Digital Logic Chapter2 Boolean Algebra and Logic Gates

Digital Logic Chapter-2 Boolean Algebra and Logic Gates

Outline of Chapter -2 • 2. 1 • 2. 2 • 2. 3 • 2. 4 • 2. 5 • 2. 6 • 2. 7 • 2. 8 • 2. 9 Introduction Basic definitions Axiomatic Definition of Boolean Algebra Basic Theorems & Properties of Boolean Algebra Boolean Functions Canonical and Standard Forms Sum of Min-terms & Product of Max-terms Digital Logic Gates – Extension of Multiple gates Integrated Circuits – Logic Families

Today’s Lecture Objectives Ø In this Lecture you will learn about Boolean algebra with Theorems and Boolean Functions. Ø 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 0 0 1 1 y x · y 0 0 1 0 0 0 1 1 x 0 0 1 1 y x + y 0 0 1 1 1 x 0 1 x’ 1 0

Basic Theorems & Properties of Boolean Algebra (a) x+0 = x (a) x+x| = 1 (a) x+x = x (a) x+1 = 1 (x|)| = x (a) x+y=y+x (a) x+(y+z)=(x+y)+z (a) x(y+z)=xy+xz (a) (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 1. 2. 3. 4. • 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: – F 1(x, y, z) = x + y’ z – F 2(x, y, z) = x’ y’ z + x’ y z + xy’ x y z F 1 F 2 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 0

Logic Circuit Diagram of F 1(x, y, z) = x + y’ z x y x + y’ z z y’z Gate Implementation of F 1 = x + y’ z

Logic Circuit Diagram of F 2 x F 2 = x’ y’ z + x’ y z + xy’ y F 2 z – Algebraic manipulation – F 2 = x’ y’ z + x’ y z + xy’ F 2 = x’ z + xy’

Alternative Implementation of F 2 x y z F 2 = x’ y’ z + x’ y z + xy’ x y z F 2

Today’s Lecture Objectives ØIn this Lecture you will learn about Basic Logic Gates, Universal Gates with Two Level implementation. ØIn this Lecture you will learn about Standard forms as Sum of Min Terms and Max Terms with Integrated Circuits. ØAt 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 NOT TRANSFER AND XOR OR XNOR NAND NOR

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 – x 0 0 1 1 So can NOR gate y (xy)’ x’ y ’ 0 1 1 1 0 0 0 (x’ y’ )’

NAND Gate x NOT x y OR x AND y

NOR Gate x 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 → m 0 x’y → 01 → m 1 xy’ → 10 → m 2 xy → 11 → m 3 – m 0, m 1, m 2, m 3 (minterms or AND terms, standard product) – n variables can be combined to form 2 n minterms

Canonical & Standard Forms • Maxterms (OR terms, standard sums) – M 0 = x + y → 00 – M 1 = x + y’ → 01 – M 2 = x’ + y → 10 – M 3 = x’ + y’ → 11 – n variables can be combined to form 2 n maxterms • • m 0’ = M 0 m 1’ = M 1 m 2’ = M 2 m 3’ = M 3

Min- & Maxterms with n = 3 x y Minterms z term designation 0 0 1 1 0 x’y’z’ 1 x’y’z 0 x’yz’ 1 x’yz 0 xy’z’ 1 xy’z 0 xyz’ 1 xyz m 0 m 1 m 2 m 3 m 4 m 5 m 6 m 7 Maxterms term designation x+y+z x + y + z’ x + y’ + z’ x’ + y’ + z’ M 0 M 1 M 2 M 3 M 4 M 5 M 6 M 7 *

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 + xy’z + xyz’ – How do we find the complement of F’? – F = (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: – 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 1. the sum of products 2. the product of sums – Examples: • • F 1 = y’ + xy + x’yz’ F 2 = x(y’ + z)(x’ + y + z’)

Standard Forms • F 1 = y’ + xy + x’yz’ • F 2 = x(y’ + z)(x’ + y + z’) ‘ *

Nonstandard Forms A B C D D E – – F 3 = AB(C+D) + C(D + E) This hybrid form yields threelevel implementation F 3 C – The standard form: F 3 = ABC + ABD + CE A B C A B D C E F 3

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