Boolean Algebra Part I.pdf

Dr. Samantha Rajapaksha Head/Department of IT BOOLEAN ALGEBRA Senior Lecturer(HG) B.Sc. M.Sc. Ph.D. 7/24/2023 DR. SAMANTHA RAKAPAKSHA 1 BOOLEAN ALGEBRA...

Dr. Samantha Rajapaksha Head/Department of IT BOOLEAN ALGEBRA Senior Lecturer(HG) B.Sc. M.Sc. Ph.D. 7/24/2023 DR. SAMANTHA RAKAPAKSHA 1 BOOLEAN ALGEBRA AND Multiplication Basic OR Addition Developed by Gerge Bool in 1854. Operators Use only two operands True(1) and False(0). NOT Boolean NAND Operators Universal Operators NOR XOR XNOR Manufacturing cost and power requirements for NAND and NOR gates are low. Therefore NAND and NOR 7/24/2023 gates are called as universal gates. DR. SAMANTHA RAKAPAKSHA 2  A B Bulb(A and B) AND OPERATOR(* ) OFF OFF OFF ON OFF OFF ON OFF OFF ON ON ON AND operator is similar to the multiplication in Mathematics. ON=True OFF=False Bulb will be on when both switch A and Switch B are on. Truth Table A B Bulb(A and B) False False False A B False True False True False False Bulb True True True ON=True=1 OFF=False=0 A B Bulb(A and B) Truth Table: 0 0 0 It represents the relationship between the inputs and outputs. 0 1 0 1 0 0 7/24/2023 1 1 DR. SAMANTHA RAKAPAKSHA 3 1 LOGIC GATE FOR THE AND OPERATOR A B A.B A 0 0 0 A.B B 0 1 0 1 0 0 1 1 1 7/24/2023 DR. SAMANTHA RAKAPAKSHA 4  A B Bulb(A or B) OR OPERATOR(+ ) OFF OFF OFF ON OFF ON ON OFF ON OR operator is similar to the addition in Mathematics. ON ON ON Bulb will be on when either switch A or Switch B are on. ON=True OFF=False Truth Table A B Bulb(A or B) A False False False False True True B True False True Bulb True True True ON=True=1 OFF=False=0 Truth Table: A B Bulb(A or B) It represents the relationship between the inputs and outp`uts. 0 0 0 0 1 1 1 0 1 7/24/2023 1 1 DR. SAMANTHA RAKAPAKSHA 5 1 LOGIC GATE FOR THE OR OPERATOR A B A+B A 0 0 0 A+B B 0 1 1 1 0 1 1 1 1 7/24/2023 DR. SAMANTHA RAKAPAKSHA 6 NOT OPERATOR A ഥ 𝑨 𝑁𝑂𝑇 𝐴 = 𝐴ҧ False True True False A ഥ 𝐀 A ഥ 𝑨 0 1 1 0 7/24/2023 DR. SAMANTHA RAKAPAKSHA 7 UNIVERSALS GATE (NAND/NOR) ❑Manufacturing cost is low. ❑Electricity needed to operator is low. ❑All basic gates can be developed using universal gates. ❑NAND=NOT(AND) ❑NOR=NOT(OR) ❑NAND = 𝐴. 𝐵 ❑NOR = 𝐴 + 𝐵 7/24/2023 DR. SAMANTHA RAKAPAKSHA 8 NAND A 0 B 0 A.B 0 𝑨. 𝑩 1 0 1 0 1 1 0 0 1 NANAD=NOT(AND) 1 1 1 0 NAND = 𝐴. 𝐵 A A 𝐴. 𝐵 B B 𝐴. 𝐵 7/24/2023 DR. SAMANTHA RAKAPAKSHA 9 NOR A 0 B 0 A+B 0 𝑨+𝑩 1 0 1 1 0 1 0 1 0 NOR=NOT(OR) 1 1 1 0 NOR= 𝐴 + 𝐵 A A 𝐴+𝐵 B B 𝐴+𝐵 7/24/2023 DR. SAMANTHA RAKAPAKSHA 10 FIND THE BOOLEAN EXPRESSION FOR THE OUTPUT. A B 𝑿 C 7/24/2023 DR. SAMANTHA RAKAPAKSHA 11 FIND THE BOOLEAN EXPRESSION FOR THE OUTPUT. A 𝑨+𝑩 B 𝑿 = 𝑨 + 𝑩. 𝑩𝑪 C 𝑩. 𝑪 7/24/2023 DR. SAMANTHA RAKAPAKSHA 12 FIND THE BOOLEAN EXPRESSION FOR THE OUTPUT. A B 𝑿 C 7/24/2023 DR. SAMANTHA RAKAPAKSHA 13 FIND THE BOOLEAN EXPRESSION FOR THE OUTPUT. A 𝑨+𝑩 B 𝑨+𝑩 ഥ 𝑿 = 𝑨 + 𝑩. 𝐵. 𝑪 ഥ 𝑩. 𝑪 C ഥ 𝑪 7/24/2023 DR. SAMANTHA RAKAPAKSHA 14 FIND THE BOOLEAN EXPRESSION FOR THE OUTPUT. 7/24/2023 DR. SAMANTHA RAKAPAKSHA 15 FIND THE BOOLEAN EXPRESSION FOR THE OUTPUT. 7/24/2023 DR. SAMANTHA RAKAPAKSHA 16  OPERATOR PRECEDENCE Eg: Boolean Algebra: Mathematics: 𝟑 + 𝟐 × 𝟏𝟎 = 𝟑 + 𝟐𝟎 = 𝟐𝟑 Bracket () B-Brackets 𝟑 + 𝟐 × 𝟏𝟎 = 𝟓 × 𝟏𝟎 =50 NOT O-Of/Order AND(×) D-Division OR(+) M-Multiplication A-Addition Eg: 𝐴 + 𝐵. 𝐶 S-Subtraction First B and C multiply together then add with A. 7/24/2023 DR. SAMANTHA RAKAPAKSHA 17 DRAW THE LOGIC CIRCUITS. 𝐴 + 𝐵ത 𝐴𝐵 7/24/2023 DR. SAMANTHA RAKAPAKSHA 18 DRAW THE LOGIC CIRCUITS. A 𝐴 + 𝐵ത 𝐴 + 𝐵ത B A B 𝐴. 𝐵 𝐴𝐵 7/24/2023 DR. SAMANTHA RAKAPAKSHA 19  𝐴 + 𝐵ത. 𝐶 7/24/2023 DR. SAMANTHA RAKAPAKSHA 20  𝐴 + 𝐵ത. 𝐶 A 𝐴 + 𝐵ത B C 7/24/2023 DR. SAMANTHA RAKAPAKSHA 21  ҧ + 𝐴𝐵 𝐴𝐵 ഥ 7/24/2023 DR. SAMANTHA RAKAPAKSHA 22  ҧ + 𝐴𝐵 𝐴𝐵 ഥ A 𝐴𝐵ത B ҧ 𝐴𝐵 7/24/2023 DR. SAMANTHA RAKAPAKSHA 23 FIND THE BOOLEAN EXPRESSION FOR THE OUTPUT. A B OUTPUT A B OUTPUT 0 0 0 0 0 1 0 1 1 0 1 0 1 0 1 1 0 0 1 1 1 1 1 0 A B OUTPUT A B OUTPUT 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 1 1 0 7/24/2023 DR. SAMANTHA RAKAPAKSHA 24 FIND THE BOOLEAN EXPRESSION FOR THE OUTPUT. A B OUTPUT A B OUTPUT 0 0 0 0 0 1 OUTPUT=A+B 𝑂𝑈𝑇𝑃𝑈𝑇 = 𝐴 + 𝐵 0 1 1 0 1 0 1 0 1 1 0 0 1 1 1 1 1 0 A B OUTPUT A B OUTPUT 0 0 0 0 0 1 0 1 0 0 1 1 𝑂𝑈𝑇𝑃𝑈𝑇 = 𝐴𝐵 OUTPUT=A.B 1 0 0 1 0 1 1 1 1 1 1 0 7/24/2023 DR. SAMANTHA RAKAPAKSHA 25  XOR(EXCLUSIVE OR) 7/24/2023 DR. SAMANTHA RAKAPAKSHA 26 XNOR (EXCLUSIVE NOR) 7/24/2023 DR. SAMANTHA RAKAPAKSHA 27 FIND THE BOOLEAN EXPRESSION FOR S AND C. 7/24/2023 DR. SAMANTHA RAKAPAKSHA 28 FIND THE BOOLEAN EXPRESSION FOR S AND C. A.B 7/24/2023 DR. SAMANTHA RAKAPAKSHA 29 FIND THE BOOLEAN EXPRESSION FOR SUM AND CARRY. 7/24/2023 DR. SAMANTHA RAKAPAKSHA 30 FIND THE BOOLEAN EXPRESSION FOR SUM AND CARRY. 7/24/2023 DR. SAMANTHA RAKAPAKSHA 31 DRAW THE LOGIC GATE DIAGRAM AND TRUTH TABLE FOR THE EXPRESSIONS 7/25/2023 DR. SAMANTHA RAKAPAKSHA 32 DRAW THE LOGIC GATE DIAGRAM AND TRUTH TABLE FOR THE EXPRESSIONS 7/25/2023 DR. SAMANTHA RAKAPAKSHA 33 IF 0 AND 1 RESPECTIVELY ARE GIVEN AS INPUTS FOR X IN THE FOLLOWING LOGIC CIRCUIT, WHAT WOULD BE THE TWO RESPECTIVE OUTPUTS AT Y? 7/25/2023 DR. SAMANTHA RAKAPAKSHA 34 LAWS IN BOOLEAN ALGEBRA 7/25/2023 DR. SAMANTHA RAKAPAKSHA 35 EXAMPLE 7/25/2023 DR. SAMANTHA RAKAPAKSHA 36 EXAMPLE 7/25/2023 DR. SAMANTHA RAKAPAKSHA 37 FOR GIVEN INPUTS, WHICH OF THE FOLLOWING LOGIC CIRCUITS PROVIDES THE SAME OUTPUT? 7/25/2023 DR. SAMANTHA RAKAPAKSHA 38 BASIC GATES USING NAND AND NOR GATES 7/25/2023 DR. SAMANTHA RAKAPAKSHA 39  THANK YOU 7/27/2023 DR. SAMANTHA RAKAPAKSHA 40