Let $L$, $M$, and $N$ are subspaces of a vector space. Prove that following is not necessarily true.
$L \cap (M + N) = (L \cap M) + (L \cap N) $
This problem is given in 'Finite dimensional vector spaces' by Halmos. I was using 'if a vector belongs to L.H.S. then it must belong to R.H.S and vice versa' argument. Neither I can disprove it using this argument nor I could find a case where this is wrong!