Download • Commutative Law of Addition:

Floyd
Digital Fundamentals, 9/e
CHAPTER 4
Boolean Algebra and Logic Simplification
Slide 1
Boolean Operations and Expressions
• Addition
• Multiplication
0+0=0
0+1=1
1+0=1
1+1=1
0*0=0
0*1=0
1*0=0
1*1 =1
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Slide 2
Floyd
Digital Fundamentals, 9/e
Laws of Boolean Algebra
• Commutative Law of Addition:
A+B=B+A
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 3
Floyd
Digital Fundamentals, 9/e
Laws of Boolean Algebra
• Commutative Law of Multiplication:
A*B=B*A
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 4
Floyd
Digital Fundamentals, 9/e
Laws of Boolean Algebra
• Associative Law of Addition:
A + (B + C) = (A + B) + C
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 5
Floyd
Digital Fundamentals, 9/e
Laws of Boolean Algebra
• Associative Law of Multiplication:
A * (B * C) = (A * B) * C
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 6
Floyd
Digital Fundamentals, 9/e
Laws of Boolean Algebra
• Distributive Law:
A(B + C) = AB + AC
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Slide 7
Floyd
Digital Fundamentals, 9/e
Rules of Boolean Algebra
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 8
Floyd
Digital Fundamentals, 9/e
Rules of Boolean Algebra
• Rule 1
OR Truth Table
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 9
Floyd
Digital Fundamentals, 9/e
Rules of Boolean Algebra
• Rule 2
OR Truth Table
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 10
Floyd
Digital Fundamentals, 9/e
Rules of Boolean Algebra
• Rule 3
AND Truth Table
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 11
Floyd
Digital Fundamentals, 9/e
Rules of Boolean Algebra
• Rule 4
AND Truth Table
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 12
Floyd
Digital Fundamentals, 9/e
Rules of Boolean Algebra
• Rule 5
OR Truth Table
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 13
Floyd
Digital Fundamentals, 9/e
Rules of Boolean Algebra
• Rule 6
OR Truth Table
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 14
Floyd
Digital Fundamentals, 9/e
Rules of Boolean Algebra
• Rule 7
AND Truth Table
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 15
Floyd
Digital Fundamentals, 9/e
Rules of Boolean Algebra
• Rule 8
AND Truth Table
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 16
Floyd
Digital Fundamentals, 9/e
Rules of Boolean Algebra
• Rule 9
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 17
Floyd
Digital Fundamentals, 9/e
Rules of Boolean Algebra
• Rule 10: A + AB = A
AND Truth Table OR Truth Table
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 18
Floyd
Digital Fundamentals, 9/e
Rules of Boolean Algebra
• Rule 11: A + AB = A + B
AND Truth Table OR Truth Table
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 19
Floyd
Digital Fundamentals, 9/e
Rules of Boolean Algebra
• Rule 12: (A + B)(A + C) = A + BC
AND Truth Table OR Truth Table
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 20
Floyd
Digital Fundamentals, 9/e
DeMorgan’s Theorems
• Theorem 1
XY = X + Y
• Theorem 2
X + Y = XY
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Slide 21
Floyd
Digital Fundamentals, 9/e
Standard Forms of Boolean Expressions
• The sum-of-product (SOP) form
Example: X = AB + CD + EF
• The product of sum (POS) form
Example: X = (A + B)(C + D)(E + F)
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 22
Floyd
Digital Fundamentals, 9/e
The Karnaugh Map
3-Variable Example
3-Variable Karnaugh Map
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 23
Floyd
Digital Fundamentals, 9/e
The Karnaugh Map
4-Variable Karnaugh Map
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 24
Floyd
Digital Fundamentals, 9/e
The Karnaugh Map
4-Variable Example
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Slide 25
Floyd
Digital Fundamentals, 9/e
The Karnaugh Map
5-Variable Karnaugh Mapping
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.