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Let $V$ be a vector space and $S(V)$ the set of all sub-spaces of $V$. Is the operation of $\cap$ distributive against $+$ in $S(V)$. Is the next statement correct $\forall A,B,C\in V$

$A\cap(B+C)=(A\cap B)+(A\cap C)$

In the answer sheet it says that this is in-correct, yet I don't see how.

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    Let $V$ be two-dimensional and consider three distinct lines through the origin. – 2017-01-04

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Let $V=\mathbb{R}^2$, let $A$ be the set of all $x,y$ such that $y=x$, let $B$ be the $x$-axis and let $C$ be the y-axis. Then $B+C = \mathbb{R}^2$, so the LHS of your equation is just $A$. But $A \cap B = \{0\}$ and $A \cap C = \{0\}$, so the RHS is $\{0\}$.

(The zero here is the zero vector.)