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Find subsets $W$ and $V$ of $\mathbb{R}^3$ such that $\mathbb{R}(W\cap V)\neq\mathbb{R}W\cap \mathbb{R}V$.

I'm not sure how to find these sets. I'm sure there is an elementary solution. Any solutions or hints are greatly appreciated.

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Take $U$ and $V$ to be non empty disjoint subsets of the $x$-axis.

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  • $\begingroup$ Are these subsets or subspaces? $\endgroup$ – Omnomnomnom Aug 29 '15 at 16:28
  • $\begingroup$ Then why do you think that they can't they be disjoint? $\endgroup$ – Omnomnomnom Aug 30 '15 at 0:11
  • $\begingroup$ You seem to be assuming that $U$ and $V$ have a particular form. How about $U=\{(1,0,0)\}$ and $V=\{(2,0,0)\}$. Surely, these two sets are disjoint. $\endgroup$ – Omnomnomnom Aug 30 '15 at 1:26
  • $\begingroup$ Answer me this: what is the set $V\cap W$ in this case? What about the set $\Bbb RV$? $\Bbb R W$? $\endgroup$ – Omnomnomnom Aug 30 '15 at 1:46
  • $\begingroup$ In other words, $\Bbb RV$ and $\Bbb RW$ are each equal to the entire $x$-axis. Do you see why? $\endgroup$ – Omnomnomnom Aug 30 '15 at 2:02

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