Need help proving this statment?

Question

Let f:X → Y be a function. Then for any A⊂Y, f^-1 (Y\A)=X\ f^-1(A).

Answer 1

Suppose x∈f⁻¹(Y\A). Then x∈X (since x is the domain of f), and f(x)∈Y\A, so f(x)∉A, so x∉f⁻¹(A), so x∈X\f⁻¹(A). Thus, f⁻¹(Y/A) ⊆ X\f⁻¹(A). Conversely, suppose x∈X\f⁻¹(A). Then x∉f⁻¹(A), so f(x)∉A, but since Y is the codomain of the function, f(x)∈Y, so f(x)∈Y\A, so x∈f⁻¹(Y\A). Thus, X\f⁻¹A ⊆ f⁻¹(Y\A). It follows that in fact X\f⁻¹A = f⁻¹(Y\A), as required.