Minterms and Maxterms
Minterm and Maxterm
Expansion
• If all variables appear as Sum of Products Form is
called minterm
Z. Aliyazicioglu
ECE
X
Y
Z
Product
Term
Symbol
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
X’Y’Z’
X’Y’Z
X’YZ’
X’YZ
XY’Z’
XY’Z
XYZ’
XYZ
m0
m1
m2
m3
m4
m5
m6
m7
Example:
• F=X’Y’Z’+X’YZ’+XY’Z+XYZ
• If all variables appear as Product of sums Form is
called maxterm
X
Y
Z
Sum
Term
Symbol
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
X+Y+Z
X+Y+Z’
X+Y’+Z
X+Y’+Z’
X’+Y+Z
X’+Y+Z’
X’+Y’+Z
X’+Y’+Z’
M0
M1
M2
M3
M4
M5
M6
M7
Ex: E=Y’+X’Z’
Exp is not sum of minterm form
E(X,Y,Z)= ∑m(0,1,2,4,5)
X
Y
or
0
0
E’(X,Y,Z)= ∑m(3,6,7)
0
0
0
0
1
1
1
1
1
1
0
0
1
1
= m0+m2+m5+m7
Or
F(X,Y,Z)=∑m(0,2,5,7)
X
Y
Z
F
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
1
0
1
0
0
1
0
1
• F=X’Y’Z+X’YZ’+XY’Z+XYZ’
Z
E
0
1
0
1
0
1
0
1
1
1
1
0
1
1
0
0
X
0
0
0
0
1
1
1
1
Y
0
0
1
1
0
0
1
1
Z
0
1
0
1
0
1
0
1
F
0
1
1
0
0
1
1
0
1
F=X’Y’Z+X’YZ’+XY’Z+XYZ’
F=X’YZ’+X’YZ+XYZ
X
Y
Z
F
Maxterm
F(A,B,C) = ∏M(0,1,2)
= (A + B + C) (A + B + C') (A + B' + C)
F’(A,B,C) = ∏ M(3,4,5,6,7)
= (A + B' + C') (A' + B + C) (A' + B + C')
(A' + B' + C) (A' + B' + C'
X
Y
Z
F
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
0
1
Example:
X
Y
Z
F
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
0
1
1
1
1
1
F=X’YZ+XY’Z’+ZY’Z+XYZ’+XYZ
F=(X+Y+Z)(X+Y+Z’)(X+Y’+Z)
Sum of Products, Products of Sums, and
DeMorgan's Law
F' = A' B' C' + A' B' C + A' B C'
Apply DeMorgan's Law to obtain F:
(F')' = (A' B' C' + A' B' C + A' B C')'
F = (A + B + C) (A + B + C') (A + B' + C)
F' = (A + B' + C') (A' + B + C) (A' + B + C') (A' + B' + C)
(A' + B' + C')
Apply DeMorgan's Law to obtain F:
(F')' = {(A + B' + C') (A' + B + C) (A' + B + C')
(A' + B' + C) (A' + B' + C')}'
F = A' B C + A B' C' + A B' C + A B C' + A B C
Mapping Between Forms
Minterm to Maxterm conversion:
Rewrite minterm shorthand using maxterm shorthand:
replace minterm indices with the indices not already
used
E.g., F(A,B,C) = Σm(3,4,5,6,7) = Π M(0,1,2)
Maxterm to Minterm conversion:
Rewrite maxterm shorthand using minterm shorthand:
replace maxterm indices with the indices not already used
E.g., F(A,B,C) = Π M(0,1,2) = Σm(3,4,5,6,7)
2
Example:
Minterm expansion of F to Minterm expansion of F':
in minterm shorthand form, list the indices not already
used in F
E.g.,
F(A,B,C) = Σm(3,4,5,6,7)
= Π M(0,1,2)
F'(A,B,C) = Σm(0,1,2)
= Π M(3,4,5,6,7)
Minterm expansion of F to Maxterm expansion of F' :
rewrite in Maxterm form, using the same indices as F
E.g.,
F(A,B,C) = Σm(3,4,5,6,7) F'(A,B,C) = Π M(3,4,5,6,7)
= Π M(0,1,2)
= Σm(0,1,2)
• Design a simple binary adder (adds two 1-bit binary
number)
A
0
0
1
1
B
0
1
0
1
X=AB
Y=A’B+AB’
sum
00 (0+0=0)
01 (0+1=1)
01 (0+1=1)
10 (0+1=1)
A
B
0
0
1
1
0
1
0
1
X Y
0
0
0
1
0
1
1
0
U2A
A
B
2
1
Y
3
X
3
74128
U1A
1
2
7408
Example:
• Add two 2-bit binary numbers
A
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
B
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
0
C
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
D
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
X
0
0
0
0
0
0
0
1
0
0
1
1
0
1
1
1
Y
0
0
1
1
0
1
1
0
1
1
0
0
1
0
0
1
Z
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
X(A,B,C,D)=Σ m(7,10,11,13,14,15)
Y(A,B,C,D)=Σ m(2,3,5,6,8,9,12,15)
Z(A,B,C,D)=Σ m(1,3,4,6,9,11,12,14)
3