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

More about Canonical Forms

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