Chapter 2 Boolean Algebra and Logic Gates Chapter

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 -2 Basic Definitions • Boolean Algebra (formulated 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; 1 • x = x • 1=x x B, x' B (complement of x) x+x'=1; x • x'=0 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. 2. 3. 4. parentheses NOT AND OR 5

Basic Theorems 6

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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. De. Morgan 8

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

Boolean Function F 2 = 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

De. Morgan’s Theorem 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 5 -(x+y)(x’+z)(y+z) = (x+y)(x’+z) by duality function 4

Complement of a Function • 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 An. D and OR operators and 1’s and 0’s -- Finding the complement of a function F Applying De. Morgan’s theorem as many times as necessary complementing each literal of the dual of F 13

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

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2 -5 Canonical and Standard Forms • Minterms and maxterms – Expressing combinations of 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

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

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 n minterms 22 n possible functions 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 2 x

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 0 + m 2 + m 3 Compute complement of F’ by De. Morgan’s Theorem F = (F’)’ = (m 0 + m 2 + m 3)‘ = (m 0’ m 2’ m 3’) = m 0’ m 2’ m 3’ = M 0 M 2 M 3 (0, 2, 3) Summary • mj ’ = Mj • Conversion between product of maxterms and sum of minterms (0, 2, 3) • Shown by truth table (Table 2 -5) (1, 4, 5, 6, 7) = 21

Example – Two Canonical Forxs of Boolean Algebra from Truth Table Boolean exprexsion: x(x, y, z) = xy + x’z Dexiving the truth xxxxe Expressing in canonical fxrms x(x, y, z) = (1, 3, 6, 7) = (0, 2, 4, 5) x 2

Stanxard Forms x Canonixal forms: eaxh xinterm xr mxxterm muxt contain all the variables x Standard forms: the terms thxt form the functixn may contain one, two, or any number of literalx (variables) • Two typxs xf standard forms (2 -level) – sum of proxucts F 1 = y’ + xy + x’yz’ – xxoduct of sumx F 2 = x(y’ + z)(x’ + y + x’) • Canxnixal forms Standard fxrms – xux of minterms, Product of maxtexms – Sum of productx, Product of suxs 23

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

Nonstandard Form and Logic Circuit Nonstandard form: F 3 = AB + C(D+E) Standard form: F 3 = AB + CD + CE 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 22 n functionn 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 2 x

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

Multiple-Inputs • NAND and NOR functions are communicative busnot Associative – Define multiple NOR (or NANs) 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 Signal value 1 H 0 L (a) Positive logic Logic value Signal value H 0 L 1 (b) Negative logic