An example of 3 subspaces of $V$ such that $w_1 \cap (w_2+w_3) \neq (w_1 \cap w_2) + (w_1 \cap w_3)$

Question

Vector space is a space consisting of vectors, together with the associative and commutative operation of addition of vectors, and the associative and distributive operation of multiplication of vectors by scalars.

For each $W \subseteq V \ $ if $\ W$ is a vector space itself (which means that it is closed under operations of addition and scalar multiplication), with the same vector space operations as $V$ has, then $W$ is a subspace of $V$.

Question:
Give an example of 3 subspaces of $V$ such that $w_1 \cap (w_2+w_3) \neq (w_1 \cap w_2) + (w_1 \cap w_3)$

Note: I tried everything i could but the two sides are equal in all of my examples ...

Thanks in advance.

Answer 1

Let $V=\mathbb R^2$ and $W_1=[x=y], W_2=[x=0], W_3=[y=0]$.