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ECE2030 Introduction to Computer Engineering Lecture 6: Canonical (Standard) Forms. Prof. Hsien-Hsin Sean Lee School of Electrical and Computer Engineering Georgia Tech. Boolean Variables. A multi-dimensional space spanned by a set of n Boolean variables is denoted by B n

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ECE2030 Introduction to Computer EngineeringLecture 6: Canonical (Standard) Forms

Prof. Hsien-Hsin Sean Lee

School of Electrical and Computer Engineering

Georgia Tech

boolean variables
Boolean Variables
  • A multi-dimensional space spanned by a set of n Boolean variables is denoted by Bn
  • A literal is an instance (e.g. A) of a variable or its complement (Ā)
sop form
SOP Form
  • A product of literals is called a product term or a cube (e.g. Ā·B·C in B3, or B·C in B3)
  • Sum-Of-Product (SOP) Form: OR of product terms, e.g. ĀB+AC
  • A minterm is a product term in which every literal (or variable) appears in Bn
    • ĀBC is a minterm in B3 but not in B4. ABCD is a minterm in B4.
  • A canonical (or standard) SOP function:
    • a sum of minterms corresponding to the input combination of the truth table for which the function produces a “1” output.
pos form dual of sop form
POS form (dual of SOP form)
  • A sum of literals is called a sum term (e.g. Ā+B+C in B3, or (B+C) in B3)
  • Product-Of-Sum (POS) Form: AND of sum terms, e.g. (Ā+B)(A+C)
  • A maxterm is a sum term in which every literal (or variable) appears in Bn
    • (Ā+B+C) is a maxterm in B3 but not in B4. A+B+C+D is a maxterm in B4.
  • A canonical (or standard) POS function:
    • a product of maxterms corresponding to the input combination of the truth table for which the function produces a “0” output.
convert a boolean to canonical sop
Convert a Boolean to Canonical SOP
  • Expand the Boolean eqn into a SOP
  • Take each product term w/ a missing literal A, “AND” () it with (A+Ā)
convert a boolean to canonical sop1
Convert a Boolean to Canonical SOP

0

1

Minterms listed

as 1’s

3

7

convert a boolean to canonical pos
Convert a Boolean to Canonical POS
  • Expand Boolean eqn into a POS
    • Use distributive property
  • Take each sum term w/ a missing variable A and OR it with A·Ā
convert a boolean to canonical pos2
Convert a Boolean to Canonical POS

Maxterms listed

as 0’s

2

4

5

6

convert a boolean to canonical sop3
Convert a Boolean to Canonical SOP

0

1

Minterms listed

as 1’s

3

7

interchange canonical sop and pos
Interchange Canonical SOP and POS
  • For the same Boolean eqn
    • Canonical SOP form is complementary to its canonical POS form
    • Use missing terms to interchange  and 
  • Examples
    • F(A,B,C) =  m(0,1,4,6,7)

Can be re-expressed by

    • F(A,B,C) = M(2,3,5)

Where 2, 3, 5 are the missing minterms in the canonical SOP form