Introduction to Boolean Algebra Part 2 - DeMorgan's Laws, Logic Gates, NOT, OR, AND, NAND, NOR, XOR, XNOR - with MCQ Quizzes



Number System

 



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  Digital Electronics : Boolean Algebra Part 2

 
boolean algebra tutorial digital electronics

 

 

An Outline of this Tutorial. (You can also try out the MCQ Quizzes at the end of this page.)


Here's a very quick outline of what we'll cover in this tutorial. The tutorial document at the end will cover these sub-topics in greater detail, with examples and truth tables.


DEMORGAN’S LAW

It states that

(X+Y)’ = X’.Y’
(X.Y)’ = X’ + Y’
Steps to perform De Morgan’s law- 
  • Complement the whole expression on which DeMorgan’s law needs
     to be applied
  • Change all ANDs to ORs and all ORs to ANDs
  • Then, Complement all the terms in the expression.


LOGIC GATES

A GATE is an electronic circuit that operates on one or more signals to give an output. Gates are two state circuits i.e. the input or output can be either low or high (0 or 1).
The three basic gates are-
  • NOT gate (inverter)
  • AND gate
  • OR gate

NOT gate

A NOT gate performs the not logic that is, it gives the inverted value of input signal as output. It gives a high output for a low input and a low output for a high input. If the input is X then output is X’ hence the output expression for not gate is F(x) = x’.

AND gate


An AND gate operates on two or more inputs and gives one 
output. It gives a low output if any of the inputs is low, and a high 
output only when all the inputs are high. For a two input AND gate, output expression is F(x,y) = x.y. For a 3 input AND gate, F(x,y,z) = x.y.z

XOR gate


An XOR gate operates on two or more inputs and gives one 
output. It gives a high output for odd number of high inputs and a low 
output for even number of high inputs. For a two input XOR gate, output expression is F(x,y) = x + y = xy’ + x’y.  For a 3 input XOR gate, F(x,y,z) = x + y + z.

XNOR gate


An XNOR gate also operates on two or more inputs and gives one output. It gives a low output for odd number of high inputs and a high output for even number of high inputs. For a two input XNOR gate, output expression is F(x,y) = (x . y) = x’y’ + xy

NAND gate

A NAND gate also operates on two or more inputs and gives one output. It is an AND gate preceding a NOT gate. It gives a high output if any of the inputs is low, and a low output only when all the inputs are
high. For a two input NAND gate, output expression is F(x,y) = (x . y)’.
 

NOR gate

A NOR gate also operates on two or more inputs and gives one 
output. It is a NOR gate preceding a NOT gate. It gives a low output 
if any of the inputs is high, and a high output only when all the inputs are low. For a two input NOR gate, output expression is F(x,y) = (x + y)’.

Universality of NAND and NOR gates

Both NAND and NOR gate are very special gates as any Boolean expression can be realized by using only NAND or NOR gate. The tutorial document will show you how.



Realization of Boolean expressions using NAND and NOR gate


Consider the expression

• XY+Z
Now,
X.Y+Z = ((X.Y)’)’ + Z = ((X.Y)’ . Z’)’
X.Y+Z = ((X’+Y’)’+Z) = (((X’+Y’)’+Z)’)’

The above conversions can be proved by using DeMorgan’s law. But the conversions seem to be hard and non-intuitive. A good way to realize a circuit using NAND or NOR gate is the following—
(XY+Z using NAND gate)
(XY+Z using NOR gate)
  • Draw a circuit simply by using AND, NOT and OR gates
  • Replace every gate by NAND/NOR gate according to realization you want to carry out.
You might understand this better once you read the tutorial.



Complete Tutorial with Truth Tables and Examples (You can also try out the companion quizzes at the end of this page):




MCQ Quiz #1- Boolean Algebra


Companion MCQ Quiz for Boolean Algebra- test how much you know about the topic. Your score will be e-mailed to you at the address you provide.

MCQ Quiz #1- Digital Electronics: Boolean Algebra



MCQ Quiz #2- Logic Gates


Companion MCQ Quiz for Logic Gates- test how much you know about the topic. Your score will be e-mailed to you at the address you provide.

MCQ Quiz #2 - Logic Gates


Here's a list of all the tutorials we currently have in this area - Introductory Digital Electronic Circuits and Boolean logic

 Introduction to the Number System : Part 1 
Introducing number systems. Representation of numbers in Decimal, Binary,Octal and Hexadecimal forms. Conversion from one form to the other.
 Number System : Part 2 
Binary addition, subtraction and multiplication. Booth's multiplication algorithm. Unsigned and signed numbers. 
Introduction to Boolean Algebra : Part 1
 Binary logic: True and false. Logical operators like OR, NOT, AND. Constructing truth tables. Basic postulates of Boolean Algebra. Logical addition, multiplication and complement rules. Principles of duality.  Basic theorems of boolean algebra: idempotence, involution, complementary, commutative, associative, distributive and absorption laws. 
Boolean Algebra : Part 2
De-morgan's laws. Logic gates. 2 input and 3 input gates. XOR, XNOR gates. Universality of NAND and NOR gates. Realization of Boolean expressions using NAND and NOR. Replacing gates in a boolean circuit with NAND and NOR.
  Understanding Karnaugh Maps : Part 1 Introducing Karnaugh Maps. Min-terms and Max-terms. Canonical expressions. Sum of products and product of sums forms. Shorthand notations. Expanding expressions in SOP and POS Forms ( Sum of products and Product of sums ). Minimizing boolean expressions via Algebraic methods or map based reduction techniques. Pair, quad and octet in the context of Karnaugh Maps.

Karnaugh Maps : Part 2
Map rolling. Overlapping and redundant groups. Examples of reducing expressions via K-Map techniques.
 Introduction to Combinational Circuits : Part 1
Combinational circuits: for which logic is entirely dependent of inputs and nothing else. Introduction to Multiplexers, De-multiplexers, encoders and decoders.Memories: RAM and ROM.  Different kinds of ROM - Masked ROM, programmable ROM. 
 Combinational Circuits : Part 2
 Static and Dynamic RAM, Memory organization.
Introduction to Sequential Circuits : Part 1 
Introduction to Sequential circuits. Different kinds of Flip Flops. RS, D, T, JK. Structure of flip flops. Switching example. Counters and Timers. Ripple and Synchronous Counters. 
Sequential Circuits : Part 2
ADC or DAC Converters and conversion processes. Flash Converters, ramp generators. Successive approximation and quantization errors. 
 


 






Tutorials in Basic Digital Electronics: At a glance
 

Introduction to the Number System : Part 1 
Introducing number systems. Representation of numbers in Decimal, Binary,Octal and Hexadecimal forms. Conversion from one form to the other.

Number System : Part 2 Binary addition, subtraction and multiplication. Booth's multiplication algorithm. Unsigned and signed numbers.

Introduction to Boolean Algebra : Part 1Binary logic: True and false. Logical operators like OR, NOT, AND. Constructing truth tables. Basic postulates of Boolean Algebra. Logical addition, multiplication and complement rules. Principles of duality. Basic theorems of boolean algebra: idempotence, involution, complementary, commutative, associative, distributive and absorption laws.

Boolean Algebra : Part 2De-morgan's laws. Logic gates. 2 input and 3 input gates. XOR, XNOR gates. Universality of NAND and NOR gates. Realization of Boolean expressions using NAND and NOR. Replacing gates in a boolean circuit with NAND and NOR.

 Understanding Karnaugh Maps : Part 1 Introducing Karnaugh Maps. Min-terms and Max-terms. Canonical expressions. Sum of products and product of sums forms. Shorthand notations. Expanding expressions in SOP and POS Forms ( Sum of products and Product of sums ). Minimizing boolean expressions via Algebraic methods or map based reduction techniques. Pair, quad and octet in the context of Karnaugh Maps.

Karnaugh Maps : Part 2 Map rolling. Overlapping and redundant groups. Examples of reducing expressions via K-Map techniques.

Introduction to Combinational Circuits : Part 1 Combinational circuits: for which logic is entirely dependent of inputs and nothing else. Introduction to Multiplexers, De-multiplexers, encoders and decoders.Memories: RAM and ROM. Different kinds of ROM - Masked ROM, programmable ROM.

Combinational Circuits : Part 2 Static and Dynamic RAM, Memory organization.

Introduction to Sequential Circuits : Part 1  Introduction to Sequential circuits. Different kinds of Flip Flops. RS, D, T, JK. Structure of flip flops. Switching example. Counters and Timers. Ripple and Synchronous Counters.

Sequential Circuits : Part 2
ADC or DAC Converters and conversion processes. Flash Converters, ramp generators. Successive approximation and quantization errors.


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