Boolean Algebra and Logic Gates - KFUPM...maxterm with subscript j is a complement of the minterm with the same subscript j and vice versa. EE 200 – Digital Logic Circuit Design

Boolean Algebra and Logic Gates

EE 200

Digital Logic Circuit Design

Dr. Abdulaziz Tabbakh

College of Computer Sciences and Engineering

King Fahd University of Petroleum and Minerals

EE 200– Digital Logic Circuit Design – KFUPM slide 2

Outline

Introduction

Definition of Boolean Algebra

Axioms, Theorems and Properties of Boolean Algebra

Boolean Functions

Digital Logic Gates

Canonical & Standard Forms

Minterms and Maxterms

More Logic Operations


Introduction

Binary logic is used in all of today’s digital computers

and devices the cost of the circuits that implement it is

an important factor addressed by designers

Finding simpler and cheaper, but equivalent, realizations

of a circuit can reap huge payoffs in reducing the overall

cost of the design.

Mathematical methods that simplify circuits rely primarily

on Boolean algebra.

This chapter provides a basic vocabulary and a brief

foundation in Boolean algebra that will enable you to

optimize simple circuits.


Definitions

George Boole in 1854 developed an algorithmic system

called Boolean Algebra and Shannon in 1938 introduced

the 2-valued Boolean Algebra (Switching Theory).

Boolean Algebra: is an algebraic system defined by:

a set of elements, B = {0,1}

a set of binary operators (.) and (+)

satisfies (Huntington) postulates.


Definitions

Huntington postulates:

1. Closure: wrt (.) and (+)

2. Identity element:

a + 0 = 0 + a = a

b · 1 = 1 · b = b

3. Commutitive:

e.g. a + b = b + a, a · b = b · a

BbaBbBa

BbaBbBa

,

,


Definitions

Huntington postulates (Contd.):

4. Distributive : wrt (.) and (+)

a · (b+c)= (a·b) + (a·c)

a + (b·c)= (a+b) · (a+c)

5. For every element a, there exists a’ (complement) where

a + a’ =1 and a · a’ = 0

6. There are at least two elements.

baBba ,,


Two-Valued Boolean Algebra

Is defined on a set of two elements, B = {0,1}, with rules

for two binary operators, ‘+’, ‘.’

These rules follow exactly the AND, OR and NOT

operations defined previously.

See the Book Sec (2.3) to show how the Huntington postulates are valid for the

set B = {0, 1}


Theorem and Properties

1. Duality:

Duality principle states that every algebraic expression

deducible from the postulates of Boolean Algebra remains valid

if the operators and identity elements are interchanged (i.e.

replacing each 1 with a 0, each 0 with a 1, and replacing each

AND (.) with an OR (+), and each OR (+) with an AND(.)).

e.g. (X+Y+Z) and X.Y.Z (are Duals)

x+1=1 and x.0=0 (are Duals)

a·(b+c)=(a·b)+(a·c) and a+(b·c)=(a+b)·(a+c) (are Duals)


Basic Theorem

Table 2.1 lists 6 theorem and 4 postulates


Proof of Theorem 1

Identity element

Complement

Distributive

Complement

Identity element


Proof of Theorem 2


Proof of Theorem 6


Proof of Theorem 5

The theorems of Boolean algebra can be proven by

means of truth tables.

both sides of the relation are checked to see whether they

yield identical results `for all possible combinations of the

variables involved

Check the book for proof of the remaining theorems.


Operator Precedence

Given the Boolean expression X.Y + W.Z the order of

applying the operators will affect the final value of the

expression.


Operator Precedence

For Boolean Algebra, the precedence rules for various

operators are given below, in a decreasing order of

priority:

Parenthesis

NOT Operator

AND Operator

OR Operator


Boolean Functions

A Boolean function is described by an algebraic

expression that expresses the logical relationship

between binary variables and consists of:

Binary variables

Constants (0,1)

Logical operators

The function can be evaluated to a specific value for

given value of the Boolean variables.

F = x + y’z


Boolean Functions

A Boolean function can be represented by:

Truth Table. (2n rows)

Circuit Diagram.


Boolean Functions Expression

One truth table

Multiple algebraic expressions multiple circuits.

By manipulating a Boolean expression according to the

rules of Boolean algebra, it is sometimes possible to

obtain a simpler expression for the same function and

thus reduce the number of gates in the circuit and the

number of inputs to the gate.

Designers are motivated to reduce the complexity and

number of gates to reduce the cost of the circuit.


Example


Algebraic Manipulation

A literal is a single variable within a term, in

complemented or uncomplemented form

Less literals and terms simpler circuit.

Use theorem and axioms to simplify functions.

e.g.

F2 = x’y’z + x’yz + xy’

= x’z(y’+y) + xy’

= x’z(1) + xy’

= x’z + xy’


Examples

𝑥 𝑥′ + 𝑦 = 𝑥𝑥′ + 𝑥𝑦 = 0 + 𝑥𝑦 = 𝑥𝑦

𝑥 + 𝑥′𝑦 = 𝑥 + 𝑥′ 𝑥 + 𝑦 = 1. 𝑥 + 𝑦 = 𝑥 + 𝑦

𝑥 + 𝑦 𝑥 + 𝑦′ = 𝑥 + 𝑥𝑦 + 𝑥𝑦′ + 𝑦𝑦′ = 𝑥 1 + 𝑦 + 𝑦′ = 𝑥

𝑥𝑦 + 𝑥′𝑧 + 𝑦𝑧 = 𝑥𝑦 + 𝑥′𝑧 + 𝑦𝑧 𝑥 + 𝑥′

= 𝑥𝑦 + 𝑥′𝑧 + 𝑥𝑦𝑧 + 𝑥′𝑦𝑧= 𝑥𝑦 1 + 𝑧 + 𝑥′𝑧 1 + 𝑦 = 𝑥𝑦 + 𝑥′𝑧

𝑥 + 𝑦 𝑥′ + 𝑧 𝑦 + 𝑧 = (𝑥 + 𝑦)(𝑥′ + 𝑧)

Consensus Theorem


Complement of a Function

The complement of a function can be derived

algebraically using DeMorgan’s theorems for two or

more variables

(a+b+c+d)’ = a’b’c’d’

(abcd)’ = a’+b’+c’+d’

The generalized form of DeMorgan’s theorems states

that the complement of a function is obtained by

interchanging AND and OR operators and

complementing each literal.


Example

F=(x’yz’ + x’y’z)

F’=(x’yz’ + x’y’z)’ = (x’yz’)’ (x’y’z)’

= (x+y’+z)(x+y+z’)

take the dual of the function and complement each literal

Fdual = (x’+y+z’)(x’+y’+z)

F’ = (x+y’+z)(x+y+z’)


Canonical and Standard Forms Minterms or standard products (AND operation)

For a Boolean function of n variables, a product term (ANDed term) in which each variable appears once (in either its true or complemented form) is called a minterm.

There are 2n different Minterms.

Maxterms or standard sums (OR operation)

For a Boolean function of n variables, a sum term (ORedterm) in which each variable appears once (in either its true or complemented form) is called a maxterm.

There are 2n different maxterms

Each maxterm is the complement of its corresponding minterm

A term that is a minterm or a max term SHOULD INCLUDE ALL INPUT VARIABLES


Minterms and Maxterms


Miterms and Maxterms

A Boolean function can be expressed algebraically from

a given truth table by forming a minterm for each

variable combination that produce a ‘1’ in the output, and

then ORing them together. (Sum Of Products) (SOP)

Similarly, we can express the function by forming a

maxterm for each variable combination that produce a ‘0’

in the output, and then ANDing them together. (Product

Of Sums) (POS).

A function is said to be in Canonical form if it is

represented as a SOP or POS.


Example

𝑓1 = 𝑥′𝑦′𝑧 + 𝑥𝑦′𝑧′ + 𝑥𝑦𝑧 = 𝑚1 +𝑚4 +𝑚7 = σ (1,4,7)

𝑓2 = 𝑥′𝑦𝑧 + 𝑥𝑦′𝑧 + 𝑥𝑦𝑧′ + 𝑥𝑦𝑧 = 𝑚3 +𝑚5 +𝑚6 +𝑚7

= σ (3,5,6,7)


Minterms of Function Complement

Forming a minterm for each combination that produces a

0 in the function and then ORing them.

𝑓1 = 𝑚1 +𝑚4 +𝑚7 ⇒ 𝑓′1 = 𝑚0 +𝑚2 +𝑚3 +𝑚5 +𝑚6𝑓′1 = 𝑥

′𝑦′𝑧′ + 𝑥′𝑦𝑧′ + 𝑥′𝑦𝑧 + 𝑥𝑦′𝑧 + 𝑥𝑦𝑧′

Take the complement of f’1 to get f1

𝑓1 = 𝑥 + 𝑦 + 𝑧 𝑥 + 𝑦′ + 𝑧 𝑥 + 𝑦′ + 𝑧′ 𝑥′ + 𝑦 + 𝑧′ 𝑥′ + 𝑦′ + 𝑧

𝑓1 = 𝑀0𝑀2𝑀3𝑀5𝑀6 =ෑ(0,2,3,5,6)

𝑓1 = 𝑥′𝑦′𝑧 + 𝑥𝑦′𝑧′ + 𝑥𝑦𝑧 = 𝑚1 +𝑚4 +𝑚7 =𝑚(1,4,7)


Conversion to Minterm

It is sometimes convenient to express a Boolean function

in its sum of minterms (SOP) form.

Expanding the expression into a sum of AND terms

Inspect each term

If it contains all the variables (good)

If it misses one or more variables, it is ANDed with an

expression such as (x + x’)


Example

Express the Boolean function 𝐹 = 𝐴 + 𝐵′𝐶 as a sum of minterms.

The first term A is missing two variables

𝐴 = 𝐴 𝐵 + 𝐵′ = 𝐴𝐵 + 𝐴𝐵′ = 𝐴𝐵 𝐶 + 𝐶′ + 𝐴𝐵′(𝐶 + 𝐶′)𝐴 = 𝐴𝐵𝐶 + 𝐴𝐵𝐶′ + 𝐴𝐵′𝐶 + 𝐴𝐵′𝐶′

The second term B’C is missing one variables

𝐵′𝐶 = 𝐵′𝐶 𝐴 + 𝐴′ = 𝐴𝐵′𝐶 + 𝐴′𝐵′𝐶

Combine all terms

𝐹 = 𝐴 + 𝐵′𝐶 = 𝐴𝐵𝐶 + 𝐴𝐵𝐶′ + 𝐴𝐵′𝐶 + 𝐴𝐵′𝐶′ + 𝐴′𝐵′𝐶𝐹 = 𝑚1 +𝑚4 +𝑚5 +𝑚6 +𝑚7

𝐹(𝐴, 𝐵, 𝐶) =(1,4,5,6,7)


Example

An alternative procedure for deriving the minterms of a

Boolean function is to obtain the truth table of the

function directly from the algebraic expression and then

read the minterms from the truth table.

𝐹 = 𝐴 + 𝐵′𝐶𝐹 = 𝑚1 +𝑚4 +𝑚5 +𝑚6 +𝑚7


Conversion to Maxterms

Express it in form of OR terms (use distributive law)

Any missing variable x in each term is ORed with xx’

Express the Boolean function 𝐹 = 𝑥𝑦 + 𝑥’𝑧 as a product of maxterms.

𝐹 = 𝑥𝑦 + 𝑥’𝑧 = (𝑥𝑦 + 𝑥’)(𝑥𝑦 + 𝑧)= (𝑥 + 𝑥’)(𝑦 + 𝑥’)(𝑥 + 𝑧)(𝑦 + 𝑧)= (𝑥’ + 𝑦)(𝑥 + 𝑧)(𝑦 + 𝑧)

Each OR term is missing one variable.

𝑥′ + 𝑦 = 𝑥’ + 𝑦 + 𝑧𝑧’ = (𝑥’ + 𝑦 + 𝑧)(𝑥’ + 𝑦 + 𝑧’)𝑥 + 𝑧 = 𝑥 + 𝑧 + 𝑦𝑦’ = (𝑥 + 𝑦 + 𝑧)(𝑥 + 𝑦’ + 𝑧)𝑦 + 𝑧 = 𝑦 + 𝑧 + 𝑥𝑥 = (𝑥 + 𝑦 + 𝑧)(𝑥’ + 𝑦 + 𝑧)

Combine.

𝐹 = 𝑥 + 𝑦 + 𝑧 𝑥 + 𝑦’ + 𝑧 𝑥’ + 𝑦 + 𝑧 𝑥’ + 𝑦 + 𝑧’ = 𝑀0𝑀2𝑀4𝑀5

𝐹(𝑥, 𝑦, 𝑧) =ෑ(0,2,4,5)


Conversion between Canonical Forms

The minterms of F’ are the minterms missing from F

𝐹 𝐴, 𝐵, 𝐶 = σ 1,4,5,6,7 ⇒ 𝐹’ 𝐴, 𝐵, 𝐶 = σ(0,2,3)

Now, if we take the complement of F‘ by DeMorgan’s

theorem:

𝐹’ ’ = 𝐹 = 𝑚0+𝑚2+𝑚3 ’ = 𝑚0′ . 𝑚2

′ . 𝑚3′ = 𝑀0𝑀2𝑀3

=ෑ(0,2,3)

To convert from one canonical form to another,

interchange the symbols ∑ and ∏ and list those numbers

missing from the original form.

maxterm with subscript j is a complement of the minterm with the

same subscript j and vice versa.


Conversion between Canonical Forms

𝐹 =(1,3,6,7) =ෑ(0,2,4,5)


Standard Form In Canonical forms, minterms and maxterms MUST contain

all variables.

In Standard forms, SOP and POS contain any number variables.

𝐹 = (𝑥 + 𝑦 + 𝑧)(𝑥 + 𝑦’ + 𝑧)(𝑥’ + 𝑦 + 𝑧)(𝑥’ + 𝑦 + 𝑧’)𝐹 = 𝑥𝑦 + 𝑥’𝑧

The sum of products is a Boolean expression containing

AND terms, called product terms, with one or more literals

each. The sum denotes the ORing of these terms.

A product of sums is a Boolean expression containing OR

terms, called sum terms. Each term may have any number

of literals. The product denotes the ANDing of these terms.


Standard Forms

These standard forms results in a two‐level structure of gates.

𝐹1 = 𝑦’ + 𝑥𝑦 + 𝑥’𝑦𝑧’ 𝐹2 = 𝑥(𝑦’ + 𝑧)(𝑥’ + 𝑦 + 𝑧)


Standard vs. Non-Standard Form

a two‐level implementation is

preferred :

it produces the least

amount of delay through

the gates

However, the number of

inputs to a given gate

might not be practical.


Other Logic Operations

There are 22n functions for n binary variables.(16 for n=2)

All these functions can be expressed using AND, OR

and NOT operators.

We can assign special operator to some of these

functions.


More Logic Functions

Inhibition and Implication are used by logicians, but are

rarely used in computer logic.

NOR is the complement of the OR (not-OR)

NAND is the complement of AND (not‐AND)

The exclusive‐OR,(XOR) is similar to OR, but excludes the combination of both x and y equal to 1

Equivalence is a function that is 1 when the two binary

variables are equal

The exclusive‐OR and equivalence functions are the complements. Hence, the equivalence function is called

exclusive‐NOR, abbreviated XNOR


Digital Logic Gates


Digital Logic Gates

Buffer circuit is used for power amplification of the signal


Digital Logic Gates


Digital Logic Gates

NAND and NOR gates are used extensively as standard

logic gates and are more popular than the AND and OR

gates because

NAND and NOR gates are easily constructed with

transistor circuits

Digital circuits can be easily implemented with them


Extension to Multiple Inputs

A gate can be extended to have multiple inputs if the

binary operation it represents is commutative and

associative.

NAND and NOR gates are commutative but not

associative

we define the multiple NOR (or NAND) gate as a

complemented OR (or AND) gate

zyxzyx


Demonstrating the Non-Associativity of the NOR Operator



NAND

NOR

xyzzyx

zyxzyx



XOR and XNORE are commutative and associative

can be extended to more than two inputs

Definition of the function is modified for more than two

variables:

XOR is an odd function


Positive and Negative Logic

Binary signals have one of two values:

One represents Logic 0

One represents Logic 1

Choosing the high‐level H to represent logic 1 defines a positive logic system. Choosing the low‐level L to represent logic 1 defines a negative logic system.

Documents

Boolean Algebra and Logic Gates - KFUPM...maxterm with subscript j is a complement of the minterm with the same subscript j and vice versa. EE 200 – Digital Logic Circuit Design