Introduction to Boolean Algebra Part 1- Binary decisions, Logical Operations, Truth Tables, Expressions, Basic Postulates, Theorems




Number System

 



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                                                        Digital Electronics : Boolean Algebra Part 1



boolean algebra tutorial digital electronics




An Outline of this Tutorial


Here's a quick look at what this tutorial covers. Boolean algebra, the algebra of logic, is also known as switching algebra.


BINARY DECISION

Any decision which results into a YES (TRUE) or NO (FALSE) is called a binary decision.

LOGICAL OPERATIONS

• Logical function or Compound statement. Algebraic variables like a, b, c, x, y etc. are combined with help of mathematical operators like +,-,x,/. Similarly, logical statements or truth functions are combined with help of logical operators to form a logical function. For example :He prefers tea not coffee. He plays piano and she plays sitar. On Sundays, I watch TV or go for swimming. The AND, OR, NOT are the logical operators.
• Logical operators

TRUTH TABLE

A truth table is a table which represents all the possible values of logical variables along with all possible results of given combinations of values.

BOOLEAN EXPRESSIONS

Logical variables are combined by means of logical operators to form Boolean expression. For example-
o X+Y
o X + YZ’ + Z
o Y’ + Z’

Boolean expressions are evaluated by constructing the truthtable. Consider the expression- X + (Y.Z)’ To study this expression, first consider all the combinations of X, Y and Z. total number of combinations for 3 variables is 23 = 8.

BASIC POSTULATES OF BOOLEAN ALGEBRA

• If X = 0 then X = 1 and if X = 1 then X = 0
• Logical addition (OR relations)
  •  0+0=0
  •  0+1=1
  •  1+0=1
  •  1+1=1
• Logical multiplication (AND relations)
  • 0.0=0
  • 0.1=0
  • 1.0=0
  • 1.1=1
• Complement rules (NOT relations)
  • 0’ = 1
  • 1’ = 0

Principle of duality

  • Change each OR (+) to AND (.)
  • Change each AND (.) to OR (+)
  • Replace each 0 by 1 and 1 by 0

Basic theorems of Boolean algebra

( The Tutorial document contains more elaborate explanations on how these theorems can be explained using truth tables or otherwise. )

Properties of 0 and 1

  • 0+X=X
  • 1+X=1
  • 0.X=0
  • 1.X=X

Proof— These properties can be proved by creating a truth table as will be shown in the tutorial.

Indempotence law

X+X=X
X.X=X

Involution law

(X’)’ = X



Complementary law

  • X + X’ = 1
  • X . X’ = 0


Commutative law


  • X+Y=Y+X
  • X.Y=Y.X

Associative law


  • X+(Y+Z)=(X+Y)+Z
  • X.(Y.Z)=(X.Y).Z

Distributive law

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

Moving on beyond the outline, here's the complete tutorial with examples, truth tables and problems :


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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