Chapter 2 Boolean Algebra and Logic Gates - PowerPoint PPT Presentation

1 / 34
About This Presentation
Title:

Chapter 2 Boolean Algebra and Logic Gates

Description:

Huntington postulates for Boolean algebra defined on a set B with binary operators & ., and ... Basic Theorems & Properties of Boolean Algebra (Cont. ... – PowerPoint PPT presentation

Number of Views:2296
Avg rating:3.0/5.0
Slides: 35
Provided by: Alo8
Category:

less

Transcript and Presenter's Notes

Title: Chapter 2 Boolean Algebra and Logic Gates


1
Chapter 2Boolean Algebra and Logic Gates
2
Basic Definitions
  • Binary operator
  • A binary operator defined on a set S of elements
    is a rule that assigns to each pair of elements
    from S a unique element from S.
  • The most common postulates (axioms) used to
    formulate an algebraic structure (e.g., field)
    are given below. It is assumed that there is an
    equivalence relation (), which satisfies the
    principle of substitution.

3
Rules
  • Closure
  • A set S is closed w.r.t. a binary
    operator ? if
  • ? x, y ? S, x ? y ? S.
  • 2. Associate law
  • A binary operator ? on a set S is associate
    if
  • ? x, y, z ? S, (x ? y) ? z x ? (y ?
    z )
  • 3. Commutative law
  • A binary operator ? on a set S is commutative
    if
  • ? x, y ? S, x ? y y ? x

4
Rules
  • Identity Element
  • A set S is said to have an identity element
    w.r.t. a binary operation ?. on S if
  • ? e ? S ? ? x ? S, e ? x x ? e x.
  • Inverse
  • A set S having the identity element e w.r.t.
    a binary operator ? is said to have an inverse if
  • ? x ? S, ? y ? S ? x ? y e.
  • Distributive Law
  • If ? and ? are 2 binary operators on a set
    S, then ? is said to be distributive over ? if
  • x ? (y ? z) (x ? y) ? (x ?
    z).

5
Axiomatic definition of Boolean Algebra
  • Boolean algebra
  • An algebraic system of logic introduced by
    George Boole in 1854.
  • Switching algebra
  • A 2-valued boolean algebra introduced by
    Shannon in 1938.
  • Huntington postulates for Boolean algebra defined
    on a set B with binary operators ., and
  • the equivalence relation (1904)

6
Huntington Postulates
  • 1. (a) Closure w.r.t. .
  • (b) Closure w.r.t. ?.
  • 2. (a) Identity element 0 w.r.t. . x 0 0 x
    x.
  • (b) Identity element 1 w.r.t. ?. x ? 1 1 ?
    x x.
  • 3. (a) Commutative w.r.t. . x y y x.
  • (b) Commutative w.r.t. ?. x ? y y ? x.
  • 4. (a) ? is distributive over . x ? (y z) (x
    ? y) (x ? z).
  • (b) is distributive over ?. x (y ? z)
    (x y) ? (x z).
  • 5. ? x ? B, ? x ? B (called the complement of x)
    .) (a) x x 1 and (b) x ? x 0.
  • 6. There exists at least 2 distinct elements in
    B.

7
Huntington postulates (Cont.)
  • The postulates are independent none can be
    proved from the others.
  • The associative law can be derived (for both
    operators) from the other postulates.
  • 4(b) is valid for Boolean algebra, but not for
    ordinary algebra.
  • No additive or multiplicative inverses no
    subtraction or division operations.
  • Complement is not available in ordinary algebra.
  • B is as yet undefined. It is to be defined as the
    set 0, 1 (.two-valued Boolean algebra).

8
Summary of Boolean Algebra
  • (1) Set B of at least 2 elements (? variables).
  • (2) Rules of operation for the 2 binary operators
    ( ?).
  • (3) Huntington postulates satisfied by the
    elements of B and the operators.
  • (Two-valued) Boolean algebra
  • (1) B 0, 1.
  • (2) The binary operations are defined as the
    logical AND (?) and OR (). In addition, there is
    a unary operation
  • NOT (complement).
  • (3) The Huntington postulates can easily be shown
    valid
  • (read the text).

9
Basic Theorems Properties of Boolean Algebra
  • Duality principle
  • Every algebraic expression deducible from the
  • postulates of Boolean algebra remains valid if
    (1) ? ?
  • and (2) 1? 0.
  • THEOREM 1 (a) x x x (b) x ? x x.
  • Thm 1(b) is the dual of Thm 1(a), and vice versa.
  • THEOREM 2 (a) x 1 1 (b) x ? 0 0.
  • THEOREM 3 (involution) (x) x.
  • THEOREM 4 (associative) (a) x (y z) (x
    y) z
  • (b) x (yz) (xy)z.

10
Basic Theorems Properties of Boolean Algebra
(Cont.)
  • THEOREM 5 (DeMorgan) (a) (x y) x ? y (b)
    (xy)
  • x y
  • THEOREM 6 (a) x xy x (b) x (x y) x.
  • The theorems usually are proved algebraically or
    by
  • truth table.

11
Boolean Function
  • A Boolean function is an expression formed with
    Boolean variables, the operators OR, AND,and NOT,
    parentheses, and an equal sign.
  • Any Boolean function can be represented in a
    truth table (see Tab. 2-2, p. 41 for examples).
  • The number of rows in the table is 2n, where n is
    the number of variables in the function.
  • There are infinitely many algebraic expressions
    that specify a given Boolean function. It is
    important to find the simplest one.
  • Any Boolean function can be transformed in a
    straightforward manner from an algebraic
    expression into a logic diagram composed only of
    AND, OR, and NOT gates (see Fig. 2-1, p.41 for
    examples).

12
Gate Implementation
13
Gate Implementation
14
Algebraic Manipulation
  • A literal is a complemented or uncomplemented
    variable. The minimization of the number of
    literals and the number of terms usually results
    in a simpler circuit (less expensive).
  • Number of literals can be minimized by algebraic
    manipulation. Unfortunately, there are no
    specific rules to follow that will guarantee the
    final answer.
  • CAD tools for logic minimization are commonly
    used today.
  • 1. x xy x y.
  • 2. x (x y) xy.
  • 3. xy yz xz xy xz.
  • 4. (x y)(y z)(x z) (x y)(x z).

15
Complement of a Function
  • DeMorgans theorems can be extended to 3 or more
    variables, and in general to any function.
  • (x1 x2 . . . xn) x1 x2xn.
  • (x1x2. . . xn) x1 x2 . . . xn.
  • To complement a function 1) take the dual of the
    function, and 2) complement each literal.
  • Example 2-2, Example 2-3.

16
Canonical Standard Forms
  • 2 variables ? 4 combinations (x1x2, x1x2, x1x2,
    and x1x2).
  • n variables ? 2n combinations, each called a
    minterm or a standard product (denoted mi, 0 ? i
    ? 2n - 1). Their complements are called the
    maxterms or standard sums (denoted Mi, 0 ? i ? 2n
    - 1).
  • Take a real close look at Tab. 2-3, p. 45.
  • Canonical forms 1) sum of minterms 2) product
    of maxterms

17
Example
18
Example (Cont.)
19
Canonical Standard Forms
  • Any function can be represented in either of
    these 2
  • ways.
  • n variables ? 2n distinct minterms (maxterms) ?
  • possible functions.
  • F A BC ? f S(1, 4, 5, 6, 7). How? (Ex.
    2-4)
  • (1) Expansion by A A (B B)(C C).
  • (2) By constructing the truth table.

20
Conversion Between Canonical Forms
21
Two-Level Implementation
22
Three- and Two-Level Implementation
23
Other logic operations
  • Take a look at Tab. 2-7 Tab. 2-8, and read the
    text on pp. 51-53.
  • Notice that apart from AND, OR, and NOT
    (complement), the following functions also
  • are important NAND, NOR, XOR (exclusive-OR),
    XNOR (exclusive-NOR, or equivalence), Transfer.

24
Digital Logic Gates
  • Take a close look at Fig. 2-5, and memorize the
    symbols.
  • The NAND (?) and NOR (?) operations are
    commutative, but not associative, while the
  • XOR XNOR functions are both commutative and
    associative (pp. 53-55).
  • The AND, OR, NAND, NOR, XOR, XNOR gates can
    take multiple inputs, and be cascaded (see, e.g.,
    Figs. 2-7 2-8, pp. 61-62). However, the
    multi-input XOR/XNOR functions should be
    redefined in terms of parity (equality on number
    of 1s).

25
Digital Logic Gate
26
Nonassociativity
27
Multiple-Input and Cascaded Gates
28
3-Input Exclusive-OR Gate
29
Positive and Negative Logic
30
Positive and Negative Logic
31
? ? ? ?
  • ??? (p ?? n ????)
  • ??? (Transistor)
  • Chip ???????
  • ???? (Integrated Circuit, IC)
  • Small Scale IC SSI
  • Medium Scale IC MSI
  • Large Scale IC LSI
  • Very Large Scale IC VLSI
  • Ultra Large Scale IC ULSI
  • Giga Scale IC GSI
  • Kilo K ? Mega M ?? Giga G ?? Tera
    T ?

32
??????? (Layout) ?
33
(No Transcript)
34
?????????
Write a Comment
User Comments (0)
About PowerShow.com