Digital Logic Chapter-2 Boolean Algebra and Logic Gates

Outline of Chapter -2 2.1 Introduction 2.2 Basic definitions 2.3 Axiomatic Definition of Boolean Algebra 2.4 Basic Theorems & Properties of Boolean Algebra 2.5 Boolean Functions 2.6 Canonical and Standard Forms -Minterms and Maxterms 2.7 Other Logic Operations 2.8 Digital Logic Gates 2.9 Integrated Circuits

Today’s Lecture Objectives In this Lecture you will learn about Boolean algebra with Theorems and Boolean Functions. At the end of this Lecture you will be able to answer the below questions which is in Text book exercise. Define Boolean Algebra? What is Boolean Operators? Simplify the Boolean Functions in the exercise.

Boolean Algebra – Introduction Basic definitions To define a Boolean algebra The set B Rules for two binary operations The elements of B and rules should conform to our axioms Two-valued Boolean algebra B = {0, 1} x y x · y 1 x y x + y 1 x x’ 1

Basic Theorems & Properties of Boolean Algebra x+0 = x (a) x+x| = 1 x+x = x (a) x+1 = 1 (x|)| = x (a) x+y=y+x x+(y+z)=(x+y)+z (a) x(y+z)=xy+xz (x+y)| = x| + y| (a) x+xy=x (b) x.1=x (b) x. x| = 0 (b) x.x = x (b) x.0 = 0 (b) xy = yx (b)x(yz) = (xy)z (b) x+yz = (x+y)(x+z) (b) (xy)| = x| + y| (b) x(x+y) = x

Operator Precedence Parentheses NOT AND OR Example: (x + y)’ x’ · y’ x + x · y’

Boolean Functions Consists of binary variables (normal or complement form) the constants, 0 and 1 logic operation symbols, “+” and “·” Example: F1(x, y, z) = x + y’ z F2(x, y, z) = x’ y’ z + x’ y z + xy’ F2 F1 z y x 1 1 1 1 1 1 1 1

Logic Circuit Diagram of F1 F1(x, y, z) = x + y’ z x x + y’ z y z y’z Gate Implementation of F1 = x + y’ z

Logic Circuit Diagram of F2 F2 = x’ y’ z + x’ y z + xy’ x y z F2 Algebraic manipulation F2 = x’ y’ z + x’ y z + xy’ F2 = x’ z + xy’

Alternative Implementation of F2 x y z F2 F2 = x’ y’ z + x’ y z + xy’ F2 x y z

Today’s Lecture Objectives In this Lecture you will learn about Basic Logic Gates, Universal Gates with Two Level implementation. In this Lecture you will learn about Standard forms as Sum of Min Terms and Max Terms with Integrated Circuits. At the end of this Lecture you will be able to answer the below questions which is in Text book exercise. Design the Basic Logic Gates Design the Universal Logic Gates Simplify the Minterms for Boolean functions in the exercise.

Logic Gate Symbols TRANSFER NOT AND OR XOR XNOR NAND NOR

Digital Logic Gates

Universal Gates NAND and NOR gates are universal We know any Boolean function can be written in terms of three logic operations: AND, OR, NOT In return, NAND gate can implement these three logic gates by itself So can NOR gate (x’ y’ )’ y ’ x’ (xy)’ y x 1 1 1

NAND Gate x NOT OR x y x y AND

NOR Gate x x y x y

Canonical & Standard Forms Minterms A product term: all variables appear (either in its normal, x, or its complement form, x’) How many different terms we can get with x and y? x’y’ → 00 → m0 x’y → 01 → m1 xy’ → 10 → m2 xy → 11 → m3 m0, m1, m2, m3 (minterms or AND terms, standard product) n variables can be combined to form 2n minterms

Canonical & Standard Forms Maxterms (OR terms, standard sums) M0 = x + y → 00 M1 = x + y’ → 01 M2 = x’ + y →10 M3 = x’ + y’ → 11 n variables can be combined to form 2n maxterms m0’ = M0 m1’ = M1 m2’ = M2 m3’ = M3

Min- & Maxterms with n = 3 x y z Minterms Maxterms term designation x’y’z’ m0 x + y + z M0 1 x’y’z m1 x + y + z’ M1 x’yz’ m2 x + y’ + z M2 x’yz m3 x + y’ + z’ M3 xy’z’ m4 x’ + y + z M4 xy’z m5 x’ + y + z’ M5 xyz’ m6 x’ + y’ + z M6 xyz m7 x’ + y’ + z’ M7 *

Important Properties Any Boolean function can be expressed as a sum of minterms Any Boolean function can be expressed as a product of maxterms Example: F’ = Σ (0, 2, 3, 5, 6) = x’y’z’ + x’yz’ + x’yz + xy’z + xyz’ How do we find the complement of F’? F = (x + y + z)(x + y’ + z)(x + y’ + z’)(x’ + y + z’)(x’ + y’ + z) *

Canonical Form If a Boolean function is expressed as a sum of minterms or product of maxterms the function is said to be in canonical form. Example: F = x + y’z → canonical form? No But we can put it in canonical form. F = x + y’z = Σ (7, 6, 5, 4, 1) Alternative way: Obtain the truth table first and then the canonical term.

Standard Forms Fact: Alternative representation: Canonical forms are very seldom the ones with the least number of literals Alternative representation: Standard form a term may contain any number of literals Two types the sum of products the product of sums Examples: F1 = y’ + xy + x’yz’ F2 = x(y’ + z)(x’ + y + z’)

Standard Forms F1 = y’ + xy + x’yz’ F2 = x(y’ + z)(x’ + y + z’) ‘ *

Nonstandard Forms The standard form: F3 = ABC + ABD + CD + CE F3 F3 A F3 = AB(C+D) + C(D + E) This hybrid form yields three- level implementation D A F3 B C E The standard form: F3 = ABC + ABD + CD + CE A B C F3 D E

Integrated Circuits Many digital Logic families of IC(Integrated Circuit) have been introduced commercially. The following are the most popular: TTL Transistor-Transistor Logic ECL Emitter-coupled Logic MOS Metal-oxide semiconductor CMOS Complementary metal-oxide semiconductor