Which of the following statements are true, and which false? Give brief justifications for your answers.

(a) If $U$ and $W$ are subspaces of a vector space $V$, then $U \cap W$ is always a subspace of $V$.

(b) If $U$ and $W$ are distinct subspaces of a vector space $V$, then $U \cup W$ is never a subspace of $V$.

(c) If $U, W$ and $X$ are subspaces of a vector space $V$, then $U \cap(W+X)=$ $(U \cap W)+(U \cap X)$.

(d) If $U$ is a subspace of a finite-dimensional space $V$, then there exists a subspace $W$ such that $U \cap W=\{0\}$ and $U+W=V$.