1. Circuit Simplification: Boolean Algebra
Digital Electronics ™
2,1 Introduction to AOI Logic
Circuit Simplification: Boolean Algebra
Digital Electronics
© 2014 Project Lead The Way, Inc.
Project Lead The Way, Inc.
Copyright 2009

2. What is Boolean Algebra ?
Digital Electronics ™
2,1 Introduction to AOI Logic
Boolean Algebra is a mathematical technique that provides the ability to algebraically simplify logic expressions. These simplified expressions will result in a logic circuit that is equivalent to the original circuit, yet requires fewer gates.
Introductory Slide / Overview of Presentation
Simplification
With Boolean
Algebra
Project Lead The Way, Inc.
Copyright 2009

3. Boolean Algebra
George Boole
Digital Electronics ™
2,1 Introduction to AOI Logic
My name is George Boole and I lived in England in the 19th century. My work on mathematical logic, algebra, and the binary number system has had a unique influence upon the development of computers. Boolean Algebra is named after me.
A little history…
Project Lead The Way, Inc.
Copyright 2009

4. Boolean Theorems (1 of 7) Single Variable - AND Function Theorem #1
Boolean Algebra
Boolean Theorems (1 of 7)
Digital Electronics ™
2,1 Introduction to AOI Logic
Single Variable - AND Function
Theorem #1
Theorem #2
Theorem #3
Theorem #4 X X 1 X X X Y Z 1 X Y Z 1 X Y Z 1 X Y Z 1
Basic AND Boolean Theorems
Project Lead The Way, Inc.
Copyright 2009

5. Boolean Theorems (2 of 7) Single Variable - OR Function Theorem #5
Boolean Algebra
Boolean Theorems (2 of 7)
Digital Electronics ™
2,1 Introduction to AOI Logic
Single Variable - OR Function
Theorem #5
Theorem #6
Theorem #7
Theorem #8 X X 1 X X 1 X Y Z 1 X Y Z 1 X Y Z 1 X Y Z 1
Basic OR Boolean Theorems
Project Lead The Way, Inc.
Copyright 2009

6. Boolean Theorems (3 of 7) Single Variable - Invert Function Theorem #9
Boolean Algebra
Digital Electronics ™
2,1 Introduction to AOI Logic
Single Variable - Invert Function
Theorem #9
Basic INVERTER Boolean Theorems 1
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Copyright 2009

7. Example #1: Boolean Algebra
Digital Electronics ™
2,1 Introduction to AOI Logic
Example
Simplify the following Boolean expression and note the Boolean theorem used at each step. Put the answer in SOP form.
Pause the presentation and allow students to complete the example. The solution is on the next slide.
Project Lead The Way, Inc.
Copyright 2009

8. Example #1: Boolean Algebra
Digital Electronics ™
2,1 Introduction to AOI Logic
Example
Simplify the following Boolean expression and note the Boolean theorem used at each step. Put the answer in SOP form.
Solution
This slide includes the solution to example #1. If you print handouts, do not print this page.
; Theorem #3
; Theorem #4
; Theorem #1
; Theorem #5
Project Lead The Way, Inc.
Copyright 2009

9. Example #2: Boolean Algebra
Digital Electronics ™
2,1 Introduction to AOI Logic
Example
Simplify the following Boolean expression and note the Boolean theorem used at each step. Put the answer in SOP form.
Pause the presentation and allow the student to complete the example. The solution is on the next slide.
Project Lead The Way, Inc.
Copyright 2009

10. Example #2: Boolean Algebra
Digital Electronics ™
2,1 Introduction to AOI Logic
Example
Simplify the following Boolean expression and note the Boolean theorem used at each step. Put the answer in SOP form.
Solution
; Theorem #3 (twice)
; Theorem #7
; Theorem #2
; Theorem #1
; Theorem #5
This slide includes the solution to example #2. If you print handouts, do not print this page.
Project Lead The Way, Inc.
Copyright 2009

11. Boolean Theorems (4 of 7) Commutative Law Theorem #11 – AND Function
Boolean Algebra
Digital Electronics ™
2,1 Introduction to AOI Logic
Commutative Law
Theorem #11 – AND Function
Theorem #12 – OR Function
Commutative Law
Project Lead The Way, Inc.
Copyright 2009

12. Boolean Theorems (5 of 7) Associative Law Theorem #13– AND Function
Boolean Algebra
Digital Electronics ™
2,1 Introduction to AOI Logic
Associative Law
Theorem #13– AND Function
Theorem #14 – OR Function
Associative Law
Project Lead The Way, Inc.
Copyright 2009

13. Boolean Theorems (6 of 7) Distributive Law Theorem #15 – AND Function
Boolean Algebra
Digital Electronics ™
2,1 Introduction to AOI Logic
Distributive Law
Theorem #15 – AND Function
Theorem #16 – OR Function
Distributive Law
Project Lead The Way, Inc.
Copyright 2009

14. Example #3: Boolean Algebra
Digital Electronics ™
2,1 Introduction to AOI Logic
Example
Simplify the following Boolean expression and note the Boolean theorem used at each step. Put the answer in SOP form.
Pause the presentation and allow students to complete the example. The solution is on the next slide.
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Copyright 2009 14

15. Example #3: Boolean Algebra
Digital Electronics ™
2,1 Introduction to AOI Logic
Example
Simplify the following Boolean expression and note the Boolean theorem used at each step. Put the answer in SOP form.
Solution
; Theorem #15
; Theorem #4
; Theorem #5
; Theorem #14
; Theorem #8
; Theorem #6
; Theorem #2
This slide includes a solution for example #3. If you print handouts, do not print this page.
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Copyright 2009 15

16. Boolean Theorems (7 of 7) Consensus Theorem Theorem #16 Theorem #18
Boolean Algebra
Digital Electronics ™
2,1 Introduction to AOI Logic
Consensus Theorem
Theorem #16
Theorem #18
Theorem #17
Theorem #19
Project Lead The Way, Inc.
Copyright 2009

17. Example #4: Boolean Algebra
Digital Electronics ™
2,1 Introduction to AOI Logic
Example
Simplify the following Boolean expression and note the Boolean theorem used at each step. Put the answer in SOP form.
Pause the presentation and allow students to complete the example. The solution is on the next slide.
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Copyright 2009 17

18. Example #4: Boolean Algebra
Digital Electronics ™
2,1 Introduction to AOI Logic
Example
Simplify the following Boolean expression and note the Boolean theorem used at each step. Put the answer in SOP form.
Solution
; Theorem #14
; Theorem #18
; Theorem #6
; Theorem #2
This slide includes a solution to example #4. If you print handouts, do not print this page.
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Copyright 2009 18

19. Summary Boolean Theorems Commutative Law Associative Law
Boolean Algebra
Summary
Digital Electronics ™
2,1 Introduction to AOI Logic
Boolean Theorems
Commutative Law
Associative Law
Distributive Law
Print this page so that students can use it as a reference guide.
Consensus Theorem
Project Lead The Way, Inc.
Copyright 2009