Shorthand, Conversion Between Canonical Forms, Standard Forms, Conversion of sum-of-productsfrom standard to canonical form, Conversion of product-of-sums from standard to canonical form (1.12)

Shorthand:  S and P

• f1(a,b,c) = Sm(1,2,4,6), where S indicates that this is a sum-of-products form, and m(1,2,4,6) indicates that the minterms to be included are m1, m2, m4, and m6.
• f1(a,b,c) = PM(0,3,5,7), where indicates that this is a product-of-sums form, and M(0,3,5,7) indicates that the maxterms to be included are M0, M3, M5, and M7.

Conversion Between Canonical Forms

• Replace S with P (or vice versa) and replace those js that appeared in the original form with those that do not.
• Example: f1(a,b,c) = a’b’c + a’bc’ + ab’c’ + abc’ = m1 + m2 + m4 + m6= S (1,2,4,6)=

        = P (0,3,5,7)= (a+b+c)•(a+b’+c’)•(a’+b+c’)•(a’+b’+c’)

Standard Forms

• Standard forms are like canonical forms except that not all variables need appear in the individual product (SOP) or sum (POS) terms.
• Example:f1(a,b,c) = a’b’c + bc’ + ac’ is a standard sum-of-products form
• f1(a,b,c) = (a+b+c)•(b’+c’)•(a’+c’) is a standard product-of-sums form.

Conversion of sum-of-productsfrom standard to canonical form

• Expand non-canonical terms by inserting  equivalent of 1 in each missing variable:

(x + x’) = 1

• Remove duplicate minterms
• f1(a,b,c) = a’b’c + bc’ + ac’= a’b’c + (a+a’)bc’ + a(b+b’)c’= a’b’c + abc’ + a’bc’ + abc’ + ab’c’= ab’c + abc’ + a’bc + ab’c’

Conversion of product-of-sums, from standard to canonical form

• Expand noncanonical terms by adding 0 in terms of missing variables: xx’ = 0
• Remove duplicate maxterms
• f1(a,b,c) = (a+b+c)•(b’+c’)•(a’+c’)= (a+b+c)•(aa’+b’+c’)•(a’+bb’+c’)

= (a+b+c)•(a+b’+c’)•(a’+b’+c’)•(a’+b+c’)•(a’+b’+c’)= (a+b+c)•(a+b’+c’)•(a’+b’+c’)•(a’+b+c’)