Are subsets of independent events independent

You can try with $C:=A\cap B$; this is a subset of $A$ and $$\mathbb P(C\cap B)=\mathbb P( A\cap B) =\mathbb P(A)\mathbb P(B) $$ while $$\mathbb P(B)\mathbb P(C)=(\mathbb P(B))^2\mathbb P(A).$$ Unless we are in a degenerated case (that is, $\mathbb P(A)=0$ or $\mathbb P(B)\in\{0,1\}$), the set $A\cap B$ is not independent of $B$.