Boolean algebra is the study and use of system representations where each quantity has only two possible values. Because they represent many real-world variables, especially in computing machinery, Boolean systems are widely used but are not always presented in a mathematically rigorous way. Further, important mathematical tools such as differential calculus are not often formulated and applied to them. In this short book, the authors define and apply differential calculus as defined for Boolean quantities.

First, Boolean quantities, functions, and Boolean algebra are defined in a methodical and rigorous manner. Steinbach and Posthoff distinguish between a Boolean lattice and a Boolean algebra, defining a Boolean algebra as a lattice that supports distributivity, neutral elements, complementation, and DeMorgan’s laws. Readers will find this initial chapter to be a thorough development of Boolean algebra, although a solid mathematical background is required to fully benefit from it.

From this foundation, the Boolean derivative operation is defined for functions of one or several variables. The first presentation is in the form of a partial derivative, indicating whether the function value changed when a particular dependent variable changed. The derivative carries no “sign” information. Higher orders of derivation are defined and their properties are explored, with careful attention to the notation and accuracy (the book is well-written throughout with no errors detected). In a separate chapter, the Boolean differential operator is also defined and applied.

If the book were just a rigorous mathematical presentation of the Boolean differential calculus, it would be interesting but not useful to many readers. However, the authors present applications of the calculus to a variety of problems. These include solutions of Boolean equations; calculation of graphs; and analysis, synthesis, and testing of digital circuits. The latter set of applications, in particular, is thorough and persuasive of the utility of the calculus.

Although the book is short, it is a mathematically rigorous description of Boolean differential calculus that feels thorough. The applications sections are enough to hint at the power of the approach, but more examples of use would be helpful, especially as applied to graphs. Not all readers will find the book easily readable and applicable, but the overwhelming importance of differential calculus for Boolean systems makes this a useful resource.

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