Independence Is Not Hereditary to Sub-Events

Let $\Omega = \{1,2,3,4\}$ with uniform probability. Define $A = \{1,2\}$ and $B = \{1,3\}$. (a) Verify that $A$ and $B$ are independent. (b) Let $A_1 = \{1\} \subseteq A$. Determine whether $A_1$ and $B$ are independent. (c) State a general principle: if $A \perp\!\!\perp B$ and $A_1 \subseteq A$, does $A_1 \perp\!\!\perp B$ necessarily hold?