SYNERGY INSTITUTE OF ENGINEERING & TECHNOLOGY,DHENKANAL LECTURE NOTES ON DIGITAL ELECTRONICS CIRCUIT(SUBJECT CODE:PCEC4202)

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1 Lecture No:5 Boolean Expressions and Definitions Boolean Algebra Boolean Algebra is used to analyze and simplify the digital (logic) circuits. It uses only the binary numbers i.e. 0 and 1. It is also called as Binary Algebra or logical Algebra. Boolean algebra was invented by George Boole in Lows of Boolean Algebra Commutative Lows: The commutative laws are applied to addition and multiplication. For addition, the commutative law states: In terms of the result, the order in which variables are ORed makes no difference. A + B = B + A For multiplication, the commutative law states: In terms of the result, the order in which variables are ANDed makes no difference. AB = BA Associative Laws: The associative laws are also applied to addition and multiplication. For addition, the associative law states: When ORing more than two variables, the result is the same regardless of the grouping of the variables. A + (B +C) = (A + B) + C 1

For multiplication, the associative law states: When ANDing more than two variables, the result is the same regardless of the grouping of the variables.

2 For multiplication, the associative law states: When ANDing more than two variables, the result is the same regardless of the grouping of the variables. A(BC) = (AB)C Distributive Law: The distributive law is the factoring law. A common variable can be factored from an expression just as in ordinary algebra. That is: AB + AC = A(B+ C) 2

3 Rules of Boolean Algebra Rule 1: A + 0 = A, a variable ORed with 0 is always equal to the variable. Rule 2: A + 1 = 1, a variable ORed with 1 is always equal to 1. Rule 3: A. 0 = 0, a variable ANDed with 0 is always equal to 0. Rule 4: A. 1 = A, a variable ANDed with 1 is always equal to the variable. 3

4 Rule 5: A + A = A, a variable ORed with itself is always equal to the variable. Rule 6: A + A = 1, a variable ORed with its complement is always equal to 1. Rule 7: A. A = A, a variable ANDed with itself is always equal to the variable. 4

5 Rule 8: A. A = 0, a variable ANDed with its complement is always equal to 0. Rule 9: A = A, the double complement of a variable is always equal to the variable. Rule 10: A + AB = A, this rule can be proved as follows: A+AB= A (1+B) Factoring (distributive low) = A.1 Rule2: (1+B) = 1 = A Rule4: A. 1 = A Rule 11: A + A B = A + B, this rule can be proved as follows: A + A B = (A +AB) + A B Rule10 (A=A+AB) = (AA+AB) + A B Rule7: A = AA = AA + AB +AA +AB Rule8: adding AA = 0 = (A+A ) (A+B) Factoring = 1. (A+B) Rule6: A+A =1 5

6 = A+ B Rule4: 1. X = X Rule 12: (A + B)(A + C) = A + BC, this rule can be proved as follows: (A + B)(A + C) = AA + AC + AB + BC Distributive low = A + AC + AB + BC Rule7: AA = A = A (1+C) + AB + BC Factoring = A.1 + AB + BC Rule2: 1+C = 1 = A (1 + B) + BC Factoring = A.1 +BC Rule2: 1+B = 1 = A + BC Rule4: A.1 = A DeMorgan s Theorems 1 st Theorem: The complement of a product of variables is equal to the sum of the complemented variables. (XY) = X + Y 2 st Theorem: The complement of a sum of variables is equal to the product of the complemented variables. (X + Y) = X. Y 6

7 Example: Apply DeMorgan s theorem to remove the overbar covering both terms from the expression X = (C + D) Solution: To apply DeMorgan s theorem to the expression, you can break the overbar covering both terms and change the sign between the terms. This results in: X = C. D Deleting the double bar gives X = C. D 7

ENGIN 112 Intro to Electrical and Computer Engineering

ENGIN 112 Intro to Electrical and Computer Engineering ENGIN 2 Intro to Electrical and Computer Engineering Lecture 5 Boolean Algebra Overview Logic functions with s and s Building digital circuitry Truth tables Logic symbols and waveforms Boolean algebra