F_2$ and $V$ be a vector space over K.
Let $A,B,C$ be subspaces of $V$.
If $A\cup B\cup C$ is a subspace of $V$, then one of $A,B,C$ is contained in the union of the other two, ie one of the following holds: $$A \subseteq B\cup C$$ $$B \subseteq C\cup A$$ $$C \subseteq A\cup B$$
How do I show if this is right or wrong?
Take $V = K^2.$ Then pick $A = \{(0, 0), (1, 0)\}, B = \{(0, 0), (0, 1)\},$ and $C = \{(0, 0), (1,1)\}.$ Thus $A \cup B \cup C = V$ is a subspace, but none of your inclusions hold.