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Boolean Algebra: Basic Definitions, Huntington Postulates, and Theorems, Assignments of Microelectronic Circuits

Eastern Washington University (EWU)Microelectronic Circuits

An introduction to boolean algebra, including its history, basic definitions, huntington postulates, and theorems. It covers the concepts of closure, commutative law, identity element, inverse, distributive law, and duality. The document also includes examples and basic theorems.

Typology: Assignments

Pre 2010

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2-1 Basic Definitions
Boolean Algebra
introduced by George Boole in 1854
a set of elements: E = {0, 1}
a set of operators: O = {+, , ‘}}
binary operator – works on two operands
a number of axioms or postulates (assumptions – do not need to be proved)
a number of theorems (proven from the postulates)
Common postulates used to formulate algebraic structures
1. Closure
A set S is closed with respect to a binary operator if, for every pair of
elements of S, the binary operator specifies a rule for obtaining a unique
element of S.
example:
The set of natural numbers N = {1, 2, 3, 4, …} is closed with respect
to the binary operator plus (+) by the rules of arithmetic addition, since a,
b N we obtain a unique c N by the operation a + b = c.
The set of natural numbers is not closed with respect to the binary operator
minus (-) by the rules of arithmetic subtraction because 2 – 3 = -1 and
2, 3 N, while (-1) N.
2. Associative law
A binary operator * on a set S is said to be associative whenever
(x * y) * z = x * (y * z)for all x, y, z S
3. Commutative law
A binary operator * on a set S is said to be commutative whenever
x * y = y * xfor all x, y S
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2-1 Basic Definitions

Boolean Algebra

 introduced by George Boole in 1854

 a set of elements: E = {0, 1}

 a set of operators: O = {+, , ‘}}

binary operator – works on two operands

 a number of axioms or postulates (assumptions – do not need to be proved)

 a number of theorems (proven from the postulates)

 Common postulates used to formulate algebraic structures

1. Closure

A set S is closed with respect to a binary operator if, for every pair of

elements of S , the binary operator specifies a rule for obtaining a unique

element of S.

example:

The set of natural numbers N = {1, 2, 3, 4, …} is closed with respect

to the binary operator plus (+) by the rules of arithmetic addition, since a ,

b  N we obtain a unique c  N by the operation a + b = c.

The set of natural numbers is not closed with respect to the binary operator

minus (-) by the rules of arithmetic subtraction because 2 – 3 = -1 and

2, 3  N , while (-1)  N.

2. Associative law

A binary operator * on a set S is said to be associative whenever

( x * y ) * z = x * ( y * z ) for all x , y , z  S

3. Commutative law

A binary operator * on a set S is said to be commutative whenever

x * y = y * x for all x , y  S

4. Identity element

A set S is said to have an identity element with respect to a binary

operation * on S if there exists an element e  S with the property

e * x = x * e = x for any x  S

example:

The element 0 is an identity element with respect to operation + on the set

of integers I = {…, -3, -2, -1, 0, 1, 2, 3, …} since

x + 0 = 0 + x for any x  I

5. Inverse

A set S having the identity element e with respect to a binary operator * is

said to have an inverse whenever, for every x  S , there exists an element

y  S such that

x * y = e

6. Distributive law

If * and  are two binary operators on a set S , * is said to be distributive

over  whenever

x * ( y  z ) = ( x * y )  ( x * z )

Ordinary algebra

 The binary operator + defines addition.

 The additive identity is 0.

 The additive inverse defines subtraction.

 The binary operator  defines multiplication.

 The multiplicative identity is 1.

 The multiplicative inverse of a = 1/ a defines division, i.e., a  1/ a = 1.

 The only distributive law applicable is that of  over +:

a  ( b + c ) = ( a  b ) + ( a  c )

Two-Valued Boolean Algebra

 defined on a set of two elements, B = {0, 1}

 rules for the two binary operators + and  as shown in the following tables

 rule for the unary operator ‘} (for verification of postulate 5)

1. Closure

Closure is obvious from the tables since the result of each operation is

either 1 or 0 and 0 , 1  B.

2. Identity elements (from the tables)

(a) 0 + 0 = 0 0 + 1 = 1 + 0 = 1

(b) 1  1 = 1 1  0 = 0  1 = 0

3. Commutative law

Commutivity is obvious from the symmetry of the binary operator tables.

 (AND)AND))

x y z

+ (AND)OR)

x y z

‘ (AND)NOT)

x z

4. Distributive law x  ( y + z ) = ( x  y ) + ( x  z )

x y z y + z x(AND)y + z) xy xz (AND)xy) + (AND)xz)

5. Complement

(a) x + x’ = 1, since 0 + 0’ = 0 + 1 = 1 and 1 + 1’ = 1 + 0 = 1.

(b) x  x’ = 0, since 0  0’ = 0  1 = 0 and 1  1’ = 1  0 = 0

which verifies postulate 5.

6. Postulate 6

Postulate 6 is satisfied because the two-valued Boolean algebra has two

distinct elements, 1 and 0 , with 1  0.

Theorem 1(a): x + x = x

x + x = ( x + x )  1 by Postulate 2(b)

= ( x + x )( x + x ’) by Postulate 5(a)

= x + xx ’ by Postulate 4(b)

= x + 0 by Postulate 5(b)

= x by Postulate 2(a)

Theorem 2(a): x + 1 = 1

x + 1 = 1  ( x + 1) by Postulate 2(b)

= ( x + x ’)( x + 1) by Postulate 5(a)

= x + x ’ 1 by Postulate 4(b)

= x + x ’ by Postulate 2(b)

= 1 by Postulate 5(a)

Theorem 2(b): x  0 = 0 by duality

Theorem 3: ( x ’)’ = x

From postulate 5, we have x + x ’ = 1 and x  x ’ = 0, which defines

the complement of x. The complement of x ’ is x and is also ( x ’)’.

Therefore, since the complement is unique, we have that ( x ’)’ = x.

Theorem 6(a): x + xy = x

x + xy = x  1 + xy by Postulate 2(b)

= x (1 + y ) by Postulate 4(a)

= x ( y + 1) by Postulate 3(a)

= x  1 by Postulate 2(a)

= 1 by Postulate 2(b)

Theorem 6(b): x ( x + y ) = x by duality

DeMorgan’s Theorem by Truth Table

x y x + y (AND)x + y)’ x’ y’ x’y’

Operator Precedence

(1) parentheses

(2) NOT

(3) AND

(4) OR

2-4 Boolean Functions

 A Boolean variable can take the value of 0 or 1.

 A Boolean function is an expression formed with binary variables, the two binary

operators OR and AND, and unary operator NOT, parentheses, and an equal sign.

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

example: F 1 = xyz’

The function F 1 is equal to 1 if x = 1 AND y = 1 AND z’ = 1; otherwise F 1 = 0.

example: F 2 = x + y + z’

The function F 2 is equal to 1 if x = 1 OR y = 1 OR z’ = 1; otherwise F 2 = 0.

Truth Table of F 1 and F 2

x y z z’ F 1 F 2

Algebraic Manipulation (From Digital Electronics, William H. Gothman, P.E.)

 A literal is a primed or unprimed variable.

 A Boolean function may be simplified to minimize the number of literals or the

number of terms. It is not always possible to minimize both simultaneously.

General Approach:

A. Multiply all variables necessary to remove parentheses.

B. Look for identical terms. All but one can be eliminated.

C. Look for a variable and its complement in the same term. It can be

eliminated.

example: AA’C = 0  C

D. Look for pairs of terms that are identical except for one variable. If one

variable is missing, the larger term can be dropped.

example: ABCD + ABD = ABD(C + 1)

= ABD  1

= ABD

If one variable is present but complemented in the second term, it can be

reduced.

example: ABCD + AB’CD = ACD(B + B’)

= ACD  1

= ACD

example: Simplify the Boolean function F 1 = A + A’B + AB to a minimum of

literals.

F 1 = A + A’B + AB

= A + B ( A + A’ )

= A + B (1)

= A + B

2-5 Canonical and Standard Forms

Minterms and Maxterms

 Consider two binary variables x and y combined with an AND operation.

x’y’, x’y, xy’, xy

Each of these four AND terms represents one of the distinct areas in the Venn

diagram and is called a minterm or standard product.

 Consider two binary variables x and y combined with an OR operation.

x’+ y’, x’+ y, x + y’, x + y

Each of these four OR terms represents one of the distinct areas in the Venn

diagram and is called a maxterm or standard sum.

 n Variables can be combined to form 2 n^ minterms or maxterms.

Minterms and Maxterms for Three Binary Variables

Minterms Maxterms

x y z Term D)esignation Term D)esignation

0 0 0 x’y’z’ m 0 x+y+z M 0

0 0 1 x’y’z m 1 x+y+z’ M 1

0 1 0 x’yz’ m 2 x+y’+z M 2

0 1 1 x’yz m 3 x+y’+z’ M 3

1 0 0 xy’z’ m 4 x’+y+z M 4

1 0 1 xy’z m 5 x’+y+z’ M 5

1 1 0 xyz’ m 6 x’+y’+z M 6

1 1 1 xyz m 7 x’+y’+z’ M 7

 A Boolean function may be represented 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.

Functions of Three Variables

x y z Function f 1 Function f 2

f 1 = x’y’z + xy’z’ + xyz = m 1 + m 4 + m 7 f 2 = x’yz + xy’z + xyz’ + xyz = m 3 + m 5 + m 6 + m 7

 The complement of a Boolean function may be read from the truth table by

forming a minterm for each combination that produces a 0 in the function and

then ORing those terms.

f 1 ’ = x’y’z’ + x’yz’ + x’yz + xy’z + xyz’

example: Express the Boolean function F ( A , B , C ) = AB + C as a sum of minterms.

step 1 – each term must contain all variables

AB = AB ( C + C’ ) = ABC + ABC’

C = C ( A + A’ ) = AC + A’C

= AC ( B + B’ ) + A’C ( B + B’ )

= ABC + AB’C + A’BC + A’B’C

step 2 – OR all new terms, eliminating duplicates

F ( A , B , C ) = A’B’C + A’BC + AB’C + ABC’ + ABC

= m 1 + m 3 + m 5 + m 6 + m 7 = (1, 3, 5, 6, 7)

Standard Forms

 The two canonical forms of Boolean algebra are basic forms that one obtains from

reading a function from the truth table. By definition, each minterm or maxterm

must contain all variables in either complemented or uncomplemented form.

 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 types of standard forms: the sum of products and the product of

sums.

 The sum of products is a Boolean function containing AND terms, called product

terms , of one or more literals each. The sum denotes the ORing of these terms.

example: f 1 = y’ + xy + x’yz’

 The product of sums is a Boolean expression containing OR terms, called sum

terms. Each term may have one or more literals. The product denotes the

ANDing of these terms.

example: f 2 = x ( y’ + z )( x’ + y + z’ + w )

 A Boolean function may also be expressed in nonstandard form.

example: f 3 = ( AB + CD )( A’B’ + C’D’ )

2-6 Other Logic Operations

 There are sixteen possible functions for two variables. We will only concern

ourselves with the most commonly encountered ones.

 Any function can be equal to a constant, but a binary function can be equal to

only 1 or 0. Functions of this type are called constant functions.

 The transfer function does not have any effect on the variable.

example: f 1 = y

 The NAND function is equivalent to an AND function followed by a NOT

function.

example: f 2 = x  y = ( xy )’

 The NOR function is equivalent to an OR function followed by a NOT function.

example: f 3 = x  y = ( x + y )’

 The XOR (exlusive-OR) function is similar to OR but excludes the combination

where both x and y are equal to 1.

example: f 4 = x  y

 The XNOR (exclusive-NOR) function is also called the equivalence function.

example: f 5 = ( x  y )’

2-8 Integrated Circuits

 Digital circuits are constructed with integrated circuits. An integrated circuit (IC)

is a small silicon semiconductor crystal, called a chip , containing the electronic

components for the digital gates.

 The chip is mounted in a plastic or ceramic package and electrical connections are

made via external pins.

Levels of Integration

 Small-scale integration (SSI) devices contain several independent gates in a

single package. The number of gates is usually fewer than 10.

 Medium-scale integration (MSI) devices have a complexity of approximately 10

to 100 gates in a single package.

 Large-scale integration (LSI) devices contain between 100 and a few thousand

gates in a single package.

 Very large-scale integration (VLSI) devices contain thousands of gates in a single

package.

Digital Logic Families

 TTL – tranistor-transistor logic

 ECL emitter-coupled logic

 MOS metal-oxide semiconductor

 CMOS complementary metal-oxide semiconductor

Positive and Negative Logic

 The terms positive and negative logic do not refer to the polarity of the electrical

signals, but rather the assignment of logic values to the relative amplitudes of the

two signal levels.

 Positive logic: 1 = true = on

 Negative logic: 0 = true = on