Boolean Algebra

 • Boolean algebra provides a mathematical framework for digital systems.
• A basic knowledge of Boolean algebra is indispensable to the study and analysis of logic
circuits.
• Remember that in the last lecture notes,  Boolean operations and expressions in terms of their relationship to NOT, AND, OR, NAND, and NOR gates were introduced.
• Variable, complement and literal are terms used in Boolean algebra.
• A variable is a symbol used to represent a logical quantity. Any single variable can have a 1
or a 0 value.
• The complement is the inverse of a variable and is indicated by a bar over the variable (overbar).
• A literal is a variable or the complement of a variable.

Boolean Addition
• Boolean addition is equivalent to the OR operation.
• Basic rules:
– 0 + 0 = 0
– 0 + 1 = 1
– 1 + 0 = 1
– 1 + 1 = 1
• In Boolean algebra, a sum term is a sum of literals.
• In logic circuits, a sum term is produced by an OR operation with no AND operations involved.
Examples:
– A + B

– A + B + C

 Boolean Multiplication
• Boolean multiplication is equivalent to the AND operation.
• Basic rules:
– 0.0 = 0
– 0.1 = 0
– 1.0 = 0
– 1.1 = 1
• In Boolean algebra, a product term is the product of literals.
• In logic circuits, a product term is produced by an AND operation with no OR operations involved. Examples:
– AB
– ABCD

 Laws and Rules of Boolean Algebra
• As in other areas of mathematics, there are certain well-developed laws and rules that must be followed in order to properly apply Boolean Algebra

Laws of Boolean algebra
• Commutative laws:
A + B = B + A
AB = BA

• Associative laws:
A + (B + C) = (A + B) + C
A(BC) = (AB)C

• Distributive laws:
A(B + C) = AB + AC

Rules of Boolean Algebra
1. A + 0 = A

2. A + 1 = A

3. A.0 = 0

4. A.1 = A

5. A + A = A

6. A + A = 1

7. A.A = A

8. A.A = 0

9. A = A

10.A + AB = A

11.A + AB = A + B

12.(A + B)(A + C) = A + BC

 DeMorgan’s Theorems
1. The complement of a product of variable is
equal to the sum of the complements of the
variables.
XY = X + Y

2. The complement of a sum of variables is equal
to the product of the complements of the
variables.
X + Y = X Y