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# 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*