#1: Prove or provide a counterexample:

For all sets A, B, C, if A is subset of B and B is a subset of C^c (complement of C), then AC= { }.

This is my proof so far, but I don't know if it is correct?

Assume to the contrary that A ⋂ C = { }. This means that there exists x∈(A⋂C). This then means that x∈A and x∈C. By our starting assumption, if x∈A, then x∈B because A⊆B. Also, if x∈B, then x∈C^C given that B⊆C^C. This shows that there is a contradiction because x∈C and x∈C^C. That means that A⋂C = { } is false. Therefore,the original statement of A⋂C = { } is true.