This document contains lecture notes on basic concepts in digital logic and design. It defines key terms like binary operators and Boolean algebra. It outlines postulates of Boolean algebra including the principle of duality. It discusses methods of proving theorems like using postulates, induction, and duality. It also covers Boolean functions, how to represent them as algebraic expressions and truth tables, and how to minimize and find complements of Boolean functions.

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Digital Logic & Design
Dr. Sajjad Ahmed Nadeem
Department of Computer Science & IT
University of Azad Jammu & Kashmir
Muzaffarabad
Module 04
Basic Definitions
Postulates.
Principle of duality.
Theorems.
Proving a Theorem.
Algebraic Expressions.
TruthTable.
Minimization.
Complement.

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Basic Definitions
Binary Operator
A binary operator defined on a set S of
elements is a rule that assigns, to each pair of
elements from S, a unique element from S.
Boolean Algebra
Boolean algebra is an algebraic structure
defined by a set of elements, B, together with
two binary operators, + and ., provided that the
postulates of Boolean algebra are satisfied.
Postulates of Boolean Algebra

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Postulates of Boolean Algebra
The Principle of Duality

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ImportantTheorems of Boolean
Algebra
Methods of Proving Theorems

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Proving a Theorem by using Postulates
andTheorems
[ x.1 = x ]
[ x.(y+z)=(x.y) + (x.z) ]
[ x+y=y+x ]
[ x+1=1 ]
[ x.1=x ]
Proving a Theorem by Perfect
Induction / Exhaustive Enumeration
[x + x.y = x ]

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Proving a Theorem by Duality
[ x.1=x ]
[ x +x=1 ]
[ x+(y.z)=(x+y).(x+z) ]
[x.x=0 ]
[ x+0=x ]
[ x+0=x ]
[ x . x=0 ]
[ x.(y+z)=(x.y)+(x.z) ]
[ x+x=1 ]
[ x.1=x ]
Reading Assignment:Venn Diagrams

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Boolean Functions
Representation of a Boolean
Function as an Algebraic Expression

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Representation as a TruthTable
Representation as a TruthTable

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Manipulations/Minimizations of
Boolean Functions
Manipulations/Minimizations of
Boolean Functions

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Try out some minimization!
Complement of a Boolean Function

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Complement of a Boolean Function