DigitalDesign - Introduction to Digital Design Chapter...

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Beta Draft - Do not distribute ' 2001, By Randall Hyde Page 203 Intr oduction to Digital Design Chapter Three Logic circuits are the basis for modern digital computer systems. T o appreciate how computer systems operate you will need to understand digital logic and boolean algebra. This chapter provides only a basic introduction to boolean algebra. That subject alone is often the subject of an entire textbook. This chapter concentrates on those subjects that support other chapters in this text. Chapter Overview Boolean logic forms the basis for computation in modern binary computer systems. Y ou can represent any algorithm, or any electronic computer circuit, using a system of boolean equations. This chapter provides a brief introduction to boolean algebra, truth tables, canonical representa - tion, of boolean functions, boolean function simplification, logic design, and combinatorial and sequential circuits. This material is especially important to those who want to design electronic circuits or write software that controls electronic circuits. Even if you never plan to design hardware or write soft - ware than controls hardware, the introduction to boolean algebra this chapter provides is still important since you can use such knowledge to optimize certain complex conditional expressions within IF , WHILE, and other conditional statements. The section on minimizing (optimizing) logic functions uses V eitch Diagrams or Karnaugh Maps . The optimizing techniques this chapter uses reduce the number of terms in a boolean func - tion. Y ou should realize that many people consider this optimization technique obsolete because reducing the number of terms in an equation is not as important as it once was. This chapter uses the mapping method as an example of boolean function optimization, not as a technique one would regularly employ . If you are interested in circuit design and optimization, you will need to consult a text on logic design for better techniques. 3.1 Boolean Algebra Boolean algebra is a deductive mathematical system closed over the values zero and one (false and true). A binary operator ¡ defined over this set of values accepts a pair of boolean inputs and produces a single boolean value. For example, the boolean AND operator accepts two boolean inputs and produces a single boolean output (the logical AND of the two inputs). For any given algebra system, there are some initial assumptions, or postulates , that the sys - tem follows. Y ou can deduce additional rules, theorems, and other properties of the system from this basic set of postulates. Boolean algebra systems often employ the following postulates: ¥ ¥ Closur e . The boolean system is closed with respect to a binary operator if for every pair of boolean values, it produces a boolean result. For example, logical AND is closed in the boolean system because it accepts only boolean operands and produces only boolean results.
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DigitalDesign - Introduction to Digital Design Chapter...

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