# Boolean Algebra

May 19 • General • 6268 Views • 3 Comments on Boolean Algebra Notes on Boolean Algebra

In Mathematics, Boolean algebra is part of the algebra. It also includes variables and constants, the values of the variables which comes out after the calculations and using all the logic are the truth values either TRUE or FALSE. i.e. the binary values 0 or 1

Boolean algebra is widely used in computers.

Instead of other algebra in which the values are numbers which comes after doing various operations like addition, subtraction, multiplication division etc. The main operations used in Boolean algebra are conjunction which is denoted by ^ other operators are disjunction denoted by ᵥ   and negation denoted by ¬.

Basic Operations :

1. AND (conjunction) denoted by ^

2. OR (disjunction) denoted by ᵥ

3. NOT (negation) denoted by ¬

If we combine these operations with basic mathematical operations  then it can be reprsented as :

x^y = xy

xᵥy = x+y – xy

¬x = 1-x

The values of above expression can be explained with the help of truth tables. They are explained as under :

1. x^y

2. xᵥy

 x y x^y x ᵥy 0 0 0 0 1 0 0 1 0 1 0 1 1 1 1 1

3. xᵥy

 x ¬x 0 1 1 0

Some laws i.e. the MONOTONE LAWS used here are :

1. Associativity

x∨(yz) = (xy)∨z

x∧(yz) = (xy)∧z

2. Commutativity

xy = yx

xy = yx

3. Distributivity

x∧(yz) = )=(x∧y)∨(x∧z)

4. Identitivity

x∨0 = x

x∧1=x

x∧0=0

Example

1.  Verify that the following operations are commutative

a. AND  b. OR c. XOR

a. AND

A . B = B . A ( therefore AND operation is commutative)

b. OR

A + B = B + A (therefore OR operation is commutative)

2.Derive the Boolean expression for the logic circuit shown below: A.  B.  C.  D.  3. From the Truth table below, Determine the boolean expression :

From the truth table below, determine the standard SOP expression.

 Input Output A B C X 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 0

 A.  B.  C.  D.  