ch04 boolean algebra.pdf

BOOLEAN ALGEBRA

Lecture 5

Digital Design

Dr. PO Kimtho

Department of Computer Sciences

Norton University (NU)

Boolean Algebra

Contents

Boolean Operations & Expression

Laws & Rules of Boolean algebra

DeMorgan’s Theorems

Boolean analysis of logic circuits

Simplification using Boolean Algebra

Standard forms of Boolean Expressions

Boolean Expressions & truth tables

Boolean Algebra (Cont.)

The Karnaugh Map

Karnaugh Map SOP minimization

Karnaugh Map POS minimization

Programmable Logic

Boolean Operations & Expression

Expression

Variable

a symbol used to represent logical quantities (1 or 0)

E.g : A, B,..used as variable

Complement

inverse of variable and is indicated by bar over variable

E.g : Ā

Boolean Operations & Expression

Operation

Boolean Addition

– equivalent to the OR operation

X = A + B

Boolean Multiplication

– equivalent to the AND operation

X = A∙B

A

B X

A

B X

Laws of Boolean Algebra

Commutative

Addition & multiplication

Associative

Addition & multiplication

Distributive

Same as ordinary algebra

Commutative law of addition:

A+B = B+A

the order of OR-ing does not matter.

Addition

Commutative Law

Commutative Law

Commutative law of Multiplication

AB = BA

the order of ANDing does not matter.

Multiplication

Associative Law

Associative law of addition

A + (B + C) = (A + B) + C

The grouping of ORed variables does not matter

Addition

Associative Law

Associative law of multiplication

A(BC) = (AB)C

The grouping of ANDed variables does not matter

Multiplication

Distributive Law

A(B + C) = AB + AC

Question: (A+B)(C+D) ?

Rules of Boolean Algebra

Rule 1: A + 0 = A

In math if you add 0 you have changed nothing.

In Boolean Algebra ORing with 0 changes nothing.

Rule 2: A + 1 = 1

ORing with 1 must give a 1 since if any input

is 1 an OR gate will give a 1.

Rules of Boolean Algebra

Rule 3: A . 0 = 0

In math if 0 is multiplied with anything you

get 0. If you AND anything with 0 you get 0.

Rules of Boolean Algebra

Rule 4: A . 1 = A

ANDing anything with 1 will yield the anything.

Rules of Boolean Algebra

Rule 5: A + A = A

ORing with itself will give the same result

Rules of Boolean Algebra

Rule 6: A + A = 1

Either A or A must be 1 so A + A =1

Rules of Boolean Algebra

Rule 7: A . A = A

ANDing with itself will give the same result

Rules of Boolean Algebra

Rule 8: A . A = 0

In digital Logic 1 =0 and 0 =1, so AA=0 since one of the inputs must be 0.

Rules of Boolean Algebra

Rule 9: A = A

If you NOT something twice, you are back to

the beginning

Rules of Boolean Algebra

Rule 10: A + AB = A

Proof:

A + AB = A (1 + B) DISTRIBUTIVE LAW

= A∙1 RULE 2: (1+B) = 1

= A RULE 4: A∙1 = A

Rules of Boolean Algebra

Rule 11: A + AB = A + B

If A is 1 the output is 1 , If A is 0 the output is B

Proof:

A + AB = (A + AB) + AB RULE 10

= (AA + AB) + AB RULE 7

= AA + AB + AA +AB RULE 8

= (A + A)(A + B) FACTORING

= 1∙(A + B) RULE 6

= A + B RULE 4

Rules of Boolean Algebra

Rule 12: (A + B) (A + C)= A + BC

PROOF

(A + B)(A +C) = AA + AC +AB +BC DISTRIBUTIVE LAW

= A + AC + AB + BC RULE 7

= A(1 + C) +AB + BC FACTORING

= A.1 + AB + BC RULE 2

= A(1 + B) + BC FACTORING

= A.1 + BC RULE 2

= A + BC RULE 4

Rules of Boolean Algebra

SUMMARY - LAWS OF BOOLEAN ALGEBRA

[1] COMMUTATIVE:

A + B = B + A

AB = BA

[2] ASSOCIATIVE:

A + (B + C) = (A + B) + C

A(BC) = (AB)C

[3] DISTRIBUTIVE:

A(B + C) = AB + AC

(A + B)(C + D) = AC + AD + BC + BD

Rules of Boolean Algebra

SUMMARY - RULES OF BOOLEAN ALGEBRA

[1] A + 0 = A [7] A.A = A

[2] A + 1 = 1 [8] A.A = 0

[3] A.0 = 0 [9] A = A

[4] A.1 = A [10] A + AB = A

[5] A + A = A [11] A + AB = A + B

[6] A + A = 1 [12] (A + B)(A + C) = A + BC

Rules of Boolean Algebra

[1] A + 0 = A [7] A.A = A

[2] A + 1 = 1 [8] A.A = 0

[3] A.0 = 0 [9] A = A

[4] A.1 = A [10] A + AB = A

[5] A + A = A [11] A + AB = A + B

[6] A + A = 1 [12] (A + B)(A + C) = A + BC

Rules of Boolean Algebra

SUMMARY - RULES OF BOOLEAN ALGEBRA

DeMorgan’s Theorems

Two most important theorems of Boolean Algebra were

contributed by De Morgan

Extremely useful in simplifying expression in which

product or sum (POS) of variables is inverted

The TWO theorems are:

X.Y = X + Y &

X+Y = X . Y

X.Y = X + Y

(a) & (b) Equivalent circuit implied by the

theorem

(c) Alternative symbol for the NAND function

(d) Truth table that illustrates DeMorgan’s

Theorem

(d)

Input Output

X Y XY X+Y

0 0 1 1

0 1 1 1

1 0 1 1

1 1 0 0

DeMorgan’s Theorems

(a) & (b) Equivalent circuit implied by the

theorem

(c) Alternative symbol for the NOR function

(d) Truth table that illustrates DeMorgan’s

Theorem

Input Output

X Y X+Y XY

0 0 1 1

0 1 0 0

1 0 0 0

1 1 0 0 (d)

X+Y = X . Y

DeMorgan’s Theorems

Solve this

Solve this

Standard Forms of Boolean Expression

Sum of products (SOP)

Sum of products (SOP)

Sum of products (SOP)

Product of sum (POS)

Standard Forms of Boolean Expressions

Standard Forms of Boolean Expressions

Solve this

Example (Standard SOP)

Example (Standard POS)

Boolean Expressions & Truth Tables

Boolean Expressions & Truth Tables

Boolean Expressions & Truth Tables

Solve this

The Karnaugh Map (K-Map)

The K-Map

K-Map SOP Minimization

K-map SOP Minimization

K-map SOP Minimization

K-map SOP Minimization

K-map SOP Minimization

K-map SOP Minimization

K-map SOP Minimization

K-map SOP Minimization

K-map SOP Minimization

K-map SOP Minimization

K-map SOP Minimization

K-map SOP Minimization

K-map SOP Minimization

K-map SOP Minimization

K-map SOP Minimization

Example

Example

Example

K-map POS Minimization

K-map POS Minimization

K-map POS Minimization

K-map POS Minimization

K-map Minimization – Don’t cares

K-map Minimization – Don’t cares

Example

Example

Solve this