Logic Design (CE1111 ) Lecture 2 (Chapter 2) Prepared by Dr. Lamiaa Elshenawy 1

Outlines  Boolean Algebra  Basic Theorems and Properties  Boolean Functions  Canonical and Standard Forms  Other Logic Operations  Digital Logic Gates  Integrated Circuits (ICs) 2

Boolean Algebra  What is Boolean algebra? O Boolean algebra is a branch of algebra in which the values of the variables are described as true (1) and false (0) O Boolean algebra was introduced by George Boole in George Boole (2 November 1815 – 8 December 1864) was an English mathematician, educator, philosopher, and logician.

Basic Theorems and Properties  Boolean algebra can be defined by a set of elements, B, together with two binary operators, + and., provided that Huntington postulates are satisfied (1904) 4 Edward Vermilye Huntington (April 26, 1874-November 25, 1952) was an American mathematician

5 PostulatesDescription 1. Operators+,. 2. Identity 3. Commutative 4. Distributive 5. Complements 6. Associative Duality principle Huntington Postulates

6 TheoremsDescription 1. Theorem 1 2. Theorem 2 3. Theorem 3, involution 4. Theorem 4, DeMorgan 5. Theorem 5 6. Theorem 6, absorption Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician

Operator Precedence 7

Boolean Function 8 term literal

Boolean Function 9

 How implement Boolean function? 10

Simplification of Boolean Function 11

Complement of Function 12

Complement of Function 13 Example 1

Complement of Function 14 Example 2

Canonical and Standard Forms  Canonical form 1. Minterms 2. Maxterms  Standard form 1. Sum of products 2. Product of sums 15

Canonical Form  Boolean functions expressed as a sum of minterms or product of maxterms are said to be in canonical form  Minterm is complement of maxterm and vice versa 16

Canonical Form  Example: Express function as sum of minterms 17

Canonical Form  Example: Express function as product of maxterms 18

Standard Form 1. Sum of products 2. Product of sums 19 Two-level implementation

Standard Form  A Boolean function may be expressed in a nonstandard form  Example 20 Three-level implementation

Standard Form  It can be transferred to standard form 21

Other Logic Functions 22

Other Logic Functions 23

Other Logic Functions 24

Other Logic Functions 1. Two functions that produce a constant 0 or 1 2. Four functions with unary operations, complement and transfer 3. Ten functions with binary operators that define eight different operations: AND, OR, NAND, NOR, exclusive ‐ OR, equivalence, inhibition, and implication 25

Digital Logic Gates 26

Digital Logic Gates 27

Integrated Circuits-ICs  What is ICs? O Integrated circuit (IC) is fabricated on chip of silicon semiconductor crystal, containing the electronic components for constructing digital gates 28

Integrated Circuits-ICs  What are levels of integration? 1. Small ‐ scale integration (SSI) < 10 logic gates 2. Medium ‐ scale integration (MSI) 10 to 1,000 logic gates - registers and counters 3. Large ‐ scale integration (LSI) >1,000 logic gates - processors, memory chips 4. Very large ‐ scale integration (VLSI) millions of logic gates- microcomputer chips 29

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