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Chapter 2 Boolean Algebra and Logic Gates

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

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3 2-2 Basic Definitions 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 xy B 2. Associative (x+y)+z = x + (y + z);(xy)z = x(yz) 3. Commutative x+y =y+x;xy = yx 4.anidentity element 0+x = x+0 = x; 1x = x1=x ----- e,x B x B, x' B (complementof x) x+x'=1;xx'=0 (o,1 - identity elements w.r.t. +,. 6.distributive Law over + :x(y+z)=(xy)+(xz) distributiveoverx:x+ (y.z)=(x+y)(x+z) 2-3 Axiomatic Definition of Boolean Algebra

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4 Two-valued BooleanAlgebra = AND + = OR = NOT Distributive law:x(y+z)=(xy)+(xz)

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5 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

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

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6 Basic Theorems

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

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9 2-5 Boolean Functions Boolean Fxnctions F 1212 = x + (yz)F= xyz + xyz + xy Logic Circuit BooleanFunction

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10 Boolean Function F2 F2 = xyz + xyz + xy

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Example 2.1 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+xy =(x+x)(x+y) =1(x+y) = x+y Algebraic Manipulation - Simplification

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

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13 Complement of a Function Complement of a variable x is x (0 1 and 1 0) The complement of a functionFisx and is obtainedfrom an interchange of 0s for 1s and 1s for 0sin the value of F The dual of a functionis obtained from the interchange ofAND and ORoperators and1sand 0s ** Finding the complement of a function F Applying DeMorgans theorem as many times as necessary complementing each literal of the dual of F DeMorgans Theorem

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12 DeMorgans Theorem Generalized DeMorgans Theorem 3-variable DeMorgans Theorem 2-variable DeMorgans Theorem (x+ y) = xy and (xy) = x + y

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

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15 2-5 Canonical and Standard Forms Minterms andmaxterms – Expressing combinationsof 0sand 1swith binary variables Logic circuit Boolean function Truthtable – Any Boolean function can be expressed as a sum of minterms -Any Boolean functioxcan beexpressed as a product of maxterms Canonical and Standard Forms

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16 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 amaxterm is form 2minterms twoVariables: xy, xy, xy, and xy – A variable of a minterm is primed if the corresponding bit of the binary number isa0, and unprimed if a 1 n unprimed isthe corresponding bit is a 0 and primed if a 1 001 => xyz 100 => xyz 111 => xyz

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17 Expressing Truth Table in BooleanFunction 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 acanonical form xvariables 2minterms n 2possible functions 2n

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

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19 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)

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Expressing Boolean Function in Product ofMaxterms 2x

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21 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 023023 Compute complement of F by DeMorgans Theorem + m F = (F) = (m 023023023023 +m+m+m+m)= (m mm ) =m=m 023023023023 mm = MMM (0, 2, 3) Summary m = M 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)

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

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23 Standard Forms *Canonical forms: each mintermor 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 1 –Product of sums = y + xy + xyz F 2 Canonical forms Standard forms –sumof minterms, Product of maxterms – Sum of products, Product of sums = x(y + z)(x + y +x)

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

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25 Nonstandard Form and Logic Circuit Nonstandardform:Standardform: F 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 = AB + C(D+E)F= AB + CD + CE

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

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

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

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29 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 bothcommunicative and associative – uncommon, usually constructed with other gates – XOR is an odd function (Section 3-8)

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Positive and Negative logic 0 01 1 HH L L (a) Positive logic(b) Negative logic Logic value Signal value

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