Ch03_ECOA4e REVISED 10_2014 - Chapter 3 Boolean Algebra and...

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Chapter 3 Boolean Algebra and Digital Logic
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2 Chapter 3 Objectives Understand the relationship between Boolean logic and digital computer circuits. Learn how to design simple logic circuits. Understand how digital circuits work together to form complex computer systems.
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3 In the latter part of the nineteenth century, George Boole incensed philosophers and mathematicians alike when he suggested that logical thought could be represented through mathematical equations . How dare anyone suggest that human thought could be encapsulated and manipulated like an algebraic formula? Computers, as we know them today, are implementations of Boole’s Laws of Thought . John Atanasoff and Claude Shannon were among the first to see this connection. 3.1 Introduction
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4 In the middle of the twentieth century, computers were commonly known as “thinking machines” and “electronic brains.” Many people were fearful of them. Nowadays, we rarely ponder the relationship between electronic digital computers and human logic. Computers are accepted as part of our lives. Many people, however, are still fearful of them. In this chapter, you will learn the simplicity that constitutes the essence of the machine. 3.1 Introduction
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5 3.2 Boolean Algebra Boolean algebra is a mathematical system for the manipulation of variables that can have one of two values. In formal logic, these values are “true” and “false.” In digital systems, these values are “on” and “off,” 1 and 0, or “high” and “low.” Boolean expressions are created by performing operations on Boolean variables. Common Boolean operators include AND, OR, and NOT.
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6 A Boolean operator can be completely described using a truth table. The truth table for the Boolean operators AND and OR are shown at the right. The AND operator is also known as a Boolean product. The OR operator is the Boolean sum. 3.2 Boolean Algebra
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7 The truth table for the Boolean NOT operator is shown at the right. The NOT operation is most often designated by a prime mark ( X ). It is sometimes indicated by an overbar ( ) or an “elbow” ( X ). 3.2 Boolean Algebra
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8 A Boolean function has: At least one Boolean variable, At least one Boolean operator, and At least one input from the set {0,1}. It produces an output that is also a member of the set {0,1}. Now you know why the binary numbering system is so handy in digital systems . 3.2 Boolean Algebra
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9 The truth table for the Boolean function: is shown at the right. To make evaluation of the Boolean function easier, the truth table contains extra (shaded) columns to hold evaluations of subparts of the function. 3.2 Boolean Algebra
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10 As with common arithmetic, Boolean operations have rules of precedence.
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