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BOOLEAN ALGEBRA 1 P0INTS TO REMEMBER 1 .BOOLEAN ALGEBRA: Is a two state algebra or algebra of logics.It is also called Switching Algebra. It is based on Binary number system and uses the numeric constants 0 and 1. 2. BINARY DECISION: The decision which results into either ‘ TRUE ‘ or ‘FALSE’ where TRUE stands for 1 and FALSE stands for 0. 3. LOGICAL VARIABLES(Boolean Variable or Binary valued Variable):The variables which can stores values either TRUE or FALSE OR ‘0’ or ‘1’. 4 BINARY OPERATIONS :Is an operation in which for a set of variables result is the values i.e 0 or 1. 5 BOOLEAN OPERATORS:: In Boolean Operation ,operators used are of three types:

a) NOT- It is a Unary Operator.i.e it operateson single variable and operation performed by it is known as Complementation or Negation .Its symbol is “¯”or “ ’ ”.e.g. A’ or Ā b)AND – It is a Binary Operator . It operates on two variables and operation

performed by it is known as Logical Multiplication . Its Symbol is “.” Or “ ˆ ”.

e.g. A .B or Aˆ B. c) OR- It is a Binary Operator . It operates on two variables and operation performed

by it is known as Logical Addition . Its symbol is “ +” or “ˇ”. e.g. . A + B or A ˇ B. 6. TRUTH TABLE – A truth table is a table that describes the behaviour of a logic gate.It shows all input and output possibilities for logical variables or statements.The input patterns are written in Binary Progression. 7.TAUTOLOGY- If the result of a logical statement or exprssion is always true or ‘1’, it is known as Tautology. 8. FALLACY-- If the result of a logical statement or exprssion is always False or ‘0’, it is known as Fallacy. 9 .LOGIC GATES-A logic gate performs a logical operation on one or more logic inputs and produces a single logic output.

Type Distinctive shape Boolean algebra between A & B Truth table

AND

A ..B

INPUT OUTPUT A B A AND B 0 0 0 0 1 0

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1 0 0 1 1 1

OR

A + B

INPUT OUTPUT A B A OR B 0 0 0 0 1 1 1 0 1 1 1 1

NOT

Ā

INPUT OUTPUT A NOT A 0 1 1 0

NAND

INPUT OUTPUT

A B A NAND B

0 0 1 0 1 1 1 0 1 1 1 0

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NOR

INPUT OUTPUT A B A NOR B 0 0 1 0 1 0 1 0 0 1 1 0

XOR

INPUT OUTPUT A B A XOR B 0 0 0 0 1 1 1 0 1 1 1 0

XNOR

INPUT OUTPUT

A B A XNOR B

0 0 1 0 1 0 1 0 0 1 1 1

BOOLEAN POSTULATES

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

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LAWS OF BOOLEAN ALGEBRA

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)

Examples

Prove T10 : (a)

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S(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).

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(b)

DEMORGAN THEOREM

(a) Proof of (A + B)' = A' . B',

These can be proved by the use of truth tables 1(a) & 1(b)

table 1(a)

A B A+B (A+B)' 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 0

table 1(b)

A B A' B' A'.B' 0 0 1 1 1 0 1 1 0 0 1 0 0 1 0 1 1 0 0 0

The two truth tables are identical, and so the two expressions are identical.

(b)Proof of (A.B) = A' + B'

(A.B) = A' + B', These can be proved by the use of truth tables 2(a) & 2(b)

table 2(a)

A B A.B (A.B)' 0 0 0 1

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0 1 0 1 1 0 0 1 1 1 1 0

table2(b)

A B A' B' A'+B' 0 0 1 1 1 0 1 1 0 1 1 0 0 1 1 1 1 0 0 0 Canonical form: standard form for a Boolean expression provides a unique algebraic signature Minterms and Maxterms Any boolean expression may be expressed in terms of either minterms or maxterms. A literal is a single variable within a term which may or may not be complemented. For an expression with N variables, minterms and maxterms are defined as follows : A minterm is the product of N distinct literals where each literal occurs exactly once A maxterm is the sum of N distinct literals where each literal occurs exactly once

For a two-variable expression, the minterms and maxterms are as follows

X Y Minterm Maxterm 0 0 X'.Y' X+Y 0 1 X'.Y X+Y' 1 0 X.Y' X'+Y 1 1 X.Y X'+Y'

For a three-variable expression, the minterms and maxterms are as follows

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X Y Z Minterm Maxterm 0 0 0 X'.Y'.Z' X+Y+Z 0 0 1 X'.Y'.Z X+Y+Z' 0 1 0 X'.Y.Z' X+Y'+Z 0 1 1 X'.Y.Z X+Y'+Z' 1 0 0 X.Y'.Z' X'+Y+Z 1 0 1 X.Y'.Z X'+Y+Z' 1 1 0 X.Y.Z' X'+Y'+Z 1 1 1 X.Y.Z X'+Y'+Z' Sum Of Products (SOP)

The Sum of Products form represents an expression as a sum of minterms.To derive the Sum of Products form from a truth table, OR together all of the minterms which give a value of 1.

Example – SOP Consider the truth table X Y F Minterm 0 0 0 X'.Y' 0 1 0 X'Y 1 0 1 X.Y' 1 1 1 X.Y

Here SOP is f(X.Y) = X.Y' + X.Y

Product Of Sum (POS)

The Product of Sums form represents an expression as a product of maxterms.

To derive the Product of Sums form from a truth table, AND together all of the maxterms which give a value of 0.

Example – POS

Consider the truth table from the previous example.

X Y F Maxterm 0 0 1 X+Y 0 1 0 X+Y'

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1 0 1 X'+Y 1 1 1 X'+Y'

Here POS is F(X,Y) = (X+Y')

Minimisation of Boolean Functions

In mathematics expressions are simplified to understand and easier to write down, they are also less prone to error.

Minimisation can be achieved by a following methods:

1)Algebraic Manipulation of Boolean Expressions.

2)Karnaugh Maps

Algebraic Manipulation of Boolean Expressions

This is an approach where you can transform one boolean expression into an equivalent expression by applying Boolean Theorems

Karnaugh Maps

K-Maps are a convenient way to simplify Boolean Expressions. They can be used for up to 4 or 5 variables. They are a visual representation of a truth table. Expression are most commonly expressed in sum of products form.

.

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