Chapter 2 Boolean Algebra and Logic Gates

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Chapter 2Boolean Algebra and Logic Gates
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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.

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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

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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).

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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)

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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.

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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).

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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).

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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.

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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.

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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).

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Gate Implementation
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Gate Implementation
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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).

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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.

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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

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Example
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Example (Cont.)
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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.

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Conversion Between Canonical Forms
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Two-Level Implementation
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Three- and Two-Level Implementation
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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.

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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).

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Digital Logic Gate
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Nonassociativity
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Multiple-Input and Cascaded Gates
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3-Input Exclusive-OR Gate
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Positive and Negative Logic
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Positive and Negative Logic
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