# 2.2 Basic Definitions

## Full text

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數位系統 Digital Systems

### Department of Computer Science and Information Engineering, Chaoyang University of Technology

Speaker: Fuw-Yi Yang

### 伏

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Text Book: Digital Design 4th Ed.

Chap 2 Boolean Algebra and Logic Gates

### 2.9 Integrated Circuits

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Text Book: Digital Design 4th Ed.

Chap 2.1 Introduction

Because binary logic is used in all of today’s digital

computers and devices, the cost of the circuit 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.

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Text Book: Digital Design 4th Ed.

Chap 2.2 Basic Definitions

The most common postulates used to formulate various algebraic structures are as follows:

1.

## Closure. A set S is closed with respect to a binary

operator • if, for every a, b ∈ S, a • b ∈ S.

2.

## Associative law. (a

• b) • c =a • (b • c) for a, b, c ∈ S.

3.

## Commutative law. a

• b = b • a for a, b∈ S.

4.

## Identity element. If there exists an element e∈ S such

that e • b = b • e for b∈ S.

5.

## Inverse. For a∈ S, if there exists an element b∈ S such

that a • b = e, b is called the inverse of a.

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Chap 2.2 Basic Definitions

6.

## Distributive law. If • and * are two binary operators on a

set S, * is said to be distributive over • whenever

## a * (b

• c) = (a * b) • (a * c).

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Chap 2.3 Axiomatic Definition of Boolean Algebra

In 1854, George Boole developed an algebraic system now called Boolean Algebra.

Boolean algebra is an algebraic structure defined by a set

## +

and ∙, if the following postulates are satisfied:

1.

2.

## Identity element.

The element 0 is an identity element w.r.t. +;

i.e. x + 0 = 0 + x = x.

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Text Book: Digital Design 4th Ed.

Chap 2.3 Axiomatic Definition of Boolean Algebra The element 1 is an identity element w.r.t. ∙;

i.e. x ∙1 =1 ∙x = x.

3.

4.

5.

## Complement. for a∈ B, there exists an element a'∈ B

such that a + a' = 1, and a ∙a' = 0.

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Chap 2.3 Axiomatic Definition of Boolean Algebra –

## Example of an algebraic structure

A two-valued Boolean algebra is defined on a set of two

elements,

(OR) and

## *

(AND) as shown in the following tables:

0 0 0

0 1 0

1 0 0

1 1 1

0 0 0

0 1 1

1 0 1

1 1 1

## xx'

0 1

1 0

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Chap 2.3 Axiomatic Definition of Boolean Algebra

Show that the two-valued Boolean algebra defined above satisfies postulates 1~6.

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Chap 2.4 Basic Theorems and Properties of Boolean Algebra

Table 2.1 lists six theorems of Boolean algebra and four of its postulates.

Note that the property of Duality—

every algebraic expression deducible from the postulates of Boolean algebra remains valid if the operators and identity

## elements

are interchanged. (part a and part b in Table 2.1)

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Chap 2.4 Basic Theorems and Properties of Boolean Algebra

Table 2.1 lists six theorems of Boolean algebra and four of its postulates.

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Chap 2.4 Basic Theorems and Properties of Boolean Algebra

Theorem 1 (a) x + x = x

Statement Justification

## x + x = (x + x) * 1

= (x + x) * (x + x')

= x + (x * x')

= x + 0

= x

Postulate 2(b), identity Postulate 5(a), x + x' = 1 Postulate 4(b),

## x + (y * z) = (x + y) * (x + z)

Postulate 5(b), x * x' = 0

Postulate 2(a), x + 0 = x

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Chap 2.4 Basic Theorems and Properties of Boolean Algebra

Theorem 1 (b) x * x = x

Statement Justification

## x * x = (x * x) + 0

= (x * x) + (x * x')

= x * (x + x')

= x * 1

Postulate 2(a), identity Postulate 5(b), x * x' = 0 Postulate 4(a),

## x * (y + z) = (x * y) + (x * z)

Postulate 5(a), x + x' = 1

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Chap 2.4 Basic Theorems and Properties of Boolean Algebra

Theorem 2 (a) x + 1 = 1 (b) x * 0 = 0 by duality

Statement Justification

## x + 1 = (x + 1) * 1

= (x + 1) * (x + x')

= x + (1 * x')

= x + x'

= 1

Postulate 2(b), identity Postulate 5(a), x + x' = 1 Postulate 4(b),

## x + (y * z) = (x + y) * (x + z)

Postulate 2(b), 1 * x = x

Postulate 5(a), x + x' =1

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Chap 2.4 Basic Theorems and Properties of Boolean Algebra

Theorem 3 (x')' = x Postulate 5:

## x + x' = 1 and x * x' = 0 together define the complement of x.

The complement of x' is x and is also (x')'.

Since the complement is unique, we complete the proof.

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Chap 2.4 Basic Theorems and Properties of Boolean Algebra

Theorem 6 (a) x + x * y = x

## (b) x * (x + y) = x by duality Absorption

Statement Justification

## x + x * y = x * 1 + x * y

= x * (1 + y)

= x * 1

= x

Postulate 2(b), identity Postulate 4(a),

## x * (y + z) = (x * y) + (x * z)

Postulate 2(a), 1 + x = 1

Postulate 2(b), 1 * x = x

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Chap 2.4 Basic Theorems and Properties of Boolean Algebra

Theorem 4

Associative x + (y + z) = (x + y) + z

## x * (y * z) = (x * y) * z

Theorem 5

DeMorgan (x + y)' = x' * y' (x * y)' = x' + y‘

## Show its validity with truth table!!

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Chap 2.4 Basic Theorems and Properties of Boolean Algebra

The Operator Precedence for evaluating Boolean expressions is:

1. parentheses 2. NOT

3. AND 4. OR

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Chap 2.5 Boolean Functions

## Boolean algebra

is an algebra that deals with binary variables and logic operations.

A Boolean function described by an algebraic expression consists of binary variables, the constant 0 and 1, and the logic operation symbols.

For a given value of the binary variables, the function can be equal to either 0 or 1.

Example: F = x + yz, F is equal to 1 if x is equal to 1 or if

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Text Book: Digital Design 4th Ed.

Chap 2.5 Boolean Functions

A Boolean function can be represented in a truth table. The number of rows in the truth table is 2

### n

, where n is the

number of variables in the function.

Table 2.2 shows the truth table for the function F

### 1

= x + y'z.

Can we derive the Boolean function described by the column F

### 2

?

Can we draw the gate implementation of F

and F

### 2

?

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Chap 2.5 Boolean Functions

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Chap 2.5 Boolean Functions

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Chap 2.5 Boolean Functions

When a Boolean expression is implemented with logic gates, each

## term

requires a gate and each variable within the

term designates an input to the gate. We define a literal to be a single variable within a term, in complemented or un-complemented form.

## Example 2.1 Simplify the following Boolean functions to a

minimum number of literals.

1. x (x' + y) 2. x + x' y

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Text Book: Digital Design 4th Ed.

Chap 2.5 Boolean Functions

The complement of a function F is F' and is obtained from an interchanges of 0’s for 1’s and 1’s for 0’s in the value of

## F. It can be derived algebraically through DeMorgan’s

theorem. (A + B)' = A' B' ; (A B)' = A' + B'

## F1

= x' y z' + x' y' z; F

### 2

= x (y' z' + y z)

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Text Book: Digital Design 4th Ed.

Chap 2.5 Boolean Functions

## F1

= x' y z' + x' y' z and F

### 2

= x (y' z' + y z) by taking their duals and complementing each literal.

## F1

= x' y z' + x' y' z

The dual of F

### 1

is (x' + y + z')(x' + y' + z)

Complement each literal: F'

### 1

= (x + y' + z)(x + y + z')

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Text Book: Digital Design 4th Ed.

Chap 2.6 Canonical and Standard forms

A binary variable may appear either in its normal form (x) or in its complement form (x').

For two binary variables x and y combined with an AND Operation, we have four possible combinations:

## x y, x' y, x y', x' y'.

Each of these four AND terms is called a minterm, or a

## standard product.

In a similar manner, n variables can be combined to form 2

### n

minterms.

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Text Book: Digital Design 4th Ed.

Chap 2.6 Canonical and Standard forms

A binary variable may appear either in its normal form (x) or in its complement form (x').

For two binary variables x and y combined with an OR Operation, we have four possible combinations:

## x + y, x' + y, x + y', x' + y'.

Each of these four OR terms is called a maxterm, or a

## standard sum.

In a similar manner, n variables can be combined to form 2

### n

maxerms.

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Chap 2.6 Canonical and Standard forms

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Chap 2.6 Canonical and Standard forms

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Text Book: Digital Design 4th Ed.

Chap 2.6 Canonical and Standard forms

A Boolean function can be expressed algebraically from a given truth table by forming a minterm for each

combination of the variables that produces a 1 in the function and then taking the OR of all those terms.

See Table 2.3, 2.4

## f1

= x'y'z + xy'z' + xyz =m

+ m

+ m

## f2

= x'yz + xy'z + xy'z' + xyz =m

+ m

+ m

+ m

### 7

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Text Book: Digital Design 4th Ed.

Chap 2.6 Canonical and Standard forms

Now consider the complement of a Boolean function. It can be expressed algebraically from a given truth table by

forming a minterm for each combination of the variables that produces a 0 in the function and then taking the OR of all those terms.

## f1

= x'y'z + xy'z' + xyz =m

+ m

+ m

## f '1

= x'y'z' + x'yz' + x'yz + xy'z + xy'z' = m

+m

+m

+m

+m

### 6

If we take the complement of f '

### 1

we obtain the function f

:

## f 1

= (x+y+z)(x+y'+z)(x+y'+z')(x'+y+z')(x'+y+z)

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Text Book: Digital Design 4th Ed.

Chap 2.6 Canonical and Standard forms Sum of minterms

Boolean functions expressed as a sum of minterms or product of maxterms are said to be in canonical form.

sum of minterms.

## = m7

+ m

+ m

+ m

+ m

### 1

Or F(A, B, C) = Σ(1, 4, 5, 6, 7)

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Chap 2.6 Canonical and Standard forms Sum of minterms

## Example 2.4'

Deriving the minterms of a Boolean function directly from the given truth table.

See next page

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Chap 2.6 Canonical and Standard forms Sum of minterms

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Chap 2.6 Canonical and Standard forms Product of maxterms

## Example 2.5 Express the Boolean function F = xy + x'z as a

product of maxterms.

+ x'z

## = (y+x'+z') (y+x'+z)

(x+z+y') (x+z+y) (y+z+x')

## (y+z+x)

= (y+x'+z') (y+x'+z) (x+z+y')(x+z+y)

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Text Book: Digital Design 4th Ed.

Chap 2.6 Canonical and Standard forms

Conversion between Canonical Forms The complement of a function expressed as the sum of

minterms equals the sum of minterms

## missing from theoriginal function.

Example: F(A, B, C) = Σ(1, 4, 5, 6, 7)

## F'(A, B, C) = Σ(0, 2, 3) = m0

+ m

+ m

### 3

Now, take the complement of F' by DeMorgan’s theorem, we obtain F in a different form:

## F(A, B, C) = (m0

+ m

+ m

)' = m'

= M

### 0M2M3= Π(0, 2, 3)

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Text Book: Digital Design 4th Ed.

Chap 2.6 Canonical and Standard forms

Conversion between Canonical Forms A Boolean function can be expressed as product of

maxterms or sum of minterms directly from its truth table.

## F(x, y, z) = Σ(1, 3, 6, 7)F(x, y, z) = Π(0, 2, 4, 5)

### 1

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Text Book: Digital Design 4th Ed.

Chap 2.6 Canonical and Standard forms Standard Forms

The two canonical forms of Boolean algebra are basic

## forms that one obtains from reading a given function from

the truth table. These forms are very seldom the ones with

## the least number of literals, because each minterm or

maxterm must contain, by definition, all the variables, either complemented or un-complemented.

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Text Book: Digital Design 4th Ed.

Chap 2.6 Canonical and Standard forms Standard Forms

Another way to express Boolean functions is in standard

## form. In this configuration, the terms that form the function

may contain one, two, or any number of literals. There are

two type of standard forms: the sum of products (SOP) and

## products of sums

(POS).

Both POS and SOP are referred to as two-level

## implementation.

See next page

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Chap 2.6 Canonical and Standard forms Standard Forms—two level ckts

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Chap 2.6 Canonical and Standard forms Standard Forms—two level ckts

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Chap 2.6 Canonical and Standard forms Non-standard Forms

A Boolean functions may be expressed in a nonstandard form. The implementation may requires three levels of gating in this circuit.

See next page

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Chap 2.6 Canonical and Standard forms Non-standard Forms

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Chap 2.6 Canonical and Standard forms Non-standard Forms

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Chap 2.7 Other logic operations

When the binary operators AND and OR are placed

between two variables, x and y, they form two Boolean functions, xy and x+y, respectively. Previously, we stated that there are 2

see next pages

## xyf:16 combinations

0 0 0/1, two possible values 0 1 0/1, two possible values 1 0 0/1, two possible values

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Chap 2.7 Other logic operations

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Chap 2.7 Other logic operations

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Chap 2.8 Digital logic gates

Since Boolean functions are expressed in terms of AND, OR, and NOT operations, it is easier to implement a

Boolean function with these type of gates.

Still, the possibility of constructing gates for the other logic

## operations is of practical interest.

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Chap 2.8 Digital logic gates

Factors to be weighed in considering the construction of the other types of logic gates are:

1. the feasibility and economy of producing the gate 2. the possibility of extending the gate to more inputs 3. the basic properties of the binary operator, such as

commutativity and associativity

4. the ability of the gate to implement Boolean functions alone or in conjunction with other gates

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Chap 2.8 Digital logic gates

---positive and negative logic

The binary signal at the inputs and outputs of any gate has

## one of two values, except during transition.

One signal value represents logic 1 and the other logic 0.

Since two signal values are assigned to two logic values, there exist two different assignments of signal level to logic value.

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Chap 2.8 Digital logic gates

---positive and negative logic

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Chap 2.8 Digital logic gates

---positive and negative logic

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Chap 2.8 Digital logic gates

---positive and negative logic

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Chap 2.8 Digital logic gates

---positive and negative logic

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Chap 2.9 Integrated circuits

An integrated circuit (IC) is a silicon semiconductor crystal, called a chip, containing the electronic components for constructing digital gates.

Levels of Integration:

Small-scale integration (SSI)

Medium-scale integration (MSI) Large-scale integration (LSI)

Very large-scale integration (VLSI)

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Chap 2.9 Integrated circuits

Digital logic Families:

Transistor-transistor logic (TTL) Emitter-coupled logic (ECL)

Metal-oxide semiconductor (MOS)

Complementary Metal-oxide semiconductor (CMOS) Computer-Aided Design

Updating...

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