Let f: A->B be given. Prove the following:?

Question

For each subset XcA, Xc f^-1(f(X))

Answer 1

Let x∈X. Then f(x)∈f(X). But f⁻¹(f(X)) is precisely the set of those elements such that f(x)∈f(X), so x∈f⁻¹(f(X)). Since this was true for an arbitrary x∈X, it follows that X ⊆ f⁻¹(f(X)). Q.E.D.