### Canonical Forms and Standard Forms

Standard Forms

• each term in the function can have any number of literals.

Example, F1 = a +b’c + cde

• There are 5 variables in F1 (a,b,c,d,e)
• Canonical form terms should be written as m0, m1… (sum of Products) or M0, M1, M2… (Product of sums).
• Each term in the canonical form should have all the literals.

Example: F1 = ab’c’d’e’ + abcd’e’ + abcde

Minterms

• It is Sum of Products
• Canonical form is Sum of Minterms
• three variable minterms are shown below
 x a b c minterms 0 0 0 0 m0=a’.b’.c’ 1 0 0 1 m1=a’.b’.c 2 0 1 0 m2=a’.b.c’ 3 0 1 1 m3=a’.b.c 4 1 0 0 m4=a.b’.c’ 5 1 0 1 m5=a.b’.c 6 1 1 0 m6=a.b.c’ 7 1 1 1 m7=a.b.c

Maxterms

• Product of Sum (PoS)
• Canonical form is Product of MaxTerms
• three variable maxterms are shown below
 x a b c minterms 0 0 0 0 M0=(a+b+c) 1 0 0 1 M1=(a+b+c’) 2 0 1 0 M2=(a+b’+c) 3 0 1 1 M3=(a+b’+c’) 4 1 0 0 M4=(a’+b+c) 5 1 0 1 M5=(a’+b+c’) 6 1 1 0 M6=(a’+b’+c) 7 1 1 1 M7=(a’+b’+c’)

usually

 Mi = (mj)’

Express the boolean function F = A + BC in a sum of minterms.

The function has three variables,

so F = A + BC will be

F = A(B + B’) + (A +A’) BC  [since, x + x’ =1]

F = AB + AB’ + ABC + A’BC

F = AB(C+C’) + AB’(C+C’) + ABC + A’BC

F = ABC + ABC’ + AB’C + AB’C’ + ABC + A’BC

F = ABC + ABC’ + AB’C + AB’C’ + A’BC [since x + x = x]

F= m0 + m6 + m5 + m4 + m3