Boolean 1.1

Boolean Logic 1

©Paul GodinCreated September 2007Last Edit September 2009

prgodin @ gmail.com

Boolean 1.2

Boolean Simplification

◊ Boolean equations are used to describe a logic circuit’s function.

◊ Equations can become complex and require simplification.

◊ There are laws and theorems to help simplify complex Boolean problems.

◊ Manipulating Boolean equations follows many of the rules of standard algebra.

Boolean 1.3

The 3 Boolean Laws

◊ Commutative:◊ Addition: A + B = B + A◊ Multiplication: AB = BA

◊ Associative:◊ Addition: A + (B + C) = (A + B) + C◊ Multiplication: A(BC) = (AB)C

◊ Distributive:◊ A(B + C) = AB + AC◊ (A + B)(C + D) = AC + AD + BC + BD

Boolean 1.4

The 10 Basic Rules (part 1)

1. Anything ANDed with a 0 is equal to 0: A ● 0 = 0

2. Anything ANDed with a 1 is equal to itself: A ● 1 = A

3. Anything ORed with a 0 is equal to itself: A + 0 = A

4. Anything ORed with a 1 is equal to 1: A + 1 = 1

5. Anything ANDed with itself is equal to itself: A ● A = A

Boolean 1.5

The 10 Basic Rules (part 2)

6. Anything ORed with itself is equal to itself: A + A = A

7. Anything ANDed with its own complement equals 0: A ● A = 0

8. Anything ORed with its own complement equals 1: A + A = 1

9. Anything complemented twice is equal to the original: A = A

10. The two variable rules:

a) A + AB = A + Bb) A + AB = A + Bc) A + AB = A

Boolean 1.6

De Morgan’s Theorem Review

◊ De Morgan’s Theorem allows the inversion of an expression to be broken up into inversions of individual variables.

◊ Inversion of an expression: A + B

◊ Inversion of individual variables: A ● B

“Break the bar and change the sign”

Boolean 1.7

7. A ● 0 = ___

8. A + A = ____

9. A + A = ____

10. A + AB = ____

11. A ● 1 = ____

12. A = ____

Basic Boolean Rules Exercise 1

1. A + 0 = ____

2. A + AB = ____

3. A + 1 = ____

4. A + AB = ____

5. A ● A = ____

6. A ● A = ____

Determine the outcome of the following:

Boolean 1.8

Basic Boolean Rules Exercise 2

Determine the output of the following gates

?0

A

?1

?0

?1

AA

A’A

A’A

Boolean 1.9

Boolean Simplification

◊ Boolean equations can be simplified using algebraic methods, using the Boolean rules and laws to reduce the equation.

Boolean 1.10

Examples of Boolean Reduction 1

◊ Consider CD(D+DF)◊ CD(D) Rule 10a where D+DF=D◊ CDD Associative Law◊ CD Rule 5 where DD=D

◊ Consider C’D’(C+D)’◊ C’D’(C’D’) DeMorgan where (C+D)=(CD)◊ C’C’D’D’ Associative where brackets removed◊ C’D’ Rule 5 where C’C’=C’ and D’D’=D’

Boolean 1.11

Examples of Boolean Reduction 2◊ Consider (C+D)(C+D’)

◊ CC+CD’+DC+DD’ Distributive◊ C+CD’+CD+0 Rule 5: CC=C; Rule

7:DD’=0, ◊ C+CD’+CD Rule 3: (A+0=A)◊ (C+CD’)+CD Associative◊ C+CD Rule 10c: C+CD’=C◊ C+C Rule 10c: C+CD=C◊ C Rule 6: C+C=C

◊ Consider C’+CDE+E◊ (C’+CDE)+E Associative◊ C’+DE+E Rule 10b: C’+CDE=C’+DE◊ C’+(E+DE) Associative, Commutative◊ C’+E Rule 10c: E+DE=E

Boolean 1.12

Exercise 3

◊ Simplify the following:◊ A’+AB’+B

◊ A+A’B+B’C+AC

◊ AB’+A’CD+B+C’+D’

Other examples may be given in class

Boolean 1.13

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©Paul R. Godinprgodin°@ gmail.com