let A, B and C be subspaces of V . Which of the following statements are true? Let $V$ be a finite-dimensional vector space and let $A$, $B$ and $C$ be subspaces of $V$. Which of the following statements are true?
(a) $$A \cap (B + C) = (A \cap B) + (A \cap C)$$
(b) $$A \cap (B + C) \subset (A \cap B) + (A \cap C)$$
(c) $$A \cap (B + C) \supset (A \cap B) + (A \cap C)$$
My attempt: 
             I was drawing the Venn diagram. From Venn diagram I concluded  that
            $$A \cap (B + C) = (A \cap B) + (A \cap C)$$  is true ..
Is my answer is correct or not, im not sure  about my answer help Me..
 A: In fact, it is only choice c which is correct.
As a counter-example for the other two, consider the following:


*

*$V = \Bbb R^2$

*$A$ is the span of the vector $(1,1)$

*$B$ is the $x$-axis

*$C$ is the $y$-axis

A: Consider the subspaces $B=\langle\, u\,\rangle$, $C=\langle\, v\,\rangle$, $A=\langle\, u+v\,\rangle$. 
Then $A\cap(B+C)=A$, but $A\cap B=A\cap C=\{\,0\,\}$.
A: It is incorrect to make conclusions about subspace operations solely on the set theoretic properties such as De Morgan rules, as the operations on subspaces are sometimes defined differently. 
Let $M, N \leq V$. The usual definition of $M+N$ is $M+N = [M\cup N]$, where $[S]$ is the linear span of a set $S \subseteq V$.
Proof for $(c)$:
Since the intersection of subspaces is a subspace, and a span of a set is a subspace, $A\cap[B\cup C]$ is a subspace and hence equal to its span:
$A\cap[B\cup C] = \left[A\cap[B\cup C]\right] \supseteq \left[A\cap(B\cup C)\right] = \left[(A\cap B)\cup (A\cap C)\right] = (A\cap B) + (A\cap C)$
