Let f: A → B be a function. For S ⊆ A, define f(S) = {f(x) :...

Prompt

Let f: A → B be a function. For S ⊆ A, define f(S) = {f(x) : x ∈ S}. (a) Prove that f(S ∩ T) ⊆ f(S) ∩ f(T) for all S, T ⊆ A. (b) Give an example where the inclusion in (a) is strict. (c) Prove that f is injective if and only if f(S ∩ T) = f(S) ∩ f(T) for all S, T ⊆ A.