Chapter 2 Boolean Algebra and Logic Gates - 國立高雄大學資訊工程 ...

EEA091 - Digital Logic數 位 邏 輯Chapter 2Boolean Algebra and Logic Gates吳 俊 興國 立 高 雄 大 學 資 訊 工 程 學 系2006

Chapter 2. Boolean Algebra andLogic Gates2-1 Basic Definitions2-2 Axiomatic Definition of Boolean Algebra2-3 Basic Theorems and Properties2-4 Boolean Functions2-5 Canonical and Standard Forms2-6 Other Logic Operations2-7 Digital Logic Gates2-8 Integrated Circuits2

2-1/2-2 Basic and Axiomatic Definitions• Boolean Algebra (formulated by E.V. Huntington, 1904)A set of elements B={0,1} and two binary operators + and •• Huntington postulates1. Closure w.r.t. the operator + (•)x, y ∈ B ⇒ x+y ∈B;x, y ∈ B ⇒ x•y ∈B2. Associative w.r.t. + (•)(x+y)+z = x + (y + z);3. Commutative w.r.t. + (•)x+y = y+x;x•y = y•x(x•y)•z = x • (y•z)4. An identity element w.r.t. + (•)0+x = x+0 = x; 1•x = x•1= x5. ∀ x ∈ B, ∃ x' ∈ B (complement of x)x+x'=1;x•x'=06. • is distributive over + : x•(y+z)=(x•y)+(x•z)+ is distributive over •: x+ (y•z)=(x+ y)•(x+ z)Duality principle: remains valid if the operators and identity elementsare interchanged3

Two-valued Boolean Algebra•= AND+ = OR‘ = NOTDistributive law: x•(y+z)=(x•y)+(x•z)4

2-3 Basic Theorems and PropertiesOperator Precedence1. parentheses2. NOT3. AND4. OR5

Basic Theorems6

Truth TableA table of all possible combinations of the variables showing therelation between the variable values and the result of the operationTheorem 6(a) AbsorptionTheorem 5. DeMorgan8

2-4 Boolean FunctionsLogic Circuit ⇔ Boolean FunctionBoolean FunctionsF 1 = x + (y’z)F 2 = x’y’z + x’yz + xy’9

Boolean Function F2F2 = x’y’z + x’yz + xy’10

Algebraic Manipulation - Simplification11

DeMorgan’s Theorem2-variable DeMorgan’s Theorem(x + y)’ = x’y’ and (xy)’ = x’ + y’3-variable DeMorgan’s TheoremGeneralized DeMorgan’s Theorem12

Complement of a Function•Complement of a variable x is x’ (0 ⇒ 1 and 1 ⇒ 0)•The complement of a function F is F’ and is obtained from aninterchange 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 ANDand OR operators and 1’s and 0’s•Finding the complement of a function FApplying DeMorgan’s theorem as many times as necessarycomplementing each literal of the dual of F13

2-5 Canonical and Standard Forms• Minterms and Maxterms– Expressing combinations of 0’s and 1’s with binary variables(normal form x or complement form x’)• Logic circuit ⇔ Boolean function ⇔ Truth table– Any Boolean function can be expressed as a sum of minterms– Any Boolean function can be expressed as a product ofmaxterms• Canonical and Standard Forms15

Minterms and MaxtermsMinterm (or standard product):– = n variables combined with AND– n variables can be combined toform 2 n minterms• two variables: x’y’, x’y, xy’, and xy– A variable of a minterm is• primed if the corresponding bit ofthe binary number is a 0,• and unprimed if a 1Maxterm (or standard sum):– = n variables combined with OR– A variable of a maxterm is• unprimed if the correspondingbit is a 0• and primed if a 1 001 => x’y’zm j ’= M j100 => xy’z’111 => xyz16

Expressing Truth Table in Boolean Function• Any Boolean functioncan be expressed asa sum of minterms ora product of maxterms(either 0 or 1 for each term)• said to be in a canonicalform• n variables⇒ 2 n minterms⇒ 2 2n possible functions(x+y’+z’)17

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

Expressing Boolean Function in Sum ofMinterms (Method 2 – Truth Table)F(A, B, C) = Σ(1, 4, 5, 6, 7) = Π(0, 2, 3)F’(A, B, C) = Σ(0, 2, 3) = Π(1, 4, 5, 6, 7)19

Expressing Boolean Function in Product ofMaxterms20

Conversion between Canonical Forms‣Canonical conversion procedureConsider: F(A, B, C) = Σ(1, 4, 5, 6, 7)F’: Complement of F = F’(A, B, C) = Σ(0, 2, 3) = m 0 + m 2 + m 3Compute complement of F’ by DeMorgan’s TheoremF = (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 = Π(0, 2, 3)‣Summary• m j ’= M j• Conversion between product of maxterms and sum of mintermsΣ(1, 4, 5, 6, 7) = Π(0, 2, 3)• Shown by truth table (Table 2-5)21

Example – Two Canonical Forms of BooleanAlgebra from Truth Table‣Boolean expression: F(x, y, z) = xy + x’z‣Deriving the truth table‣Expressing in canonical formsF(x, y, z) = Σ(1, 3, 6, 7) = Π(0, 2, 4, 5)22

Standard Forms• Canonical forms: each minterm or maxterm mustcontain all the variables• Standard forms: the terms that form the functionmay contain one, two, or any number of literals(variables)• Two types of standard forms (2-level)– sum of productsF 1 = y’ + xy + x’yz’– product of sumsF 2 = x(y’ + z)(x’ + y + z’)• Canonical forms ⇔ Standard forms– Sum of minterms, Product of maxterms– Sum of products, Product of sums23

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

Nonstandard Form and Logic CircuitNonstandard form:F 3 = AB + C(D+E)Standard form:F 3 = AB + CD + CEA two-level implementation is preferred: produces the least amount of delaythrough the gates when the signal propagates from the inputs to the output25

2-6 Other Logic Operations• There are 2 2^n functions for n binaryvariables• For n=2– there 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 forthe 16 Functions of Two Variables27

2-7 Digital LogicGatesFigure 2-5 Digital Logic Gates1. Two are equal to a constant2. Four are repeated twice3. Two, Inhibition and implication, areimpracticalThe gates can be extended to havemore than two inputs except for theinverter and buffer28

Multiple-Inputs• NAND and NOR functions arecommunicative but not associative– Define multiple NOR (or NAND) gate as acomplemented 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)29

2-8 Integrated CircuitsDigital ICs are often categorized according to their circuitcomplexity as measured by the number of logic gates in asingle package– Small-scale integration (SSI)• the inputs and outputs of the gates are connected directly to the pinsin the package• usually fewer than 10 gates, limited by the number of pins available– Medium-scale integration (MSI)• 10 to 1,000 gates in a package• usually perform specific elementary digital operations– Large-scale integration (LSI)• Thousands of gates• Include digital systems such as processors, memory chips, andprogrammable logic devices– Very large-scale integration (VLSI)• Hundred of thousands of gates31

SummaryChapter 2 Boolean Algebra and Logic Gates2-1 Basic Definitions2-2 Axiomatic Definition of Boolean Algebra2-3 Basic Theorems and Properties2-4 Boolean Functions2-5 Canonical and Standard Forms2-6 Other Logic Operations2-7 Digital Logic Gates2-8 Integrated Circuits32