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. 2 Boolean Variables B nA multi-dimensional space spanned by a set of n Boolean variables is denoted by B n literalA literal is an instance (e.g. A) of a variable or its complement (Ā)

3. 3 SOP Form product term cube B 3 B 3A product of literals is called a product term or a cube (e.g. Ā · B · C in B 3, or B · C in B 3 ) Sum-Of-Product (SOP)ORSum-Of-Product (SOP) Form: OR of product terms, e.g. ĀB+AC minterm B nA minterm is a product term in which every literal (or variable) appears in B n –ĀBC is a minterm in B 3 but not in B 4. ABCD is a minterm in B 4. canonicalstandardSOP function:A canonical (or standard) SOP function: 1 ” –a sum of minterms corresponding to the input combination of the truth table for which the function produces a “ 1 ” output.

4. 4 B 3 Minterms in B 3

5. 5 Canonical (Standard) SOP Function

6. 6 POS form (dual of SOP form) sum term B 3 B 3A sum of literals is called a sum term (e.g. Ā+B+C in B 3, or (B+C) in B 3 ) Product-Of-Sum (POS)ANDProduct-Of-Sum (POS) Form: AND of sum terms, e.g. (Ā+B)(A+C) maxterm B nA maxterm is a sum term in which every literal (or variable) appears in B n –(Ā+B+C) is a maxterm in B 3 but not in B 4. A+B+C+D is a maxterm in B 4. canonicalstandardPOS function: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. 7 B 3 Maxterms in B 3

8. 8 Canonical (Standard) POS Function

9. 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. 10 Convert a Boolean to Canonical SOP ABCF ABC0001 0011 0100 0111 1000 1010 1100 1111 0 1 3 7 Minterms listed as 1’s

11. 11 Convert a Boolean to Canonical SOP

12. 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. 13 Convert a Boolean to Canonical POS

14. 14 Convert a Boolean to Canonical POS ABCF ABC0001 0011 0100 0111 1000 1010 1100 1111 4 6 2 5 Maxterms listed as 0’s

15. 15 Convert a Boolean to Canonical SOP ABCF ABC0001 0011 0100 0111 1000 1010 1100 1111 0 1 3 7 Minterms listed as 1’s

16. 16 Convert a Boolean to Canonical POS

17. 17 Convert a Boolean to Canonical SOP

18. 18 Interchange Canonical SOP and POS For the same Boolean eqn complementary –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