Presentation on Boolean algebra

Presentation on Boolean algebra

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Presentation on Boolean algebra - Overview

------------ PAGE 1 ------------ March 22, 2017 1 Digital Logic Design I Boolean Algebra and Logic Gate Fernaz Narin Nur ------------ PAGE 2 ------------ March 22, 2017 2 Algebras ƒÞ What is an algebra? ƒá Mathematical system consisting of ƒá Set of elements ƒá Set of operators ƒá Axioms or postulates ƒÞ Why is it important? ƒá Defines rules of “ calculations” ƒÞ Example: arithmetic on natural numbers ƒá Set of elements: N = { 1,2,3,4,…} ƒá Operator: , – , * ƒá Axioms: associativity, distributivity, closure, identity elements, etc. ƒÞ Note: operators with two inputs are called binary ƒá Does not mean they are restricted to binary numbers! ƒá Operator( s) with one input are called unary ------------ PAGE 3 ------------ March 22, 2017 3 BASIC DEFINITIONS ƒÞ A set is collection of having the same property. ƒá S : set, x and y : element or event ƒá For example: S = { 1, 2, 3, 4} ƒá If x = 2, then x „¡ S . ƒá If y = 5, then y „¢ S . ƒÞ A binary operator defines on a set S of elements is a rule that assigns, to each pair of elements from S, a unique element from S. ƒá For example: given a set S , consider a * b = c and * is a binary operator. ƒá If ( a , b ) through * get c and a , b , c „¡ S , then * is a binary operator of S . ƒá On the other hand, if * is not a binary operator of S and a , b „¡ S , then c „¢ S . ------------ PAGE 4 ------------ March 22, 2017 4 BASIC DEFINITIONS ƒÞ The most common postulates used to formulate various algebraic structures are as follows: 1. Closure : a set S is closed with respect to a binary operator if, for every pair of elements of S , the binary operator specifies a rule for obtaining a unique element of S . ƒá For example, natural numbers N={ 1,2,3,...} is closed w. r. t. the binary operator by the rule of arithmetic addition, since, for any a , b „¡ N, there is a unique c „¡ N such that ƒá a b = c ƒá But operator – is not closed for N , because 2- 3 = - 1 and 2, 3 „¡ N , but (- 1) „¢ N. 2. Associative law : a binary operator * on a set S is said to be associative whenever ƒá ( x * y ) * z = x * ( y * z ) for all x , y , z „¡ S ƒá ( x y ) z = x ( y z ) 3. Commutative law : a binary operator * on a set S is said to be commutative whenever ƒá x * y = y * x for all x , y „¡ S ƒá x y = y x ------------ PAGE 5 ------------ March 22, 2017 5 BASIC DEFINITIONS 4. Identity element : a set S is said to have an identity element with respect to a binary operation * on S if there exists an element e „¡ S with the property that ƒá e * x = x * e = x for every x „¡ S ƒá 0 x = x 0 = x for every x „¡ I . I = {…, - 3, - 2, - 1, 0, 1, 2, 3, …}. ƒá 1* x = x* 1 = x for every x „¡ I. I = {…, - 3, - 2, - 1, 0, 1, 2, 3, …}. 5. Inverse : a set having the identity element e with respect to the binary operator to have an inverse whenever, for every x „¡ S , there exists an element y „¡ S such that ƒá x * y = e ƒá The operator over I , with e = 0, the inverse of an element a is (- a ), since a (- a ) = 0. 6. Distributive law : if * and D are two binary operators on a set S, * is said to be distributive over . whenever ƒá x * ( y D z ) = ( x * y ) D ( x * z ) ------------ PAGE 6 ------------ March 22, 2017 6 George Boole ƒÞ Father of Boolean algebra ƒÞ He came up with a type of linguistic algebra, the three most basic operations of which were ( and still are) AND, OR and NOT. It was these three functions that formed the basis of his premise, and were the only operations necessary to perform comparisons or basic mathematical functions. ƒÞ Boole’s system ( detailed in his ' An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities', 1854) was based on a binary approach, processing only two objects - the yes- no, true- false, on- off, zero- one approach. ƒÞ Surprisingly, given his standing in the academic community, Boole's idea was either criticized or completely ignored by the majority of his peers. ƒÞ Eventually, one bright student, Claude Shannon ( 1916- 2001), picked up the idea and ran with it George Boole ( 1815 - 1864) ------------ PAGE 7 ------------ March 22, 2017 7 Axiomatic Definition of Boolean Algebra ƒÞ We need to define algebra for binary values ƒá Developed by George Boole in 1854 ƒÞ Huntington postulates for Boolean algebra ( 1904): ƒÞ B = { 0, 1} and two binary operations, and D ƒá Closure with respect to operator and operator · ƒá Identity element 0 for operator and 1 for operator · ƒá Commutativity with respect to and · x y = y x, x · y = y · x ƒá Distributivity of · over , and over · x · ( y z) = ( x · y) ( x · z) and x ( y · z) = ( x y) · ( x z) „X Complement for every element x is x’ with x x’= 1 , x · x’= 0 ƒá There are at least two elements x, y „¡ B such that x „j y ------------ PAGE 8 ------------ March 22, 2017 8 Boolean Algebra ƒÞ Terminology: ƒá Literal: A variable or its complement ƒá Product term: literals connected by • ƒá Sum term: literals connected by ------------ PAGE 9 ------------ March 22, 2017 9 Postulates of Two- Valued Boolean Algebra ƒÞ B = { 0, 1} and two binary operations, and D ƒÞ The rules of operations: AND A OR and NOT. 1. Closure ( and ¡E ) 2. The identity elements ( 1) : 0 ( 2) D : 1 x y x D y 0 0 0 0 1 0 1 0 0 1 1 1 x y x y 0 0 0 0 1 1 1 0 1 1 1 1 x x ' 0 1 1 0 AND OR NOT ------------ PAGE 10 ------------ March 22, 2017 10 Postulates of Two- Valued Boolean Algebra 3. The commutative laws 4. The distributive laws x y z y z x D ( y z) x D y x D z ( x D y) ( x D z) 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 ------------ PAGE 11 ------------ March 22, 2017 11 Postulates of Two- Valued Boolean Algebra 5. Complement ƒá x x'= 1 ¨ 0 0'= 0 1= 1; 1 1'= 1 0= 1 ƒá x D x'= 0 ¨ 0 D 0'= 0 D 1= 0; 1 D 1'= 1 D 0= 0 6. Has two distinct elements 1 and 0, with 0 ‚ 1 ƒÞ Note ƒá A set of two elements ƒá : OR operation; D : AND operation ƒá A complement operator: NOT operation ƒá Binary logic is a two- valued Boolean algebra ------------ PAGE 12 ------------ March 22, 2017 12 Duality ƒÞ The principle of duality is an important concept. This says that if an expression is valid in Boolean algebra, the dual of that expression is also valid. ƒÞ To form the dual of an expression, replace all operators with . operators, all . operators with operators, all ones with zeros, and all zeros with ones. ƒÞ Form the dual of the expression a ( bc) = ( a b)( a c) ƒÞ Following the replacement rules… a( b c) = ab ac ƒÞ Take care not to alter the location of the parentheses if they are present. ------------ PAGE 13 ------------ March 22, 2017 13 Basic Theorems ------------ PAGE 14 ------------ March 22, 2017 14 Boolean Theorems ƒÞ Huntington’s postulates define some rules ƒÞ Need more rules to modify algebraic expressions ƒá Theorems that are derived from postulates ƒÞ What is a theorem? ƒá A formula or statement that is derived from postulates ( or other proven theorems) ƒÞ Basic theorems of Boolean algebra ƒá Theorem 1 ( a): x x = x ( b): x · x = x ƒá Looks straightforward, but needs to be proven ! Post. 1: closure Post. 2: ( a) x 0= x , ( b) x · 1= x Post. 3: ( a) x y= y x , ( b) x · y= y · x Post. 4: ( a) x( y z) = xy xz , ( b) x yz = ( x y)( x z) Post. 5: ( a) x x’= 1 , ( b) x · x’= 0 ------------ PAGE 15 ------------ March 22, 2017 15 Proof of x x= x ƒÞ We can only use Huntington postulates: ƒÞ Show that x x= x . x x = ( x x) · 1 by 2( b) = ( x x)( x x’) by 5( a) = x xx’ by 4( b) = x 0 by 5( b) = x by 2( a) Q. E. D. ƒÞ We can now use Theorem 1( a) in future proofs Huntington postulates : Post. 2 : ( a) x 0= x , ( b) x · 1= x Post. 3 : ( a) x y= y x , ( b) x · y= y · x Post. 4 : ( a) x( y z) = xy xz , ( b) x yz = ( x y)( x z) Post. 5 : ( a) x x’= 1 , ( b) x · x’= 0 ------------ PAGE 16 ------------ March 22, 2017 16 Proof of x · x= x ƒÞ Similar to previous proof ƒÞ Show that x · x = x . x · x = xx 0 by 2( a) = xx xx’ by 5( b) = x( x x’) by 4( a) = x · 1 by 5( a) = x by 2( b) Q. E. D. Huntington postulates : Post. 2 : ( a) x 0= x , ( b) x · 1= x Post. 3 : ( a) x y= y x , ( b) x · y= y · x Post. 4 : ( a) x( y z) = xy xz , ( b) x yz = ( x y)( x z) Post. 5 : ( a) x x’= 1 , ( b) x · x’= 0 Th. 1 : ( a) x x= x ------------ PAGE 17 ------------ March 22, 2017 17 Proof of x 1 = 1 ƒÞ Theorem 2( a): x 1 = 1 x 1 = 1 D ( x 1) by 2( b) =( x x' )( x 1) 5( a) = x x' 1 4( b) = x x' 2( b) = 1 5( a) ƒÞ Theorem 2( b): x D 0 = 0 by duality ƒÞ Theorem 3: ( x' ) ' = x ƒá Postulate 5 defines the complement of x, x x' = 1 and x x' = 0 ƒá The complement of x' is x is also ( x' ) ' Huntington postulates : Post. 2 : ( a) x 0= x , ( b) x · 1= x Post. 3 : ( a) x y= y x , ( b) x · y= y · x Post. 4 : ( a) x( y z) = xy xz , ( b) x yz = ( x y)( x z) Post. 5 : ( a) x x’= 1 , ( b) x · x’= 0 Th. 1 : ( a) x x= x ------------ PAGE 18 ------------ March 22, 2017 18 DeMorgan’s Theorem ƒÞ Theorem 5( a): ( x y )’ = x ’y’ ƒÞ Theorem 5( b): ( x y)’ = x ’ y ’ ƒÞ By means of truth table x y x’ y’ x y ( x y)’ x’y’ xy x’ y' ( xy)’ 0 0 1 1 0 1 1 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 ------------ PAGE 19 ------------ March 22, 2017 19 Operator Precedence ƒÞ The operator precedence for evaluating Boolean Expression is ƒá Parentheses ƒá NOT ƒá AND ƒá OR ƒÞ Examples ƒá x y' z ƒá ( x y z ) ' ------------ PAGE 20 ------------ March 22, 2017 20 Boolean Functions ƒÞ A Boolean function ƒá Binary variables ƒá Binary operators OR and AND ƒá Unary operator NOT ƒá Parentheses ƒÞ Examples ƒá F 1 = x y z' ƒá F 2 = x y'z ƒá F 3 = x' y' z x' y z x y' ƒá F 4 = x y' x' z ------------ PAGE 21 ------------ March 22, 2017 21 Boolean Functions „Z The truth table of 2 n entries ƒÞ Two Boolean expressions may specify the same function ƒá F 3 = F 4 x y z F 1 F 2 F 3 F 4 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 0 1 1 1 1 1 0 1 1 0 0 1 1 1 0 1 0 0 ------------ PAGE 22 ------------ March 22, 2017 22 Boolean Functions ƒÞ Implementation with logic gates ƒá F 4 is more economical F 4 = x y' x' z F 3 = x' y' z x' y z x y' F 2 = x y'z ------------ PAGE 23 ------------ March 22, 2017 23 Algebraic Manipulation ƒÞ To minimize Boolean expressions ƒá Literal : a primed or unprimed variable ( an input to a gate) ƒá Term : an implementation with a gate ƒá The minimization of the number of literals and the number of terms ¨ a circuit with less equipment ƒá It is a hard problem ( no specific rules to follow) ƒÞ Example 2.1 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 yzx yzx' = xy ( 1 z ) x'z ( 1 y ) = xy x'z 5.( x y )( x' z )( y z ) = ( x y )( x' z ), by duality from function 4. ( consensus theorem with duality) ------------ PAGE 24 ------------ March 22, 2017 24 2.6 Canonical and Standard Forms Minterms and Maxterms ƒÞ A minterm ( standard product): an AND term consists of all literals in their normal form or in their complement form. ƒá For example, two binary variables x and y, ƒá xy, xy', x'y, x'y' ƒá It is also called a standard product. ƒá n variables con be combined to form 2 n minterms. ƒÞ A maxterm ( standard sums): an OR term ƒá It is also call a standard sum. ƒá 2 n maxterms. ------------ PAGE 25 ------------ March 22, 2017 25 Minterms and Maxterms „Z Each maxterm is the complement of its corresponding minterm , and vice versa. ------------ PAGE 26 ------------ March 22, 2017 26 Minterms and Maxterms ƒÞ An Boolean function can be expressed by ƒá A truth table ƒá Sum of minterms ƒá f 1 = x'y'z xy'z' xyz = m 1 m 4 m 7 ( Minterms) ƒá f 2 = x'yz xy'z xyz' xyz = m 3 m 5 m 6 m 7 ( Minterms) ------------ PAGE 27 ------------ March 22, 2017 27 Minterms and Maxterms ƒÞ The complement of a Boolean function ƒá The minterms that produce a 0 ƒá f 1 ' = m 0 m 2 m 3 m 5 m 6 = x'y'z' x'yz' x'yz xy'z xyz' ƒá f 1 = ( f 1 ' ) ' = ( x y z )( x y' z ) ( x y' z' ) ( x' y z' )( x' y' z ) = M 0 M 2 M 3 M 5 M 6 ƒá f 2 = ( x y z )( x y z' )( x y' z )( x' y z ) = M 0 M 1 M 2 M 4 ƒÞ Any Boolean function can be expressed as ƒá A sum of minterms (“ sum” meaning the ORing of terms). ƒá A product of maxterms (“ product” meaning the ANDing of terms). ƒá Both boolean functions are said to be in Canonical form. ------------ PAGE 28 ------------ March 22, 2017 28 Sum of Minterms ƒÞ Sum of minterms: there are 2 n minterms and 2 2n combinations of function with n Boolean variables. ƒÞ Example 2.4: express F = A BC' as a sum of minterms. ƒá F = A B'C = A ( B B' ) B'C = AB AB' B'C = AB ( C C' ) AB' ( C C' ) ( A A' ) B'C = ABC ABC' AB'C AB'C' A'B'C ƒá F = A'B'C AB'C' AB'C ABC' ABC = m 1 m 4 m 5 m 6 m 7 ƒá F( A , B , C ) = S ( 1, 4, 5, 6, 7) ƒá or, built the truth table first ------------ PAGE 29 ------------ March 22, 2017 29 Product of Maxterms ƒÞ Product of maxterms: using distributive law to expand. ƒá x yz = ( x y )( x z) = ( x y zz' )( x z yy' ) = ( x y z )( x y z' )( x y' z ) ƒÞ Example 2.5: express F = xy x'z as a product of maxterms. ƒá F = xy x'z = ( xy x' )( xy z ) = ( x x' )( y x' )( x z )( y z ) = ( x' y )( x z )( y z ) ƒá x' y = x' y zz' = ( x' y z )( x' y z' ) ƒá F = ( x y z )( x y' z )( x' y z )( x' y z' ) = M 0 M 2 M 4 M 5 ƒá F ( x, y, z ) = P ( 0, 2, 4, 5) ------------ PAGE 30 ------------ March 22, 2017 30 Conversion between Canonical Forms ƒÞ The complement of a function expressed as the sum of minterms equals the sum of minterms missing from the original function. ƒá F ( A , B , C ) = S ( 1, 4, 5, 6, 7) ƒá Thus, F' ( A , B , C ) = S ( 0, 2, 3) ƒá By DeMorgan's theorem F ( A , B , C ) = P ( 0, 2, 3) F' ( A , B , C ) = P ( 1, 4, 5, 6, 7) ƒá m j ' = M j ƒá Sum of minterms = product of maxterms ƒá Interchange the symbols S and P and list those numbers missing from the original form ƒá S of 1' s ƒá P of 0' s ------------ PAGE 31 ------------ March 22, 2017 31 ƒÞ Example ƒá F = xy x „S z ƒá F ( x , y , z ) = S ( 1, 3, 6, 7) ƒá F ( x , y , z ) = P ( 0, 2, 4, 6) ------------ PAGE 32 ------------ March 22, 2017 32 Standard Forms ƒÞ Canonical forms are very seldom the ones with the least number of literals. ƒÞ Standard forms: the terms that form the function may obtain one, two, or any number of literals. ƒá Sum of products: F 1 = y' xy x'yz' ƒá Product of sums: F 2 = x ( y' z )( x' y z' ) ƒá F 3 = A'B'CD ABC'D' ------------ PAGE 33 ------------ March 22, 2017 33 Implementation ƒÞ Two- level implementation ƒÞ Multi- level implementation F 1 = y' xy x'yz' F 2 = x ( y' z )( x' y z' ) ------------ PAGE 34 ------------ March 22, 2017 34 Figure 2.5 Digital logic gates Summary of Logic Gates ------------ PAGE 35 ------------ March 22, 2017 35 Figure 2.5 Digital logic gates Summary of Logic Gates ------------ PAGE 36 ------------ March 22, 2017 36 Multiple Inputs ƒÞ Extension to multiple inputs ƒá A gate can be extended to multiple inputs. ƒá If its binary operation is commutative and associative. ƒá AND and OR are commutative and associative. ƒá OR „X x y = y x „X ( x y ) z = x ( y z ) = x y z ƒá AND „X xy = yx „X ( x y ) z = x ( y z ) = x y z ------------ PAGE 37 ------------ March 22, 2017 37 Multiple Inputs ƒá NAND and NOR are commutative but not associative ¨ they are not extendable. Figure 2.6 Demonstrating the nonassociativity of the NOR operator; ( x « y ) « z ‚ x «( y « z ) ------------ PAGE 38 ------------ March 22, 2017 38 Multiple Inputs ƒá Multiple NOR = a complement of OR gate, Multiple NAND = a complement of AND. ƒá The cascaded NAND operations = sum of products. ƒá The cascaded NOR operations = product of sums. Figure 2.7 Multiple- input and cascated NOR and NAND gates ------------ PAGE 39 ------------ March 22, 2017 39 Multiple Inputs ƒá The XOR and XNOR gates are commutative and associative. ƒá Multiple- input XOR gates are uncommon? ƒá XOR is an odd function: it is equal to 1 if the inputs variables have an odd number of 1' s. Figure 2.8 3- input XOR gate ------------ PAGE 40 ------------ March 22, 2017 40 Positive and Negative Logic ƒÞ Positive and Negative Logic ƒá Two signal values <=> two logic values ƒá Positive logic: H= 1; L= 0 ƒá Negative logic: H= 0; L= 1 ƒÞ Consider a TTL gate ƒá A positive logic AND gate ƒá A negative logic OR gate ƒá The positive logic is used in this book Figure 2.9 Signal assignment and logic polarity ------------ PAGE 41 ------------ March 22, 2017 41 Figure 2.10 Demonstration of positive and negative logic Positive and Negative Logic ------------ PAGE 42 ------------ March 22, 2017 42 2.9 Integrated Circuits Level of Integration ƒÞ An IC ( a chip) ƒÞ Examples: ƒá Small- scale Integration ( SSI): < 10 gates ƒá Medium- scale Integration ( MSI): 10 ~ 100 gates ƒá Large- scale Integration ( LSI): 100 ~ xk gates ƒá Very Large- scale Integration ( VLSI): > xk gates ƒÞ VLSI ƒá Small size ( compact size) ƒá Low cost ƒá Low power consumption ƒá High reliability ƒá High speed ------------ PAGE 43 ------------ March 22, 2017 43 Digital Logic Families ƒÞ Digital logic families: circuit technology ƒá TTL: transistor- transistor logic ( dying?) ƒá ECL: emitter- coupled logic ( high speed, high power consumption) ƒá MOS: metal- oxide semiconductor ( NMOS, high density) ƒá CMOS: complementary MOS ( low power) ƒá BiCMOS: high speed, high density ------------ PAGE 44 ------------ March 22, 2017 44 Digital Logic Families ƒÞ The characteristics of digital logic families ƒá Fan- out: the number of standard loads that the output of a typical gate can drive. ƒá Power dissipation. ƒá Propagation delay: the average transition delay time for the signal to propagate from input to output. ƒá Noise margin: the minimum of external noise voltage that caused an undesirable change in the circuit output.
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