Image of union under function
Let \(f : X \longrightarrow Y\) be a function and \(A, B \subseteq X\). Then \(f(A \cup B) = f(A) \cup f(B)\).

Image of intersection under injection
Let \(f : X \longrightarrow Y\) be injective and \(A, B \subseteq X\). Then \(f(A \cap B) = f(A) \cap f(B)\).

Image of intersection under function
Let \(f : X \longrightarrow Y\) be a function and \(A, B \subseteq X\). Then \(f(A \cap B) \subseteq f(A) \cap f(B)\).

Preimage of union of sets
Let \(f : X \longrightarrow Y\) be a function and \(A, B \subseteq Y\). Then \(f^{-1}(A \cup B) \subseteq f^{-1}(A) \cup f^{-1}(B)\).

Preimage of intersection of sets
Let \(f : X \longrightarrow Y\) be a function and \(A, B \subseteq Y\). Then \(f^{-1}(A \cap B) \subseteq f^{-1}(A) \cap f^{-1}(B)\).