Here is the question:
Let X and Y be sets, and let f : X → Y be a function defined in X with values in Y. Prove, that if A, B ⊂ Y, then
f^−1 (A \ B) = f^-1 (A) \ f^−1 (B).
My attempt to solve the problem:
I need to show that:f^−1 (A \ B) ⊆ f^-1 (A) \ f^−1 (B) and f^-1 (A) \ f^−1 (B) ⊆ f^−1 (A \ B)
if A⊂Y and B⊂Y => A\B⊂Y => {x∈X: f(x)∈A\B} => f(x)∈A and f(x)∉B
if A⊂Y => {m∈X: f(m)∈A} and if B⊂Y => {n∈X: f(n)∈B}
f^-1 (A) \ f^−1 (B) = {m∈X: f(m)∈A} \ {n∈X: f(n)∈B} = {l∈X: f(l)∈A and f(l)∉B} => f(l)∈A\B
After here, I could not find a good way to explicitly show my proof. I am having some troubles with scientific mathematical notation. Could you please help me?
Regards