Boolean algebra and logic gates
- 1. 7/26/2020LCWU 1 CHAPTER - 2 BOOLEAN ALGEBRA & LOGIC GATES
- 2. COMPLEMENT OF A FUNCTION The complement of a function F is F′ Obtained by Changing every 0 to 1 and every 1 to 0 Derived by using De-Morgan’s Laws 7/26/2020LCWU 2 Digital Logic Design
- 3. DE – MORGAN’S LAWS 1. A + B = A . B 2. A . B = A + B 7/26/2020LCWU 3 Digital Logic Design
- 4. EXAMPLE F = x′.y.z′ + x′.y′.z Complementing BOTH sides: F = x′.y.z′ + x′.y′.z Applying De-Morgan’s Laws: = (x′.y.z′) . (x′.y′.z) = (x′′ + y′ + z′′) . (x′′ + y′′ + z′) = (x + y′ + z) . (x + y + z′) 7/26/2020LCWU 4 Digital Logic Design
- 6. EXAMPLES Find the Complement and reduce them to a minimum no. of literals: F1 = x + x.y F2 = AB′ + C′D′ G = x.y + x.y′ + y′.z 7/26/2020LCWU 6 Digital Logic Design
- 7. HOMEWORK Example 2.2 – 2.3 Problem 2.7 7/26/2020LCWU 7 Digital Logic Design
- 8. READING MATERIAL *extra’s Applications Of Boolean Functions – Circuit Chip Design QUESTION A fire sprinkler system should spray water if high heat is sensed and the system is set to enabled. ANSWER Let Boolean variable h represent “high heat is sensed,” e represent “enabled,” and F represent “spraying water.” Then an equation is: F = h AND e Symbolically, F = h . e 7/26/2020LCWU 8 Digital Logic Design h e F
- 9. SEAT BELT WARNING LIGHT SYSTEMDesign circuit for warning light • Sensors s=1: seat belt fastened k=1: key inserted p=1: person in seat Capture Boolean equation person in seat, and seat belt not fastened, and key inserted • Convert equation to circuit 7/26/2020LCWU 9 Digital Logic Design w = p AND NOT (s) AND k
- 10. REPRESENTATION OF BOOLEAN EXPRESSIONS Canonical Forms (Not Simplified) Standard Forms (Simplified) 7/26/2020LCWU 10
- 11. CANONICAL FORMS Directly obtained from the Truth table Not Simplified Two types: SUM of MINTERMS PRODUCT of MAXTERMS 7/26/2020LCWU 11
- 12. MINTERMS Obtained by AND-ing all variables 1 = variables without bar/prime 0 = variables with bar/prime NOTE: Each Minterm must contain all variables !! 7/26/2020LCWU 12
- 13. Example Variables = x , y Possible Combinations = 22 = 4 Index Bin Val Minterms MVal 0 00 x’ . y’ m0 1 01 x’ . y m1 2 10 x . y’ m2 3 11 x . y m3 7/26/2020LCWU 13
- 14. 4 Minterms for 2 variables. ? Minterms for 3 variables. ? Minterms for 4 variables. 7/26/2020LCWU 14
- 15. MAXTERMS Obtained by OR-ing all variables 1 = variables with bar/prime 0 = variables without bar/prime NOTE: Each Maxterm must contain all variables !! 7/26/2020LCWU 15
- 16. Example Variables = x , y Possible Combinations = 22 = 4 Index Bin Val Maxterms MVal 0 00 x + y M0 1 01 x + y’ M1 2 10 x’ + y M2 3 11 x’ + y’ M3 7/26/2020LCWU 16
- 17. RELATIONSHIP BTW MINTERMS AND MAXTERMS Minterms = mi Maxterms = Mj mi = Mj and Mj = mi 7/26/2020LCWU 17