Applied Set Theory: Foundations and Practice for Computing, Data, and Engineering

This pocket-size book introduces set theory and its applications to computing, data, and engineering. We start with basic concepts such as membership, equality, subsets, relations, and functions. We then consider equivalence, order, cardinality, induction, and recursion. We also discuss infinite sets and the axioms of set theory, along with their consequences for representation and computation.

The applications include Boolean logic, relational databases, programming collections and types, formal specification with Z, B, and Alloy, combinatorial counting, probability, and graphs. The same mathematical concepts apply across contexts. For example, the distinction between sets and multisets helps explain how SQL handles duplicates. Relations model dependencies and connectivity. Different independence assumptions lead to different conclusions about collision probabilities. We also consider fuzzy sets, rough sets, and multisets as generalizations that require separate examination of their operations and laws.

We use definitions, theorems, and worked examples throughout. We pay particular attention to the assumptions behind mathematical results and the differences between abstract models and their software implementations.

The book is intended for software developers, data analysts, engineers, and students interested in the mathematical foundations of their work. A notation reference, an annotated bibliography, and an index are included for further study and practical reference.

Dmitry Vostokov is an internationally recognized expert, speaker, educator, scientist, inventor, and author. He founded the pattern-oriented software diagnostics, forensics, and prognostics discipline (Systematic Software Diagnostics) and Software Diagnostics and Observability Institute (DA+TA: DumpAnalysis.org + TraceAnalysis.org). Vostokov has also authored over 50 books on software diagnostics, anomaly detection and analysis, software and memory forensics, root cause analysis and problem solving, memory dump analysis, debugging, software trace and log analysis, reverse engineering, and malware analysis. He has over 30 years of experience in software architecture, design, development, and maintenance in various industries, including leadership, technical, and people management roles. Dmitry founded OpenTask Iterative and Incremental Publishing (OpenTask.com) and Software Diagnostics Technology and Services (former Memory Dump Analysis Services) PatternDiagnostics.com. In his spare time, he explores Software Narratology, Software Pathology, Software Morphology, Quantum Software Diagnostics, and Pattern-Oriented AI. His interest areas are theoretical software diagnostics and its mathematical and computer science foundations, application of formal logic, semiotics, artificial intelligence, machine learning, and data mining to diagnostics and anomaly detection, software diagnostics engineering and diagnostics-driven development, diagnostics workflow and interaction. Recent interest areas also include functional programming, cloud native computing, monitoring, observability, visualization, security, automation, applications of category theory to software diagnostics, development and big data, and diagnostics of artificial intelligence.