chapter 2 boolean algebra and logic...

Department of Computer ScienceNational Tsing Hua University

Chih-Tsun Huang (黃稚存)

Chapter 2Boolean Algebra and

Logic Gates

Outline

Axiomatic Definition of Boolean Algebra Basic Theorems and Properties of Boolean

Algebra Boolean Functions Canonical and Standard Forms Other Logic Operations Digital Logic Gates Integrated Circuits

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Basic Definitions

Set: a collection of objects with a common property.

Binary operator on a set S is a rule that assigns to a unique element, on each pair of element in S.

Axioms (postulates) of an algebra are the basic assumptions from which all theorems of the algebra can be proved.

It is assumed that there is an equivalent relation (=), which satisfies principle of substitution. Reflexive, symmetric, and transitive

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Most Common Axioms to Formulate an Algebra Structure (1/2)

Closure

Associative law

Commutative law

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Most Common Axioms to Formulate an Algebra Structure (2/2)

Identity element

Inverse

Distributive law

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Axiomatic Definition

Boolean algebra An algebraic system of logic introduced by

George Boole in 1854

Switching algebra A 2-valued Boolean algebra introduced by Claude

Shannon in 1938

Huntington postulates A formal definition of Boolean algebra in 1904 Defined on a set B with binary operators + and •

with the equivalence relation =

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Huntington Postulates (1/2)

Defined by a set B with binary operators + and • Closure with respect to + and • (P1)

An identity element with respect to + and • (P2)

Commutative with respective to + and • (P3)

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Huntington Postulates (2/2)

Distributive law (P4) ‘•’ is distributive over ‘+’

‘+’ is distributive over ‘•’

(called the complement of x) such that ________________________ (P5)

There are at least 2 distinct elements in B (P6)

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Notes The axioms are independent, none can be

proved from others. Associative law is not included, since it can be

derived (both + and •) from the given axioms. In ordinary algebra, + is not distributive over •. No additive or multiplicative inverses; no

subtraction or division operations. Complement is not available in ordinary algebra. We are going to define B as the set {0, 1} (two-

valued Boolean Algebra). In ordinary algebra, the set S can contain an infinite

set of elements.

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Two-Valued Boolean Algebra(Switching Algebra) Defined on a set of two elements, The binary operators are the logical AND (•) and OR (+) The unary operation NOT (complement) is also included

for basic operations

The Huntington Postulates are valid We will use the term Boolean algebra for the

2-valued Boolean algebra in the following

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}1,0{B

x y xy x y x+y x x 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 1 1 1 1

Huntington Postulates Test (1/3)

Closure Results of operators (from truth tables) are still in

{0, 1}

Identity elements

Commutative law Obvious from the symmetry of the truth tables

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Huntington Postulates Test (2/3)

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Distributive law Holds both for • over + and for + over •.

x y z y + z x·(y + z) x·y x·z (x·y) + (x·z)

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

Huntington Postulates Test (3/3)

Complement

The two-valued Boolean algebra has two distinct elements, 1 and 0, with 1 ≠ 0.

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Duality Principle

Every algebraic expression deducible from the postulates of Boolean algebra remains valid if the operators and identity elements are interchanged: Binary operators: AND OR

Identity elements: 1 0

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Postulates and Theorems of Boolean Algebra

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Basic Theorems (1/5)Theorem 1 (Idempotency)

(a) x + x = x

(b) x • x = x

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Basic Theorems (2/5)Theorem 2

(a) x + 1 = 1

(b) x • 0 = 0 by duality

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Basic Theorems (3/5)

Theorem 3 (Involution) (x’)’ = x

P5 defines the complement of x, and the complement of x’ is both x and (x’)’

Theorem 4 (Associativity) (a) x + (y + z) = (x + y) + z,

(b) x(yz) = (xy)z

Can be proved by truth table

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Basic Theorems (4/5)Theorem 5 (DeMorgan’s Theorem)

(x+y)’ = x’y’

(x y)’ = x’ + y’(Duality principle)

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x y x + y (x + y)’ x’ y’ x’y’

0 0

0 1

1 0

1 1

Basic Theorems (5/5)Theorem 6 (Absorption)

(a) x + xy = x, x + xy = x • 1 + xy

Or proved by the truth table

(b) x(x + y) = x by duality

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Operator Priority

Operator precedence Parentheses

NOT

AND

OR

Examples xy’ + z

(xy + z)’

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Boolean Functions

A Boolean function is an algebraic expression formed with Binary variables Logic operators AND, OR Unary NOT Parentheses An equal sign Constant 0 or 1

Examples F2 = xy’+x’z F2 = x’y’z + x’yz + xy’

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Boolean Functions

Can be represented by a truth table, with 2n

rows in the table (n: # of variables in the function)

There are infinite number of algebraic expressions to specify a given Boolean function Simplest one

Transform the Boolean function An algebraic expression a logic diagram

F1 = x + y’z

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Gate Implementation

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Boolean Functions

A literal is a variable or its complement in a Boolean expression F2 = x’y’z + x’yz + xy’ _____ literals,

1 OR term (sum term) and 3 AND terms (product terms)

Literal: an input to a gate

Term: implementation with a gate

Minimize the number of literals and terms A simpler (less expensive) circuit

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Complement of A Function F is F’ DeMorgan’s theorem (A + B + C + D + … + F)’

(ABCD…F)’

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Algebraic Manipulation1) x (x’ + y) =

2) x + x’y =

3) (x + y)(x + y’) =

4) xy + x’z + yz =

5) (x + y)(x’ + z)(y + z) = (x + y)(x’ + z) by duality from 4

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Canonical and Standard FormsMinterms and Maxterms Minterm (mi) An AND term consists of all literals in their normal form or

in their complement form (standard product) E.g., two binary variable x and y,

xy, xy’, x’y, x’y’ n variables can be combined to form 2n minterms

Maxterm (Mi) An OR term consists of all literals in their normal form or in

their complement form (standard sum) E.g., x + y, x + y’, x’ + y, x’ + y’ n variables: 2n maxterms

Each maxterm is the complement of its corresponding minterm and vice versa. mi’ = Mi

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Minterms and Maxterms Canonical forms

Sum-of-minterms (som)

Product-of-maxterms (pom)

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x y z Minterms Notation Maxterms Notation

0 0 0 0 x’y’z’ m0 x + y + z M0

1 0 0 1 x’y’z m1 x + y + z’ M1

2 0 1 0 x’yz’ m2 x + y’ + z M2

3 0 1 1 x’yz m3 x + y’ + z’ M3

4 1 0 0 xy’z’ m4 x’ + y + z M4

5 1 0 1 xy’z m5 x’ + y + z’ M5

6 1 1 0 xyz’ m6 x’ + y’ + z M6

7 1 1 1 xyz m7 x’ + y’ + z’ M7

Examples (1/4)A Boolean function can be expressed by

A truth table Sum-of-minterms

f1 =

Product-of-maxterms

f1 =

f1’ =

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x y z f1 f1’

0 0 0 0 0

1 0 0 1 1

2 0 1 0 0

3 0 1 1 0

4 1 0 0 1

5 1 0 1 0

6 1 1 0 0

7 1 1 1 1

Examples (2/4)A Boolean function can be expressed by

A truth table Sum-of-minterms

f2 = x’yz + xy’z + xyz

Product-of-maxtermsf2 = (x+y+z)(x+y+z’)(x+y’+z)(x’+y+z)

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x y z f2 f2’

0 0 0 0 0

1 0 0 1 0

2 0 1 0 0

3 0 1 1 1

4 1 0 0 0

5 1 0 1 1

6 1 1 0 1

7 1 1 1 1

Examples (3/4)

F = A + B’C Using truth table first

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A B C F

0 0 0 0

1 0 0 1

2 0 1 0

3 0 1 1

4 1 0 0

5 1 0 1

6 1 1 0

7 1 1 1

Examples (4/4)

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

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Canonical Forms

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Standard Forms (1/2)

Canonical forms are seldom used

Standard forms Sum-of-products (sop) Product terms (implicants) are the AND terms, which

can have fewer literals than the minterms

Product-of-sums (pos) Sum terms are the OR terms, which can have fewer

literals than the maxterms

Standard forms are not unique!

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Standard Forms (2/2)

Standard form examples f1 = xy + xy’z + x’yz (sop form)

f1’ = (x’ + y’)(x’ + y + z’)(x + y’ + z’) (pos form)

Nonstandard forms can have even fewer literals than standard forms xy + xy’z + xy’w = x(y + y’z + y’w) = x(y + y’(z + w))

xy + yz + zx = xy + (x + y)z = x(y + z) + yz = xz + y(x +z)

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Logic Diagrams Two-level implementation

Multi-level implementation

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Other Logic Operations

2n rows I the truth table of n binary variables

22nfunctions for n binary variables

16 functions of two variables

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16 Boolean Functions of Two Variables

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Digital Logic Gates (1/3)

Consider 16 functions Two functions are constants: 0 and 1

Four are unary (one-variable) functions Complement (inverter), transfer (buffer)

Inhibition and implication are neither commutative nor associative (seldom used in computer logic)

8 other binary functions AND, OR, NAND, NOR, XOR, Equivalence (XNOR),

inverter and buffer

Standard gates in digital logic systems

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Digital Logic Gates (2/3)

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Digital Logic Gates (3/3)

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

Commutative and associative

OR: (x + y) + z = x + (y + z) = x + y + z

AND: (xy)z = x(yz) = xyz

NAND and NOR are commutative but not associative => not extendable

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Multi-Input NOR and NAND Gates Multi-input NOR gate = complement of multi-input OR gate

Multi-input NAND gate = complement of multi-input AND gate

The cascaded NAND operations = sum of products

The cascaded NOR operations = product of sums

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Multi-Input XOR and XNOR Gates The XOR and XNOR gates are commutative and associative

Two-input XOR gates are more common

XOR is an odd function: it is equal to 1 if the inputs variables have an odd number of 1's

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Positive and Negative Logic

Two signal values two logic values

Positive logic: H = 1; L = 0

Negative logic: H = 0; L = 1

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Integrated Circuits (IC) A device with multiple electrical components and

their interconnects manufactured on a single silicon substrate

Levels of integration Small-scale integration (SSI): <=10 gates Medium-scale integration (MSI): 10-100 gates Large-scale integration (LSI): hundreds-thousands gates Very large-scale integration (VLSI): > tens of thousands

gates Digital logic families: TTL: transistor-transistor logic ECL: emitter-coupled logic MOS: metal-oxide semiconductor CMOS: complementary MOS BiCMOS

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Computer-Aided Design

Millions to billions of transistors Computer-based representation and aid Automate the design process Design entry Schematic capture Hardware description language (HDL)

Verilog, VHDL Synthesis Simulation, verification and testing Physical realization

ASIC: Application Specific IC FPGA: Field-Programmable Gate Array

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