Let \(f: X ightarrow Y\) be a map, \(A \subset X\) and \(B \subset Y\). Show that, in general,

\[f \circ f^{-1}(B) \varsubsetneqq B \text { and } f^{-1} \circ f(A) \supsetneq A \text {. }\]

When does ' \(=\) ' hold in these relations? Provide an example showing that the above inclusions are strict.