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

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

φ

D

(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.