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

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

3. 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 1854 3 George Boole (2 November 1815 – 8 December 1864) was an English mathematician, educator, philosopher, and logician.

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

6. 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

7. Operator Precedence 7

8. Boolean Function 8 term literal

9. Boolean Function 9

10.  How implement Boolean function? 10

11. Simplification of Boolean Function 11

12. Complement of Function 12

13. Complement of Function 13 Example 1

14. Complement of Function 14 Example 2

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

16. 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

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

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

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

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

21. Standard Form  It can be transferred to standard form 21

22. Other Logic Functions 22

23. Other Logic Functions 23

24. Other Logic Functions 24

25. 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

26. Digital Logic Gates 26

27. Digital Logic Gates 27

28. 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

29. 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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