Chapter 2

Boolean Algebra and Logic Gates

2-2Digital Circuits

Boolean Algebra (E.V. Huntington, 1904)

A set of elements B and two binary operators + and .

1. Closure w.r.t. the operator + (.) x, y B x+y B

2. An identity element w.r.t. + (.) 0+x = x+0 = x 1.x = x.1= x

3. Commutative w.r.t. + (.) x+y = y+x x.y = y.x

2-3Digital Circuits

4. . is distributive over +: x.(y+z)=(x.y)+(x.z) + is distributive over.: x+(y.z)=(x+y).(x+z)

5. x B, x' B (complement of x) x+x'=1 and x.x'=0

6. at least two elements x, y B x ≠ y Note

the associative law can be derived no additive and multiplicative inverses complement

2-4Digital Circuits

Two-valued Boolean Algebra

B = {0,1} The rules of operations

Closure The identity elements

– +: 0– .: 1

x y xy x y x+y x x‘0 0 0 0 0 0 0 10 1 0 0 1 1 1 01 0 0 1 0 11 1 1 1 1 1

2-5Digital Circuits

The commutative laws The distributive laws

2-6Digital Circuits

Complement x+x'=1: 0+0'=0+1=1; 1+1'=1+0=1 x.x'=0: 0.0'=0.1=0; 1.1'=1.0=0

Has two distinct elements 1 and 0, with 0 ≠ 1 Note

a set of two elements + : OR operation; . : AND operation a complement operator: NOT operation Binary logic is a two-valued Boolean algebra

2-7Digital Circuits

Basic Theorems and Properties

Duality the binary operators are interchanged; AND <-> OR the identity elements are interchanged; 1 <-> 0

P2 x+0 =x x 1 = xp5 x+x‘ = 1 x x‘ = 0T1 x + x = x x x = xT2 x + 1 = 1 x 0 = 0T3, involution (x‘)’ = xp3 x+y = y+x xy = yxT4 x+(y+z)=(x+y)+z x(yz) =(xy)zP4 x(y+z)=xy+xz x+yz=(x+y)(x+z)T5, DeMorgan (x+y)‘ =x’y‘ (xy)‘=x’+y‘T6, absorption x+xy = x x(x+y) =x

2-8Digital Circuits

• Theorem 1(a): x+x = x– x+x = (x+x) 1 by postulate:

2(b) = (x+x) (x+x')5(a) = x+xx'

4(b) = x+05(b) = x

2(a)

– Theorem 1(b): x x = x– xx = x x + 0

= xx + xx'= x (x + x')

= x 1= x

2-9Digital Circuits

Theorem 2 x + 1 = 1 (x + 1)

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

x 0 = 0 by duality Theorem 3: (x')' = x

Postulate 5 defines the complement of x, x + x' = 1 and x x' = 0

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

2-10Digital Circuits

Theorem 6 x + xy = x 1 + xy =

x (1 +y) = x 1 = x

x (x + y) = x by duality By means of truth table

x y xy x + xy0 0 0 00 1 0 01 0 0 1

1 1 1 1

2-11Digital Circuits

DeMorgan's Theorems (x+y)' = x' y' (x y)' = x' + y'

x y x+y (x+y)‘ x‘ y‘ x‘y’0 0 0 1 1 1 10 1 1 0 1 0 01 0 1 0 0 1 01 1 1 0 0 0 0

2-12Digital Circuits

• Operator Precedence– parentheses– NOT– AND– OR– examples

– x y' + z

– (x y + z)'

2-13Digital Circuits

Boolean Functions

A Boolean function binary variables binary operators OR and AND unary operator NOT parentheses

Examples F1= x y z'

F2 = x + y'z

F3 = x' y' z + x' y z + x y'

F4 = x y' + x' z

2-14Digital Circuits

The truth table of 2n entries

Two Boolean expressions may specify the same function

F3 = F4

x y z F1 F2 F3 F40 0 0 0 0 0 00 0 1 0 1 1 10 1 0 0 0 0 00 1 1 0 0 1 11 0 0 0 1 1 11 0 1 0 1 1 11 1 0 1 1 0 01 1 1 0 1 0 0

2-15Digital Circuits

• Implementation with logic gates

• F4 is more economical

x y z F1 F2 F3 F40 0 0 0 0 0 00 0 1 0 1 1 10 1 0 0 0 0 00 1 1 0 0 1 11 0 0 0 1 1 11 0 1 0 1 1 11 1 0 1 1 0 01 1 1 0 1 0 0

2-16Digital Circuits

Algebraic Manipulation

To minimize Boolean expressions literal: a primed or unprimed variable (an input to a

gate) term: an implementation with a gate The minimization of the number of literals and the

number of terms => a circuit with less equipment It is a hard problem (no specific rules to follow) 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

2-17Digital Circuits

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

xy + x'z + yz = xy + x'z + yz(x+x') = xy + x'z + yzx + yzx' = xy(1+z) + x'z(1+y) = xy +x'z

(x+y)(x'+z)(y+z) = (x+y)(x'+z) by duality from the previous result

2-18Digital Circuits

Complement of a Function

an interchange of 0's for 1's and 1's for 0's in the value of F

by DeMorgan's theorem (A+B+C)' = (A+X)' let B+C = X =

A'X' by DeMorgan's = A'(B+C)' = A'(B'C') by DeMorgan's = A'B'C' associative

generalizations (A+B+C+ ... +F)' = A'B'C' ... F' (ABC ... F)' = A'+ B'+C'+ ... +F'

2-19Digital Circuits

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

[x(y'z'+yz)]' = x' + ( y'z'+yz)' = x' + (y'z')' (yz)' = x' + (y+z) (y'+z')

A simpler procedure take the dual of the function and complement each

literal x'yz' + x'y'z => (x'+y+z') (x'+y'+z) (the dual)

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

2-20Digital Circuits

Canonical and Standard Forms

Minterms and Maxterms A minterm: an AND term consists of all literals in

their normal form or in their complement form For example, two binary variables x and y,

xy, xy', x'y, x'y' It is also called a standard product n variables con be combined to form 2n minterms A maxterm: an OR term It is also call a standard sum 2n maxterms

2-21Digital Circuits

each maxterm is the complement of its corresponding minterm, and vice versa

x y z term Term0 0 0 x‘y’z‘ m0 x+y+z M0

0 0 1 x‘y’z m1 x+y+z‘ M1

0 1 0 x‘yz‘ m2 x+y‘+z M2

0 1 1 x‘yz m3 x+y‘+z‘ M3

1 0 0 xy’z‘ m4 x‘+y+z M4

1 0 1 xy’z m5 x‘+y+z‘ M5

1 1 0 xyz‘ m6 x‘+y‘+z M6

1 1 1 xyz m7 x‘+y‘+z‘ M7

2-22Digital Circuits

An Boolean function can be expressed by a truth table sum of minterms f1 = x'y'z + xy'z' + xyz = m1 + m4 +m7

f2 = x'yz+ xy'z + xyz'+xyz = m3 + m5 +m6 + m7x y z f1 f2

0 0 0 0 00 0 1 1 00 1 0 0 00 1 1 0 11 0 0 1 01 0 1 0 11 1 0 0 11 1 1 1 0

2-23Digital Circuits

The complement of a Boolean function the minterms that produce a 0 f1' = m0 + m2 +m3 + m5 + m6

= x'y'z'+x'yz'+x'yz+xy'z+xyz' f1 = (f1')'

= (x+y+z)(x+y'+z) (x+y'+z') (x'+y+z')(x'+y'+z) = M0 M2 M3 M5 M6

Any Boolean function can be expressed as a sum of minterms a product of maxterms canonical form

2-24Digital Circuits

Sum of minterms F = A+B'C

= A (B+B') + B'C= AB +AB' + B'C

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

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

F(A,B,C) = (1, 4, 5, 6, 7) or, built the truth table first

2-25Digital Circuits

Product of maxterms x + yz = (x + y)(x + z)

= (x+y+zz')(x+z+yy') =(x+y+z)(x+y+z’)(x+y'+z)

F = xy + x'z= (xy + x') (xy +z)

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

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

F = (x+y+z)(x+y'+z)(x'+y+z)(x'+y+z')= M0M2M4M5

F(x,y,z) = (0,2,4,5)

2-26Digital Circuits

Conversion between Canonical Forms F(A,B,C) = (1,4,5,6,7) F'(A,B,C) = (0,2,3) By DeMorgan's theorem

F(A,B,C) = (0,2,3) mj' = Mj

sum of minterms = product of maxterms interchange the symbols and and list those

numbers missing from the original form of 1's of 0's

2-27Digital Circuits

Standard Forms Canonical forms are seldom used sum of products F1

= y' + zy+ x'yz' product of sums F2

= x(y'+z)(x'+y+z'+w) F3 = A'B'CD+ABC'D'

2-28Digital Circuits

Two-level implementation

Multi-level implementation

2-29Digital Circuits

Digital Logic Gates

Boolean expression: AND, OR and NOT operations

Constructing gates of other logic operations the feasibility and economy the possibility of extending gate's inputs the basic properties of the binary operations the ability of the gate to implement Boolean

functions

2-30Digital Circuits

2-31Digital Circuits

Extension to multiple inputs A gate can be extended to multiple inputs

if its binary operation is commutative and associative AND and OR are commutative and associative

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

2-32Digital Circuits

Multiple NOR = a complement of OR gate Multiple NAND = a complement of AND

The cascaded NAND operations = sum of products The cascaded NOR operations = product of sums

2-33Digital Circuits

The XOR and XNOR gates are commutative and associative

Multiple-input XOR gates are uncommon? XOR is an odd function: it is equal to 1 if the inputs

variables have an odd number of 1's