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Lecture No. 9. Computer Logic Design Boolean Algebra and Logic Simplification. Veracity of DeMorgan's Theorems. First Theorem Second Theorem Alternative Method – use Truth Tables. Application of DeMorgan's Theorems. Apply to any number of variables Apply to combination of variables.

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lecture no 9
Lecture No. 9

Computer Logic Design

Boolean Algebra and Logic Simplification

veracity of demorgan s theorems
Veracity of DeMorgan's Theorems
  • First Theorem
  • Second Theorem
  • Alternative Method – use Truth Tables
application of demorgan s theorems
Application of DeMorgan's Theorems
  • Apply to any number of variables
  • Apply to combination of variables
application of demorgan s theorems1
Application of DeMorgan's Theorems
  • Finding Complement of a Function
  • Example:
application of demorgan s theorems2
Application of DeMorgan's Theorems

Shortcut for finding Complement of a Function

  • Take dual of the function
    • swap 1’s and 0’s
    • Swap AND and OR gates
    • Helpful to add parenthesis
  • Complement each literal
types of boolean expressions
Types of Boolean Expressions
  • Define Domain of an expression
    • set of all variables (complemented or otherwise)
  • Boolean expressions may be expressed as:
    • Sum-of-Products (SOP) Form
    • Product-of-Sums (POS) Form
    • Each form may contain single variable terms
    • May contain complemented and un-complemented terms
    • A SOP and POS expression can’t have a term of more than one variable having an over bar extending over the entire term
sum of products sop form
Sum-of-Products (SOP) Form
  • Two or more product terms summed by Boolean addition
  • Any expression -> SOP using Boolean algebra
  • Examples:

* A + BC

sum of products sop form1
Sum-of-Products (SOP) Form
  • Conversion to SOP Form:
standard sop form minterms
Standard SOP Form & Minterms
  • SOP expressions containing all Variables in the Domain in each term are in Standard Form.
  • Standard product terms are also called Minterms.
  • Any non-standard SOP expression may be converted to Standard form by applying Boolean Algebra Rule 6 to it.
  • Example:
standard sop form
Standard SOP Form
  • Example: Determine Standard SOP expression

SHORTCUT: Introduce all possible combinations of the missing variables AND’ed with the original term

characteristics of a minterm
Characteristics of a Minterm
  • Minterm is a standard product term in which all variables appear exactly once (complemented or uncomplemented)
  • Represents exactly one combination of the binary variables in a truth table for which the function produces a “1” output. That is the binary representation or value.
  • Has value of 1 for that combination and 0 for all others
  • For n variables, there are 2n distinct minterms
  • Example:
product of sums pos form
Product-of-Sums (POS) Form
  • Two or more sum terms multiplied by Boolean multiplication
  • Any expression -> POS using Boolean algebra
  • Examples:

(A+B)(B+C)(A+B+C)

product of sums pos form1
Product-of-Sums (POS) Form
  • Conversion to POS Form:
standard pos form maxterms
Standard POS Form & Maxterms
  • POS expressions containing all Variables in the Domain in each term are in Standard Form.
  • Standard sum terms are also called Maxterms. A Maxterm is a NOT Minterm.
  • Any non-standard POS expression may be converted to Standard form by applying Boolean Algebra Rule 8 and Rule 12A+BC=(A+B)(A+C) to it.
standard pos form
Standard POS Form
  • Example:

{Rule 8}

{Rule 12}

SHORTCUT: Introduce all possible combinations of the missing variables OR’ed with the original term

characteristics of a maxterm
Characteristics of a Maxterm
  • Maxterm is a standard sum term in which all variables appear exactly once (complemented or uncomplemented)
  • Represents exactly one combination of the binary variables in a truth table for which the function produces a “0” output. That is the binary representation or value.
  • Has value of 0 for that combination and 1 for all others
  • For n variables, there are 2n distinct maxterms
  • Example:
why standard sop and pos forms
Why Standard SOP and POS Forms?
  • Direct mapping of Standard Form expressions and Truth Table entries.
  • Alternate Mapping methods for simplification of expressions
  • Minimal Circuit implementation by switching between Standard SOP or POS
  • PLD based function implementation