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Applications of Formal Methods, Modeling, and Testing Strategies for Safe Software Development
Published in Qamar Mahboob, Enrico Zio, Handbook of RAMS in Railway Systems, 2018
Alessandro Fantechi, Alessio Ferrari, Stefania Gnesi
The Z notation (Spivey 1989) is a formal specification language used for describing and modeling computing systems. Z is based on the standard mathematical notation used in axiomatic set theory, lambda calculus, and first-order predicate logic. All expressions in Z notation are typed, thereby avoiding some of the paradoxes of naive set theory. Z contains a standardized catalog (called the mathematical toolkit) of commonly used mathematical functions and predicates. The Z notation has been at the origin of many other systems such as, for example, Alloy2 (Jackson 2012) and its related tool Alloy Analyzer, which adapts and extends Z to bring in fully automatic (but partial) analysis.
Preliminaries
Published in Aliakbar Montazer Haghighi, Indika Wickramasinghe, Probability, Statistics, and Stochastic Processes for Engineers and Scientists, 2020
Aliakbar Montazer Haghighi, Indika Wickramasinghe
The Zermelo–Fraenkel set theory was based on symbolic mathematics, as we do nowadays. However, Bertrand Russel raised a question through an example in 1901 that later became known as Russell’s paradox. He tried to show that some attempted formalizations of the set theory created by Cantor lead to a contradiction. Russell’s paradox (which apparently had been discovered a year before by Ernst Zermelo, but never published) appeared in his celebrated paper Principles of Mathematics in 1903. The classic Russel’s paradox is given through the following example.
Set Theory and General Topology
Published in Kenneth Kuttler, Modern Analysis, 2017
To begin with we review some basic set theory. There are axioms of set theory which may be found in the book by Haimos, Naive Set Theory [23], which seem very obvious. Of course much can be said about these axioms and we refer to the above book for more discussion.
A Novel Method for Solving Simultaneous Equations in Boolean/Switching Algebra
Published in IETE Journal of Education, 2018
In Set algebra, the symbol “∪” is used to denote the “union” operation of two sets A and B. Thus if A = {1, 2, 3, 4, 5} and B = {2, 5, 7, 8, 9, 10}, then we have, A ∪ B = {1, 2, 3, 4, 5, 7, 8, 9, 10}. Also, in the Set theory, the symbol “∩” is used to denote the “intersection” operation of two sets. Thus considering again the sets A = {1, 2, 3, 4, 5} and B = {2, 5, 7, 8, 9, 10}, we have, A ∩ B = {2, 5}.