 Solution

True. The two operations retain complementary membership information. For $x\in C$, equality of the intersections makes $x\in A$ equivalent to $x\in B$. For $x\notin C$, equality of the unions makes $x\in A$ equivalent to $x\in B$. These cases exhaust every possible element. Set extensionality therefore gives $A=B$. Equivalently, reconstruct either set fromSubstituting the two given equalities into this expression gives $B$.