# Boolean Algebra

Introduction
Laws
of Boolean Algebra

Examples
Problems

On-line
Quiz

### Introduction

The most obvious way to simplify Boolean
expressions is to manipulate them in the same way as normal algebraic
expressions are manipulated. With regards to logic relations in digital forms, a
set of rules for symbolic manipulation is needed in order to solve for the
unknowns.

A set of rules formulated by the English mathematician *George Boole* describe certain propositions whose
outcome would be either *true or false*. With regard to digital logic, these rules are
used to describe circuits whose state can be either,* 1
(true) or 0 (false)*. In order to fully
understand this, the relation between the AND
gate, OR
gate and NOT
gate operations should be appreciated. A number of rules can be derived from
these relations as Table 1 demonstrates.
**P1: X = 0 or X = 1 **
**P2: 0 . 0 = 0 **

**P3: 1 + 1 = 1 **

**P4: 0 + 0 = 0 **

**P5: 1 . 1 = 1 **

**P6: 1 . 0 = 0 . 1 = 0 **

**P7: 1 + 0 = 0 + 1 = 1 **

**Table 1: Boolean
Postulates**

### Laws of Boolean
Algebra

Table
2 shows the basic Boolean laws. Note that every law has two expressions, (a)
and (b). This is known as *duality*. These are
obtained by changing every AND(.) to OR(+), every OR(+) to AND(.) and all 1's to
0's and vice-versa.

It has become conventional to drop the
**.** (AND symbol) i.e. A**.**B is written as AB.
**T1 : ****Commutative Law**
- (a)
*A + B = B + A*

(b) *A B = B A*
**T2 : Associate Law **
- (a)
*(A + B) + C = A + (B + C)*

(b) *(A B) C = A (B C)*
**T3 : Distributive Law**
- (a)
*A (B + C) = A B + A C*

(b) *A + (B C) = (A + B) (A + C)*
**T4 : Identity Law**
- (a)
*A + A = A*

(b) *A A = A*
**T5 : **
- (a)

(b)
**T6 : Redundance Law **
- (a)
*A + A B = A*

(b) *A (A + B) = A*
**T7 : **
- (a)
*0 + A = A*

(b) *0 A = 0*
**T8 : **
- (a)
*1 + A = 1*

(b) *1 A = A*
**T9 : **
- (a)

(b)
**T10 : **
- (a)

(b)
**T11 : De Morgan's
Theorem **
- (a)

(b)

**Table 2: Boolean Laws**

### Examples

Prove T10 : (a)
(1) Algebraically:

(2) Using the truth table:

Using the laws given above, complicated expressions can be simplified.

### Problems

(a) Prove T10(b).
(b) Copy or print out the truth table below and use it to prove T11: (a) and
(b).

Click here
for answers.

Click here
for* on-line Boolean Algebra quiz*.

To submit your questions and queries please click here:

*Composed by David Belton - April 98*