Let $V$ be a finite dimensional Vector Space and let $A$ $B$ and $C$ be sub spaces of $V$ Which of the following is/are true?

Question

Let $V$ be a finite dimensional Vector Space and let $A$ $B$ and $C$ be sub spaces of $V$ Which of the following is/are true ?

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

(ii) $A \cap (B + C) \subset A \cap B + A \cap C$

(iii) $A \cap (B + C) \supset A \cap B + A \cap C$

I have proved that statement(iii)is correct but for options (i) and (ii) I need proper counterexamples to disprove them. '

If I approach geometrically Then I think we can generate a counterexamples to both parts. But I am not able to think of any counter example at the moment .

Can anyone please help me here ?

Answer 1

A counterexample to (ii):

Consider a $2$-dimensional vector space $V$ and distinct $1$-dimensionsal subspaces $A, B,C$. The $B+C=V$, so $A\cap(B+C)=A$, but $A\cap B, \:A\cap C=\{0\}$, so that $A\cap B+A\cap C=\{0\}$.

Of course, this is also a counterexample to (i).