Boolean Algebra

Introduction

• An algebra useful in designing logic circuits of processors

• Formalized by George Boole in 1854

• In 1938, Claude Shannon recognized how boolean algebra could be applied to on-and off circuits , where all signals are characterized as high(1) or low(0)

Binary Logic

• Variable take 2 discrete values– True & False– Yes & No– 1 & 0

• Operations assume logical meaning• 3 basic logical operations

– AND• multiplication

– OR• addition

– NOT• inversion

LOGIC GATES

LOGIC GATES

• Logic operations involved in design of digital systems are:– AND– OR– NOT– NAND– EXCLUSIVE-OR

• Each of these operation is performed by electronic circuits known as gates.

• Logic gates are basic building blocks of digital systems

Gates

• Digital Circuit has two logic values:– 0 for low voltage– 1 for high voltage

• Other voltages are not permitted

• Gates, tiny electronic devices can compute functions of these voltages

• They form the basis of computers

Gates

• Logic gates are small structures that accept 1 or more inputs and produce 1 output

• Each gate has a distinct graphics symbol and it’s operation can be described by means of an algebraic expression or in a form of a table called the truth table.

• Truth table

• Table of all possible combination of i/ps & their corresponding o/ps

AND gate

• Corresponds to multiplication

• o/p=1 iff both i/p=1

A

BY

Y = A . B

A B Y

0 0 0

0 1 0

1 0 0

1 1 1

OR gate

• Corresponds to binary addition

• o/p=1 if atleast one i/p=1A

B

Y = A + B

This is read as Y equals A or B.

A B Y

0 0 0

0 1 1

1 0 1

1 1 1

Y

NOT gate

• Gate with a single transistor

• Also called as an inverter

• Symbol

A A

Input Output

0 1

1 0Y = A

NAND Gate

• AND gate with output inverted

Y = A . B

A

B

Y

A B Y

0 0 1

0 1 1

1 0 1

1 1 0

NOR gate

A

B

Y = A + B

A B Y

0 0 1

0 1 0

1 0 0

1 1 0

• OR gate with output inverted

Exclusive OR

• Output is high if inputs are at different logic levels. I.e either 0 and 1 or 1 and 0

• Output is low if inputs are same

Y = A + B+

A

B

YA B Y

0 0 0

0 1 1

1 0 1

1 1 0

Exclusive NOR

A

B

Y

Y = A B+

A B Y

0 0 1

0 1 0

1 0 0

1 1 1

• To add A and A together

• Sum of a variable and its complement

AA + A

A

A

A + A

Universal gates

• NAND and NOR are called as Universal gates because any digital circuit can be represented by using

• NAND gates alone• Nor gates alone

• To prove it, we have to show that AND, OR and NOT can be implemented using NAND or NOR

NAND gate implementation

X

X

F = X’

XY

Y

F= X.Y

F = X + Y

COMBINATIONAL &

SEQUENTIAL CIRCUITS

Combinational & Sequential circuits

• Two types of digital circuits– Combinational– Sequential

• Combinational – o/p depends entirely on the i/ps present at that moment– Design requires AND, OR & NOT operations– e.g. adder/subtractor, digital comparator, multiplexers/selectors,

demultiplexers, encoders, decoders, code converters

• Sequential – o/p depends on past o/p as well as the inputs present at that

moment– Flip-Flops are also required for the dsesign– e.g. registers, shift registers, counters, latches, memory

Basic Identities

A + 0 = A 0 . A = 0

A + 1 = 1 1 . A = A

A + A = A A . A = A

A + A = 1 A . A = 0

A = A Double complementation

Boolean Algebra Rules

• Commutative lawsX + Y = Y + X

X . Y = Y . X

• Associative lawsX + ( Y + Z ) = ( X + Y ) + Z

X ( YZ ) = ( XY ) Z

Boolean Algebra Rules

• Distributive lawsX ( Y + Z ) = XY + XZ

• Other rules (derived from the above)X + XZ = X

X ( X + Y ) = X

X+X’.Y = X+Y

Proof by perfect induction

• Proving with the help of truth tables

• Write all possible combinations for the inputs and check the truth values of the result (both left hand and right hand side)

De Morgan’s Theorem

• Useful in simplifying expressions in which a product or sum of variables is inverted(X + Y)‘ = X’ . Y’(X. Y)’ = X’ + Y’

• To get the complement of any expression – Replace + by . and . by +– Each of the terms in the expression is

complemented

Canonical & Standard Forms

MINTERM

• Binary variable can appear in– Normal form (x)– Complement form (x’)

• Binary variables x and y combined with an AND operation

MINTERM

• 4 possible combination– x’y’– x’y– xy’– xy

• Each of these is called a minterm or standard product.

• n variables can be combined to form 2n

minterms

MAXTERM

• n variables forming an OR term

• 2n possible combination called maxterm or standard sums.

CANONICAL FORM

• Any boolean function can be expressed as a product of maxterms or as sum of minterms.

• Such a form is called as canonical form.

STANDARD FORMS

• 2 types of standard forms:– Sum of products– Product of sums

• Sum of products– F1= y’+ xy+ x’yz’

• Product of sums– F2= x (y’ + z) (x’ + y + z’)

Karnaugh Map

Karnaugh map

• Simple, straightforward procedure for simplifying boolean expressions

• Pictorial arrangement of truth table which chooses the minimum number of terms needed to express the function algebraically

• The rows and columns of the K-map correspond to the possible values of the function's input

• Each cell in the K-map represents a minterm (i.e. a three variables function has: x’y’z’, x’y’z, x’yz’, x’yz, xy’z’, xy’z, xyz’ and xyz)

• Boolean functions having upto six variables can be minimized

• K-maps consists of squares where each square represents a minterm.

• Each combination of variables in a truth table is called a minterm.

• A function of n variables will have 2n

minterms

• Minterm is a product term of the variables

• Eg: 011 is written as A’BC

Labeling Karnaugh map

• Incorrectly labeled K-maps will not lead to correct minimization

• The sequence used is 00, 01, 11, 10

• This is an example of gray coding where adjacent values differ by only one bit

• In terms of minterms, correct labeling is

A’B’ , A’B, AB, AB’

Minimization Procedure

• Develop the minterm form from the truth table

• Plot 1’s in the Karnaugh map for each minterm in the expression

• Loop adjacent groups of 2, 4 or 8 one’s together

Minimization procedure

• Write one minterm per loop, eliminating variables where possible

When a variable and its complement are contained inside a loop then that variable can be eliminated. Write the variables that are left.

• Logically OR the remaining minterms together to give simplified minterm expression

Looping in K-maps

• Loops must contain 2n cells set to 1

• A single cell cannot be simplified

• A loop of 2n cells is independent of n variables

• Using the largest loops possible will give the smallest or simplest function

Looping (contd…)

• All cells in K-map set to 1 must be included in atleast one loop

• Loops may overlap if they contain atleast one other unlooped cell

• Any loop that has all of its cells included in other loops is redundant

• Loops must be square or rectangular

Diagonal or L-shaped loops are invalid

• Edges of K-map are considered adjacent

A loop can leave at the top of a K-map and re-enter at the bottom. Similarly for the two sides.

• There may be different ways of looping a K-map since for any given circuit there may not be a unique minimal form

A B C D f(A,B,C,D)

0 0 0 0 1

0 0 0 1 0

0 0 1 0 0

0 0 1 1 0

0 1 0 0 1

0 1 0 1 1

0 1 1 0 0

0 1 1 1 1

1 0 0 0 1

1 0 0 1 0

1 0 1 0 0

1 0 1 1 0

1 1 0 0 1

1 1 0 1 1

1 1 1 0 0

1 1 1 1 1

f(A,B,C,D) = A’B’C’D’+A’BC’D’+A’BC’D+ A’BCD+AB’C’D’+ABC’D’+ABC’D+ABCDf(A,B,C,D) = A’B’C’D’+A’BC’D’+A’BC’D+ A’BCD+AB’C’D’+ABC’D’+ABC’D+ABCD

• Grouping– Largest Group First

• BD

– Next Largest• C’D’

• Determine Final Expression

– BD + C’D’

Don’t Care condition

• Two reasons for don't-care: – It does not matter whether the output is 0 or 1; – The corresponding combinations of inputs are

impossible, therefore they never occur.

• denoted by ‘d’ or a ‘X’ in Karnaugh maps

• Need not be covered by subcubes, but used to enlarge subcubes

Don’t Care example

•

Product-of-sums

• Each sum in a product-of sums expression is called a maxterm.

• Simplification– Solve for 0’s– Complement the resulting expression