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Introduction

Titel: Introduction Boolean Algebra

Objectives:

- Able to express the basic operations in Boolean algebra
- Know how to describe a logic circuit in a Boolean equation
- Understand the basic Boolean theorems.

Time

Method

Topic

Way

Remark


Q/A

* Review Lesson 1

B



S

* Introduction - George Boole

B



S

* Boolean algebra

B,Ex




S,E


- NOT operation







- OR operation







- AND operation





S,E


- Boolean equations of logic circuits

B,Ex






- NOR gates Truth table

B




S,D


- De Morgan's 1. theorem

B






- NAND gates Truth table

B




E


- De Morgan's 2. theorem

B,HO



E

* Review Exercise

WS

Worksheet No. 2


S: Speech
D: Discussion
Q/A: Question/Answer
E: Exercise


B: Boardscript
P: Picture
Ex: Example
HO: Hands-On
WS: Worksheet
HT: Hand-Out


Boolean algebra

George Boole (1854) invented a new kind of algebra that could be used to analyse and design digital and computer circuits.

NOT operation


Fig. 2-1: Inverter symbol and Boolean notation

Ex: If A is 0 (low) ® X = NOT 0 = 1

In Boolean algebra the overbar stands for NOT operation.

OR operation


Fig. 2-2: OR symbol and Boolean notation

Ex: If A = 0, B = 1 ® X = A or B = 0 or 1 = l

In Boolean algebra the + sign stans for the OR operation:

X = A + B

Ex: If A = 1, B = 0 ® X = A + B = 1 + 0 = 1

AND operation


Fig. 2-3: AND symbol and Boolean notation

In Boolean algebra the multiplication sign stands for the AND operation:

X = A · B

or simply:

X = A B

Ex: If A = 1, B = 0 ® X = A B = 1 · 0 = 0

Boolean equations of logic circuits

You can use Boolean algebra as a shorthand notation for digital circuits.

Ex:


Fig. 2-4: Digital circuit

Output of the first gate:

X3 = A + B

Output of the second gate:

X6 = X3 + C = A + B + C

The final output is:

X8 = X6 + D = A + B + C + D = X

HO: Find the Boolean equation for the following circuit.


Fig. 2-5: Logic circuit

Solution:

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

NOR and NAN gates

NOR gate

Based on the three fundamental logic operations it is possible to design additionel logic devices.


Fig. 2-6: NOR gate, symbol and truth table

Formula:

NAND gate


Fig. 2-7: NAND grate, symbol and truth table

Formula:

De Morgan's theorems

Augustus De Morgan was the first who found the link between logic and mathematics.


Fig. 2-8: Logic circuits with the same output

Although the circuits are different but the output, as we can see in the truth table, is equal. Therefore we can write:

De Morgan's First Theorem

Also if we compare the two circuits on the next page we can easily see that the output is the same, although the circuits are different.

(see Fig. 2-9)


Fig. 2-9: Logic circuits with the same output

So we can write:

De Morgan's Second Theorem

HO: How can you connect a NAND gate to get an inverter

Solution:


Figure

HO: How can you connect NAND gates to get an OR gate?

Solution:


Figure