Inverses Unions Intersections and Complements
Why
The inverse of a function interacts nicely with family unions, family intersections and complements.
Results
Let $f: X \to Y$. Throughout this sheet, let $f^{-1}: \powerset{Y} \to \powerset{X}$. And take $\set{B_i}$ to be a family of subsets of $Y$.1
$f^{-1}(\cup_i B_i) = \cup_{i} f^{-1}(B_i)$
$f^{-1}(\cup_i B_i) = \cap_{i} f^{-1}(B_i)$
$f^{-1}(Y - B) = X - f^{-1}(B)$
Properties for function image
Notice that $f(\cup_i A_i) = \cup_i f(A_i)$ but not for interesctions. Nor is there a similar correspondence for complements. There are some relations, which we list below.2 % Notice that $f(x) = f(x) \iff x = y$ means that $f$ is one-to-one.
$f(A \cap B) = f(A) \cap f(B)$ if and only if
$f$ is one-to-one.
For all $A \subset X$, $f(X - A) = Y -
f(A)$ if and only if $f$ is one-to-one.
For all $A \subset X$, $Y - f(A) \subset f(X
- A)$ if and only if $f$ is onto.