1. Digital Logic Circuits, Digital Component and Data Representation Course: BCA-2 nd Sem Subject: Computer Organization And Architecture Unit-1 1

2. Basic Definitions Binary Operators – AND z = x y = x yz=1 if x=1 AND y=1 – OR z = x + y z=1 if x=1 OR y=1 – NOT z = x = x’ z=1 if x=0 Boolean Algebra – Binary Variables: only ‘0’ and ‘1’ values – Algebraic Manipulation

3. Boolean Algebra Postulates Commutative Law x y = y xx + y = y + x Identity Element x 1 = x x + 0 = x Complement x x’ = 0 x + x’ = 1

4. Boolean Algebra Theorems Duality – The dual of a Boolean algebraic expression is obtained by interchanging the AND and the OR operators and replacing the 1’s by 0’s and the 0’s by 1’s. – x ( y + z ) = ( x y ) + ( x z ) – x + ( y z ) = ( x + y ) ( x + z ) Theorem 1 – x x = xx + x = x Theorem 2 – x 0 = 0x + 1 = 1 Applied to a valid equation produces a valid equation

5. Boolean Algebra Theorems Theorem 3: Involution – ( x’ )’ = x ( x ) = x Theorem 4: Associative & Distributive – ( x y ) z = x ( y z )( x + y ) + z = x + ( y + z ) – x ( y + z ) = ( x y ) + ( x z ) x + ( y z ) = ( x + y ) ( x + z ) Theorem 5: De Morgan – ( x y )’ = x’ + y’ ( x + y )’ = x’ y’ – ( x y ) = x + y ( x + y ) = x y Theorem 6: Absorption – x ( x + y ) = x x + ( x y ) = x

6. Boolean Functions[1] Boolean Expression Example:F = x + y’ z Truth Table All possible combinations of input variables Logic Circuit xyzF 0000 0011 0100 0110 1001 1011 1101 1111

7. Algebraic Manipulation Literal: A single variable within a term that may be complemented or not. Use Boolean Algebra to simplify Boolean functions to produce simpler circuits Example: Simplify to a minimum number of literals F = x + x’ y( 3 Literals) = x + ( x’ y ) = ( x + x’ ) ( x + y ) = ( 1 ) ( x + y ) = x + y ( 2 Literals) Distributive law (+ over )

8. Complement of a Function DeMorgan’s Theorm Duality & Literal Complement

9. Canonical Forms[1] Minterm – Product (AND function) – Contains all variables – Evaluates to ‘1’ for a specific combination Example A = 0 A B C B = 0(0) (0) (0) C = 0 1 1 1 = 1 A B CMinterm 00 0 0m0m0 10 0 1m1m1 20 1 0m2m2 30 1 1m3m3 41 0 0m4m4 51 0 1m5m5 61 1 0m6m6 71 1 1m7m7

10. Canonical Forms[1] Maxterm – Sum (OR function) – Contains all variables – Evaluates to ‘0’ for a specific combination Example A = 1 A B C B = 1(1) + (1) + (1) C = 1 0 + 0 + 0 = 0 A B CMaxterm 00 0 0M0M0 10 0 1M1M1 20 1 0M2M2 30 1 1M3M3 41 0 0M4M4 51 0 1M5M5 61 1 0M6M6 71 1 1M7M7

11. Canonical Forms[1] Truth Table to Boolean Function A B CF 0 0 00 0 0 11 0 1 00 0 1 10 1 0 01 1 0 11 1 1 00 1 1 11

12. Canonical Forms Sum of Minterms Product of Maxterms

13. Standard Forms Sum of Products (SOP)

14. Standard Forms Product of Sums (POS)

15. Two - Level Implementations Sum of Products (SOP) Product of Sums (POS)

16. Logic Operators AND NAND (Not AND) xyAND 000 010 100 111 xyNAND 001 011 101 110

17. Logic Operators OR NOR (Not OR) xyOR 000 011 101 111 xyNOR 001 010 100 110

18. Logic Operators XOR (Exclusive-OR) XNOR (Exclusive-NOR) (Equivalence) xyXOR 000 011 101 110 xyXNOR 001 010 100 111

19. Logic Operators NOT (Inverter) Buffer xNOT 01 10 xBuffer 00 11

20. Multiple Input Gates 

21. De Morgan’s Theorem on Gates AND Gate – F = x yF = (x y) F = x + y OR Gate – F = x + yF = (x + y) F = x y  Change the “Shape” and “bubble” all lines

22. References 1.Computer Organization and Architecture, Designing for performance by William Stallings, Prentice Hall of India. 2.Modern Computer Architecture, by Morris Mano, Prentice Hall of India. 3.Computer Architecture and Organization by John P. Hayes, McGraw Hill Publishing Company. 4.Computer Organization by V. Carl Hamacher, Zvonko G. Vranesic, Safwat G. Zaky, McGraw Hill Publishing Company.