Question

Let f be a function from A to B, and let C be a subset of B. The inverse image of C is f −1 (C) = {a ∈ A | f(a) ∈ C}. Give a a proof of the following theorem: for any subsets S and T of B, f −1 (S ∪ T) = f −1 (S) ∪ f −1 (T). (You should give an informal proof, meaning a direct proof, proof by contraposition, proof by contradiction, etc.

Answer 1

Given f be a function from A to B,  S and T are two subsets of B.

Proof: let x f^-1(S T), then either f(x) S or f(x) T

So, for f(x) S, x f^-1(S) or for f(x) T, x f^-1(T)

hence x f^-1(S) f^-1(T), that implies f^-1(S T) f^-1(S) f^-1(T)

Conversely, let x f^-1(S) f^-1(T).

if x f^-1(S), then f(x) S S T

if x f^-1(T), then f(x) T S T

So x f^-1(S T)

hence f^-1(S) f^-1(T) f^-1(S T)

So f^-1(S) f^-1(T) f^-1(S T)