ECE2030 Introduction to Computer Engineering Lecture 5: Boolean Algebra

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ECE2030 Introduction to Computer
EngineeringLecture 5 Boolean Algebra
Prof. Hsien-Hsin Sean Lee School of Electrical
and Computer Engineering Georgia Tech
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What is Boolean Algebra

• An algebra dealing with
• Binary variables by alphabetical letters
• Logic operations OR, AND, XOR, etc
• Consider the following Boolean equation

• A Boolean function can be represented by a truth
  table which list all combinations of 1s and 0s
  for each binary value

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Fundamental Operators

• NOT
• Unary operator
• Complements a Boolean variable represented as A,
  A, or A
• OR
• Binary operator
• A OR-ed with B is represented as A B
• AND
• Binary operator
• A AND-ed with B is represented as AB or AB
• Can perform logical multiplication

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Binary Boolean Operations

• All possible outcomes of a 2-input Boolean
  function

A B F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
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Precedence of Operators

• Precedence of Operator Evaluation (Similar to
  decimal arithmetic)
• () Parentheses
• NOT
• AND
• OR

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Function Evaluation
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Basic Identities of Boolean Algebra
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Derivation of Simplification
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Derivation of Consensus Theorem
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Duality Principle

• A Boolean equation remains valid if we take the
  dual of the expressions on both sides of the
  equals sign
• Dual of expressions
• Interchange 1s and 0s
• Interchange AND (?) and OR ()

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Duality Principle
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Simplification Examples
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DeMorgans Law
?
?
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Example in Lecture 4
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Another Way to Draw It