# Boolean Algebra and Logic Gates

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Boolean Algebra and Logic Gates
Chapter 2 Boolean Algebra and Logic Gates

Chapter 2. Boolean Algebra and Logic Gates
2-1 Introduction 2-2 Basic Definitions 2-3 Axiomatic Definition of Boolean Algebra 2-4 Basic Theorems and Properties 2-5 Boolean Functions 2-6 Canonical and Standard Forms 2-7 Other Logic Operations 2-8 Digital Logic Gates 2

2-3 Axiomatic Definition of Boolean Algebra
2-2 Basic Definitions 2-3 Axiomatic Definition of Boolean Algebra • Boolean Algebra forrmulated by E.V. Huntington, 1904) A set of elements B={0,1} and tow binary operators + and• 1. Closure x, y  B  x+y B; x, y  B  x•y B 2. Associative (x+y)+z = x + (y + z); (x•y)•z = x • (y•z) 3. Commutative x+y = y+x; x•y = y•x 4. an identity element 0+x = x+0 = x; •x = x•1=x  e,x B   x  B,  x'  B (complement of x) x+x'=1; x•x'=0 (o,1 - identity elements w.r.t. +, . 6. distributive Law over + : x•(y+z)=(x•y)+(x•z) distributive over x: x+ (y.z)=(x+ y)•(x+ z) 3

Two-valued Boolean Algebra
•= AND + = OR ‘ = NOT Distributive law: x•(y+z)=(x•y)+(x•z) 4

2-4 Basic Theorems and Properties
Duality Principle: Using Huntington rules, one part may be obtained from the other if the binary operators and the identity elements are interchanged 5

2-4 Basic Theorems and Properties
Operator Precedence 1. parentheses 2. NOT 3. AND 4. OR 5

Basic Theorems 6

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Verification by Truth Table
A table of all possible combinations of x and y variables showing the relation between the variable values and the result of the operation Theorem 6(a) Absorption Theorem 5. DeMorgan 8

2-5 Boolean Functions Logic Circuit  Boolean Function
Boolean Fxnctions F = x + (y’z) F = x‘y’z + x’yz + xy’ 1 2 9

Boolean Function F2 F2 = x’y’z + x’yz + xy’ 10

Algebraic Manipulation - Simplification
Example 2.1 Simplify the following Boolean functions to a minimum number of literals: 1- x(x’+y) =xx’ + xy =0+xy=xy 2- x+x’y =(x+x’)(x+y) =1(x+y) = x+y

3-(x+y)(x+y’) =x+xy+xy’+yy’ =x (1+ y + y’) =x 4- xy +x’z+yz = xy+x’z+yz(x+x’) = xy +x’z+xyz+x’yz =xy(1+z) + x’z (1+y) = xy + x’z

Complement of a Function
DeMorgan’s Theorem •Complement of a variable x is x’ (0  1 and 1  0) •The complement of a function F is x’ and is obtained from an interchange of 0’s for 1’s and 1’s for 0’s in the value of F •The dual of a function is obtained from the interchange of AND and OR operators and 1’s and 0’s ** Finding the complement of a function F Applying DeMorgan’s theorem as many times as necessary complementing each literal of the dual of F 13

DeMorgan’s Theorem 2-variable DeMorgan’s Theorem
(x + y)’ = x’y’ and (xy)’ = x’ + y’ 3-variable DeMorgan’s Theorem Generalized DeMorgan’s Theorem 12

H.W: 1,2,3,4,6,8,9 14

2-5 Canonical and Standard Forms
• Minterms and maxterms – Expressing combinationsof 0’s and 1’s with binary variables • Logic circuit  Boolean function  Truth table – Any Boolean function can be expressed as a sum of minterms - Any Boolean functiox can be expressed as a product of maxterms • Canonical and Standard Forms 15

Minterms and Maxterms Minterm (or standard product): Maxterm (or standard sum): – n variables combined with AND – n variables combined with OR – n variables can be combined to – A variable of a maxterm is form 2 minterms n • unprimed is the corresponding • two Variables: x’y’, x’y, xy’, and xy bit is a 0 – A variable of a minterm is • and primed if a 1 • primed if the corresponding bit of the binary number is a 0, 001 => x’y’z • and unprimed if a 1 100 => xy’z’ 16 111 => xyz

Expressing Truth Table in Boolean Function
• Any Boolean function can be expressed a sum of minterms or a product of maxterms (either 0 or 1 for each term) • said to be in a canonical form • x variables  2 minterms n  2 possible functions 2n 17

Expressing Boolean Function in Sum of
Minterms (Method 1 - Supplementing) 18

Expressing Boolean Function in Sum of
Minterms (method 2 – Truth Table) F(A, B, C) = (1, 4, 5, 6, 7) = (0, 2, 3) F’(A, B, C) = (0, 2, 3) = (1, 4, 5, 6, 7) 19

Expressing Boolean Function in Product of Maxterms

Conversion between Canonical Forms
 Canonical conversion procedure Consider: F(A, B, C) = ∑(1, 4, 5, 6, 7) F‘: complement of F = F’(A, B, C) = (0, 2, 3) = m + m + m 0 2 3 Compute complement of F’ by DeMorgan’s Theorem F = (F’)’ = (m + m + m )‘ = (m ’  m ’  m ’) = m ’ m ’ m ’ = M M M (0, 2, 3)  Summary m ’ = M j j • Conversion between product of maxterms and sum of minterms (1, 4, 5, 6, 7) = (0, 2, 3) • Shown by truth table (Table 2-5) 21

Example:– Two Canonical Forms of Boolean Algebra from Truth Table
Boolean expression: x(x, y, z) = xy + x’z Deriving the truth table Expressing in canonical forms x(x, y, z) = (1, 3, 6, 7) = (0, 2, 4, 5) x2

Standard Forms * Canonical forms: each minterm or maxterm must
contain all the variables * Standard forms: the terms that form the function may contain one, two, or any number of literals (variables) • Two types of standard forms (2-level) – sum of products F = y’ + xy + x’yz’ 1 – Product of sums F = x(y’ + z)(x’ + y + x’) 2 • Canonical forms  Standard forms – sum of minterms, Product of maxterms – Sum of products, Product of sums 23

Standard Form and Logic Circuit
= y’ + xy + x’yz’ F = x(y’ + z)(x’ + y + z) 1 2 24

Nonstandard Form and Logic Circuit
Nonstandard form: Standard form: F = AB + C(D+E) F = AB + CD + CE 3 3 A two-level implementation is preferred: produces the least amount of delas Through the gates when the signal propagates from the inputs to the output 25

2-7 Other Logic Operations
• There are 2 function for n binary variables • For n=2 – where are 16 possible functions – AND and OR operators are two of them: xy and x+y • Subdivided into three categories: 26

Truth Tables and Boolean Expressions for
the 16 Functions of Two Variables 2x

2-8 Digital Logic Gates 28 Figure 2-5 Digital Logic Gates

Extension to Multiple-Inputs
• NAND and NOR functions are communicative but not Associative – Define multiple NOR (or NAND) gate as a complemented OR (or AND) gate (Section 3-6) XOR and equivalence gates are both communicative and associative – uncommon, usually constructed with other gates – XOR is an odd function (Section 3-8) 29

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Positive and Negative logic
Logic value Logic value Signal value Signal value H H 1 L L 1 (a) Positive logic (b) Negative logic