Question

For any structure function, we define the dual structure \(\phi^{\mathrm{D}}\) by $$ \phi^{\mathrm{D}}(\mathbf{x})=1-\phi(\mathbf{1}-\mathbf{x}) $$ (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.

Short answer

In summary, we have shown that: (a) the dual of parallel (series) systems is a series (parallel) system; (b) the dual of a dual structure is the original structure; (c) the dual of a k-out-of-n structure is given by \( \Omega^{\mathrm{D}}(\mathbf{x}) = 1 - \sum_{j=k}^n C(n,j) \prod_i^j (1 - x_i) \); and (d) the minimal path (cut) set of the dual system corresponds to the minimal cut (path) set of the original system.

Step-by-step solution

1. Dual of Parallel and Series Systems

To show that the dual of a parallel (series) system is a series (parallel) system, let's first recall the structure function representation for parallel and series systems. For a parallel system Φ(X), it can be represented as: \( \Phi(\mathbf{x}) = 1 - \prod_i^n (1 - x_i) \) And for a series system Ψ(X), it can be represented as: \( \Psi(\mathbf{x}) = \prod_i^n x_i \) Let's find the dual structure function for both systems: For the parallel system: \( \Phi^{\mathrm{D}}(\mathbf{x}) = 1 - \Phi(\mathbf{1} - \mathbf{x}) = 1 - \Big(1 - \prod_i^n (1 - (1 - x_i))\Big) = \prod_i^n x_i \) For the series system: \( \Psi^{\mathrm{D}}(\mathbf{x}) = 1 - \Psi(\mathbf{1} - \mathbf{x}) =1 - \prod_i^n (1 - x_i) \) As we can see, the dual structure function for the parallel system is equivalent to the structure function for the series system, and vice versa.

2. Dual of a Dual Structure

To show that the dual of a dual structure is the original structure, let's consider the dual structure of a given structure function Φ(X): \( \Phi^{\mathrm{D}}(\mathbf{x}) = 1 - \Phi(\mathbf{1} - \mathbf{x}) \) Now let's find the dual of the dual structure function: \( (\Phi^{\mathrm{D}})^{\mathrm{D}}(\mathbf{x}) = 1 - \Phi^{\mathrm{D}}(\mathbf{1} - \mathbf{x}) = 1 - \big(1 - \Phi(\mathbf{1} - (\mathbf{1} - \mathbf{x}))\big) \) Simplify to get: \( (\Phi^{\mathrm{D}})^{\mathrm{D}}(\mathbf{x}) = \Phi(\mathbf{x}) \) Thus, the dual of a dual structure is the original structure.

3. Dual of a k-out-of-n Structure

To find the dual of a k-out-of-n structure, we need to write down the structure function for the k-out-of-n structure. Let's represent the structure function as Ω(X): \( \Omega(\mathbf{x}) = \sum_{j=k}^n C(n,j) \prod_i^j x_i \) Now find its dual: \( \Omega^{\mathrm{D}}(\mathbf{x}) = 1 - \Omega(\mathbf{1} - \mathbf{x}) = 1 - \sum_{j=k}^n C(n,j) \prod_i^j (1 - x_i) \)

4. Minimal Path and Cut Sets of Dual Systems

Let M be a minimal path set of the original system (Φ) with its corresponding structure function. To prove that the minimal cut set of the dual system (Φ^D) is derived from the minimal path set of the original system, consider the following: For all \( \mathbf{x} \in M \), we have \( \Phi(\mathbf{x}) = 1 \). Now, we'll find the corresponding minimal cut set for the dual system. \( \Phi^{\mathrm{D}}(\mathbf{1} - \mathbf{x}) = 1 - \Phi(\mathbf{1} - (\mathbf{1} - \mathbf{x})) = 1 - \Phi(\mathbf{x}) = 0 \) The above result signifies that any minimal path set for the original structure system results in a minimal cut set for the dual system. Similarly, for a minimal cut set in the original structure, we can show that it results in a minimal path set for the dual system.