3. For any structure function, we define the dual structure D by D(x) = 1 (1...

3. For any structure function, we define the dual structure φD by

φD(x) = 1 − φ(1 − x)

(a) Show that the dual of a parallel (series) system is a series (parallel) system.

(b) Show that the dual of a dual structure is the original structure.

(c) What is the dual of a k-out-of-n structure?

(d) Show that a minimal path (cut) set of the dual system is a minimal cut

(path) set of the original structure.