ECE2030 Introduction to Computer Engineering Lecture 6: Canonical (Standard) Forms

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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
• 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
• 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)
• 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
• 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 SOP

0

1

Minterms listed

as 1’s

3

7

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 POS

Maxterms listed

as 0’s

2

4

5

6

Convert a Boolean to Canonical SOP

0

1

Minterms listed

as 1’s

3

7

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