1. ECE2030 Introduction to Computer Engineering Lecture 6: Canonical (Standard) Forms
Prof. Hsien-Hsin Sean Lee
School of Electrical and Computer Engineering
Georgia Tech

2. 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)

3. 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.

4. Minterms in B3

5. Canonical (Standard) SOP Function

6. 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.

7. Maxterms in B3

8. Canonical (Standard) POS Function

9. 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)

10. Convert a Boolean to Canonical SOPF
ABC 1 1
Minterms listed
as 1’s 3 7

11. Convert a Boolean to Canonical SOP

12. 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

13. Convert a Boolean to Canonical POS

14. Convert a Boolean to Canonical POSF
ABC 1
Maxterms listed
as 0’s 2 4 5 6

15. Convert a Boolean to Canonical SOPF
ABC 1 1
Minterms listed
as 1’s 3 7

16. Convert a Boolean to Canonical POS

17. Convert a Boolean to Canonical SOP

18. 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