Lecture 3

# Lecture 3 - 9/15/2011 CNIT 17600 IT Architectures Digital...

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9/15/2011 1 CNIT 17600 IT Architectures Digital Logic and Boolean Math Readings Chapter 3 for Digital Logic NOT 3.6.6 AND 3.7 Appendix A.1 A.2 Computer Organization & Architecture

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9/15/2011 2 3.1 Introduction 3 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 You should learn the simplicity that constitutes the essence of the machine 3.2 Boolean Algebra 4 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
9/15/2011 3 A Boolean operator can be completely described using a truth table The AND operator is also known as a Boolean product while the OR operator is the Boolean sum 5 3.2 Boolean Algebra The truth table for the Boolean NOT operator is shown at the right The NOT operation is most often designated by an overbar 6 3.2 Boolean Algebra Logical NOT

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9/15/2011 4 3.2 Boolean Algebra 7 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}. The truth table for the Boolean function: To make evaluation of the Boolean function easier, the truth table contains extra (shaded) columns to hold evaluations of subparts of the function. 8 3.2 Boolean Algebra
9/15/2011 5 As with common arithmetic, Boolean operations have rules of precedence The NOT operator has highest priority, followed by AND and then OR This is how we chose the (shaded) function subparts in our table. 9 3.2 Boolean Algebra 3.2 Boolean Algebra 10 Digital computers contain circuits that implement Boolean functions The simpler that we can make a Boolean function, the smaller the circuit that will result Simpler circuits are cheaper to build, consume less power, and run faster than complex circuits With this in mind, we always want to reduce our Boolean functions to their simplest form There are a number of Boolean identities that help us to do this

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9/15/2011 6 3.2 Boolean Algebra 11 Most Boolean identities have an AND (product) form as well as an OR (sum) form. We give our identities using both forms. Our first group is rather intuitive: 3.2 Boolean Algebra 12 Our second group of Boolean identities should be familiar to you from your study of algebra:
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## This note was uploaded on 01/30/2012 for the course CNIT 176 taught by Professor Hansen during the Fall '09 term at Purdue University.

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Lecture 3 - 9/15/2011 CNIT 17600 IT Architectures Digital...

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