Let $n \geq 1$. $V=Z_2^n \setminus \{0\}$. $A \cup B = V$. $A \cap B = \Phi$.
Is it true that either A or B contains a sub-space of dimension $n-1$ (without the zero element)?
(I reduced a homework question to this one).
Let $n \geq 1$. $V=Z_2^n \setminus \{0\}$. $A \cup B = V$. $A \cap B = \Phi$.
Is it true that either A or B contains a sub-space of dimension $n-1$ (without the zero element)?
(I reduced a homework question to this one).
This is not true. For $n=4$ let $A=\{(0,1,0,0), (0,0,1,0), (0,0,0,1), (1,1,0,0), (1,1,1,0), (1,1,0,1), (1,0,1,1), (0,1,1,1)\}$ $B=\{(1,0,0,0), (1,0,1,0), (1,0,0,1), (0,1,1,0), (0,1,0,1), (0,0,1,1), (1,1,1,1) \}$ $B \cup \{0\}$ is not a subspace, and its size (with the zero vector) is $2^3$, so it doesn't contain a subspace of dimension 4-1=3. Notice that if you remove one vector from A, then it will not be a subspace (with the zero vector) so again $A\cup \{0\}$ doesn't contain a 3 dimensional subspace.