Let f: A→B be a function, and X⊆A.Prove or disprove: f(f^−1(f(X)))=f(X).

Question

More clearly written: Let f: A→B be a function, and X⊆A.Prove or disprove: f(f^-1(f(X)))=f(X).

I'm not sure where to even begin with this question...

Answer 1

Recall, f(X) = {f(x) : x ∈ X} and if Y ⊆ B then f^-1(Y) = {x ∈ A : f(x) ∈ Y} by definition. The statement in the question is true. Here is how to prove it.

First suppose y ∈ f(f^-1(f(X))). Then y = f(x) for some x ∈ f^-1(f(X)). Then by definition of this set, f(x) ∈ f(X), so y ∈ f(X). This proves f(f^-1(f(X))) ⊆ f(X).

Conversely, suppose y ∈ f(X). Then y = f(x) for some x ∈ X. Since f(x) ∈ f(X), we get x ∈ f^-1(f(X)), so

y ∈ f(f^-1(f(X))), which proves the reverse inclusion, so the two sets are equal.