boolean algebra and logic gates logic and computer design fundamentals

22
Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Upload: esther-dixon

Post on 28-Dec-2015

256 views

Category:

Documents


2 download

TRANSCRIPT

Page 1: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra

and Logic Gates

Logic and Computer Design Fundamentals

Page 2: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 2

Binary Logic and Gates

Binary variables take on one of two values. Logical operators operate on binary values and

binary variables. Basic logical operators are the logic functions

AND, OR and NOT. Logic gates implement logic functions. Boolean Algebra: a useful mathematical system

for specifying and transforming logic functions. We study Boolean algebra as a foundation for

designing and analyzing digital systems!

Page 3: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 3

Binary Variables Recall that the two binary values have

different names:• True/False• On/Off• Yes/No• 1/0

We use 1 and 0 to denote the two values. Variable identifier examples:

• A, B, y, z, or X1 for now• RESET, START_IT, or ADD1 later

Page 4: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 4

Logical Operations The three basic logical operations are:

• AND • OR• NOT

AND is denoted by a dot (·). OR is denoted by a plus (+). NOT is denoted by an overbar ( ¯ ), a

single quote mark (') after, or (~) before the variable.

Page 5: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 5

Examples:• is read “Y is equal to A AND B.”• is read “z is equal to x OR y.”• is read “X is equal to NOT A.”

Notation Examples

Note: The statement: 1 + 1 = 2 (read “one plus one equals two”)

is not the same as1 + 1 = 1 (read “1 or 1 equals 1”).

BAY yxz

AX

Page 6: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 6

Operator Definitions

Operations are defined on the values "0" and "1" for each operator:

AND 

0 · 0 = 00 · 1 = 01 · 0 = 01 · 1 = 1

OR

0 + 0 = 00 + 1 = 11 + 0 = 11 + 1 = 1

NOT

1001

Page 7: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 7

01

10

X

NOT

XZ

Truth Tables

Tabular listing of the values of a function for all possible combinations of values on its arguments

Example: Truth tables for the basic logic operations:

111

001

010

000

Z = X·YYX

AND OR

X Y Z = X+Y

0 0 0

0 1 1

1 0 1

1 1 1

Page 8: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 8

Truth Tables – Cont’d

Used to evaluate any logic function Consider F(X, Y, Z) = X Y + Y Z

X Y Z X Y Y Y Z F = X Y + Y Z0 0 0 0 1 0 00 0 1 0 1 1 10 1 0 0 0 0 00 1 1 0 0 0 01 0 0 0 1 0 01 0 1 0 1 1 11 1 0 1 0 0 11 1 1 1 0 0 1

Page 9: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 9

Using Switches• Inputs:

logic 1 is switch closed logic 0 is switch open

• Outputs: logic 1 is light on logic 0 is light off.

• NOT input: logic 1 is switch open logic 0 is switch closed

Logic Function Implementation

Switches in series => AND

Switches in parallel => OR

CNormally-closed switch => NOT

Page 10: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 10

Example: Logic Using Switches

Light is on (L = 1) for L(A, B, C, D) =

and off (L = 0), otherwise.

Useful model for relay and CMOS gate circuits, the foundation of current digital logic circuits

Logic Function Implementation – cont’d

B

A

D

C

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

Page 11: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 11

Logic Gates

In the earliest computers, switches were opened

and closed by magnetic fields produced by

energizing coils in relays. The switches in turn

opened and closed the current paths.

Later, vacuum tubes that open and close

current paths electronically replaced relays.

Today, transistors are used as electronic

switches that open and close current paths.

Page 12: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 12

Logic Gate Symbols and Behavior

Logic gates have special symbols:

And waveform behavior in time as follows:

X 0 0 1 1

Y 0 1 0 1

X · Y(AND) 0 0 0 1

X + Y(OR) 0 1 1 1

(NOT) X 1 1 0 0

OR gate

X

YZ = X + Y

X

YZ = X · Y

AND gate

X Z= X

NOT gate orinverter

Page 13: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 13

Logic Diagrams and Expressions

Boolean equations, truth tables and logic diagrams describe the same function!

Truth tables are unique, but expressions and logic diagrams are not. This gives flexibility in implementing functions.

X

Y F

Z

Logic Diagram

Logic Equation

ZY X F

Truth Table

11 1 1

11 1 0

11 0 1

11 0 0

00 1 1

00 1 0

10 0 1

00 0 0

X Y Z Z Y X F

Page 14: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 14

1.

3.

5.

7.

9.

11.

13.

15.

17.

Commutative

Associative

Distributive

DeMorgan’s

2.

4.

6.

8.

X . 1 X=

X . 0 0=

X . X X=

0=X . X

Boolean Algebra

10.

12.

14.

16.

X + Y Y + X=

(X + Y) Z+ X + (Y Z)+=X(Y + Z) XY XZ+=

X + Y X . Y=

XY YX=

(XY) Z X(YZ)=

X + YZ (X + Y) (X + Z)=

X . Y X + Y=

X + 0 X=

+X 1 1=

X + X X=

1=X + X

X = X

Invented by George Boole in 1854 An algebraic structure defined by a set B = {0, 1}, together with two

binary operators (+ and ·) and a unary operator ( )

Idempotence

Complement

Involution

Identity element

Page 15: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 15

Some Properties of Boolean Algebra

Boolean Algebra is defined in general by a set B that can have more than two values

A two-valued Boolean algebra is also know as Switching Algebra. The Boolean set B is restricted to 0 and 1. Switching circuits can be represented by this algebra.

The dual of an algebraic expression is obtained by interchanging + and · and interchanging 0’s and 1’s.

The identities appear in dual pairs. When there is only one identity on a line the identity is self-dual, i. e., the dual expression = the original expression.

Sometimes, the dot symbol ‘’ (AND operator) is not written when the meaning is clear

Page 16: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 16

Boolean Operator Precedence

The order of evaluation is:

1. Parentheses2. NOT3. AND4. OR

Consequence: Parentheses appear around OR expressions

Example: F = A(B + C)(C + D)

Page 17: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 17

Boolean Algebraic Proof – Example 1

A + A · B = A (Absorption Theorem)Proof Steps Justification A + A · B= A · 1 + A · B Identity element: A · 1 = A

= A · ( 1 + B) Distributive

= A · 1 1 + B = 1

= A Identity element

Our primary reason for doing proofs is to learn:• Careful and efficient use of the identities and theorems of

Boolean algebra, and• How to choose the appropriate identity or theorem to apply

to make forward progress, irrespective of the application.

Page 18: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 18

AB + AC + BC = AB + AC (Consensus Theorem)Proof Steps Justification= AB + AC + BC= AB + AC + 1 · BC Identity element= AB + AC + (A + A) · BC Complement= AB + AC + ABC + ABC Distributive= AB + ABC + AC + ACB Commutative= AB · 1 + ABC + AC · 1 + ACB Identity element= AB (1+C) + AC (1 + B) Distributive= AB . 1 + AC . 1 1+X = 1= AB + AC Identity element

Boolean Algebraic Proof – Example 2

Page 19: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 19

Useful Theorems

MinimizationX Y + X Y = Y

AbsorptionX + X Y = X

SimplificationX + X Y = X + Y

DeMorgan’s X + Y = X · Y

Minimization (dual)(X+Y)(X+Y) = Y

Absorption (dual)X · (X + Y) = X

Simplification (dual)X · (X + Y) = X · Y

DeMorgan’s (dual) X · Y = X + Y

Page 20: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 20

Truth Table to Verify DeMorgan’s

X Y X·Y X+Y X Y X+Y X · Y X·Y X+Y

0 0 0 0 1 1 1 1 1 10 1 0 1 1 0 0 0 1 11 0 0 1 0 1 0 0 1 11 1 1 1 0 0 0 0 0 0

X + Y = X · Y X · Y = X + Y

Generalized DeMorgan’s Theorem:

X1 + X2 + … + Xn = X1 · X2 · … · Xn

X1 · X2 · … · Xn = X1 + X2 + … + Xn

Page 21: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 21

Complementing Functions

Use DeMorgan's Theorem:

1. Interchange AND and OR operators

2. Complement each constant and literal   

Example: Complement F =

F = (x + y + z)(x + y + z) Example: Complement G = (a + bc)d + e

G = (a (b + c) + d) e

x zyzyx

Page 22: Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 22

Expression Simplification An application of Boolean algebra Simplify to contain the smallest number

of literals (variables that may or may not be complemented)

= AB + ABCD + A C D + A C D + A B D

= AB + AB(CD) + A C (D + D) + A B D

= AB + A C + A B D = B(A + AD) +AC

= B (A + D) + A C (has only 5 literals)

DCBADCADBADCABA