1. **Problem (a): Analyze parallel and serialized tasks from a Hasse diagram** Given a Hasse diagram representing a partially ordered set (POSET) of tasks, we identify which tasks can run in parallel and which must be serialized. 2. **Understanding POSET and Hasse diagrams** - A POSET is a set with a partial order relation $\leq$ that is reflexive, antisymmetric, and transitive. - In task scheduling, $a \leq b$ means task $a$ must complete before task $b$ starts. - The Hasse diagram shows cover relations without transitive edges. 3. **Parallel vs Serialized tasks** - Tasks are serialized if there is a chain $a \leq b$ or $b \leq a$. - Tasks are parallel if they are incomparable, i.e., neither $a \leq b$ nor $b \leq a$. 4. **Reasoning** - Identify chains in the Hasse diagram: tasks connected by edges or paths. - Tasks on the same chain must be serialized. - Tasks on different branches without order relations can run in parallel. --- 5. **Problem (b): Design a POSET for software modules A, B, C, D** - Modules: Input Validation (A), Encryption (B), Logging (C), Report Generation (D). 6. **Establish relations based on data dependency and secure workflow** - Input Validation (A) must precede Encryption (B) and Logging (C) because data must be validated first. - Encryption (B) must precede Report Generation (D) to ensure secure data. - Logging (C) can occur after Input Validation (A) and before or after Encryption (B), but before Report Generation (D) to record events. Relations: $$ A \leq B, \quad A \leq C, \quad B \leq D, \quad C \leq D $$ 7. **Hasse diagram** - Bottom: A - Middle level: B and C (both depend on A, incomparable to each other) - Top: D (depends on both B and C) 8. **Justification** - This partial order ensures data is validated before encryption and logging. - Encryption and logging can proceed in parallel after validation. - Report generation waits for both encryption and logging, ensuring secure and complete workflow.