CHAPTER 1 : INTRODUCTION EKT 121 / 4 ELEKTRONIK DIGIT 1 CHAPTER 1 : INTRODUCTION

4.0 BOOLEAN ALGEBRA Boolean Operations & expression Laws & rules of Boolean algebra DeMorgan’s Theorems Boolean analysis of logic circuits Simplification using Boolean Algebra Standard forms of Boolean Expressions Boolean Expressions & truth tables The Karnaugh Map

Karnaugh Map SOP minimization Karnaugh Map POS minimization 5 Variable K-Map Programmable Logic

Boolean Operations & expression Variable – a symbol used to represent logical quantities (1 or 0) ex : A, B,..used as variable Complement – inverse of variable and is indicated by bar over variable ex : Ā

Operation : Boolean Addition – equivalent to the OR operation X = A + B Boolean Multiplication – equivalent to the AND operation X = A∙B A X B A X B

Laws & rules of Boolean algebra

Commutative law of addition A+B = B+A the order of ORing does not matter.

Commutative law of Multiplication AB = BA the order of ANDing does not matter.

Associative law of addition A + (B + C) = (A + B) + C The grouping of ORed variables does not matter

Associative law of multiplication A(BC) = (AB)C The grouping of ANDed variables does not matter

(A+B)(C+D) = AC + AD + BC + BD Distributive Law A(B + C) = AB + AC (A+B)(C+D) = AC + AD + BC + BD

Boolean Rules 1) A + 0 = A In math if you add 0 you have changed nothing In Boolean Algebra ORing with 0 changes nothing

Boolean Rules 2) A + 1 = 1 ORing with 1 must give a 1 since if any input is 1 an OR gate will give a 1

Boolean Rules 3) A • 0 = 0 In math if 0 is multiplied with anything you get 0. If you AND anything with 0 you get 0

Boolean Rules 4) A • 1 = A ANDing anything with 1 will yield the anything

Boolean Rules 5) A + A = A ORing with itself will give the same result

Boolean Rules 6) A + A = 1 Either A or A must be 1 so A + A =1

Boolean Rules 7) A • A = A ANDing with itself will give the same result

Boolean Rules 8) A • A = 0 In digital Logic 1 =0 and 0 =1, so AA=0 since one of the inputs must be 0.

Boolean Rules 9) A = A If you not something twice you are back to the beginning

Boolean Rules 10) A + AB = A Proof: A + AB = A(1 +B) DISTRIBUTIVE LAW = A∙1 RULE 2: (1+B)=1 = A RULE 4: A∙1 = A

Boolean Rules 11) A + AB = A + B If A is 1 the output is 1 , If A is 0 the output is B Proof: A + AB = (A + AB) + AB RULE 10 = (AA +AB) + AB RULE 7 = AA + AB + AA +AB RULE 8 = (A + A)(A + B) FACTORING = 1∙(A + B) RULE 6 = A + B RULE 4

Boolean Rules 12) (A + B)(A + C) = A + BC PROOF (A + B)(A +C) = AA + AC +AB +BC DISTRIBUTIVE LAW = A + AC + AB + BC RULE 7 = A(1 + C) +AB + BC FACTORING = A.1 + AB + BC RULE 2 = A(1 + B) + BC FACTORING = A.1 + BC RULE 2 = A + BC RULE 4

END OF BOOLEAN RULES & LAWS