Comprehensive K-Map Minimization Practice Set
Instructions
For each Boolean function provided, determine the Minimal Expression.
SOP (Sum-of-Products): Find the Minimal Sum-of-Products (MSP) by grouping the s
(
∑
).
POS (Product-of-Sums): Find the Minimal Product-of-Sums (MPS) by grouping the s
(
Π
).
Dont Care (
d
): Utilize the Dont Care terms strategically to form the largest possible
groups (Octets, Quads, Pairs).
Part I: 2-Variable K-map Minimization (Variables:
A , B
)
Minimize (SOP):
F1(A , B)=∑(0,3)
Minimize (SOP):
F2(A , B)=∑(1,2,3)
Minimize (POS):
F3(A , B)=Π(0,1)
Minimize (SOP):
F4(A , B)=∑(0)+d(2,3)
Minimize (POS):
F5(A , B)=Π(3)⋅d(0,2)
Part II: 3-Variable K-map Minimization (Variables:
A , B , C
)
Minimize (SOP):
F6(A , B , C )=∑(0,1,4,5)
Minimize (SOP):
F7(A , B , C)=∑(1,2,3,6,7)
Minimize (POS):
F8(A , B , C )=Π(0,2,4,6)
Minimize (POS):
F9(A , B , C)=Π(1,3,5)
Minimize (SOP):
F10(A , B ,C )=∑(0,1,2,6)
Minimize (SOP +
d
):
F11(A , B , C)=∑(2,3,4)+d(0,6)
Minimize (SOP +
d
):
F12(A , B ,C )=∑(1,5,7)+d(3)
Minimize (POS +
d
):
F13(A , B ,C )=Π(4,5)⋅d(0,1)
Minimize (SOP):
F14 (A , B , C)=∑(0,2,4,5,6)
Minimize (POS):
F15(A , B ,C )=Π(0,1,2,3,4,5)
Part III: 4-Variable K-map Minimization (Variables:
A , B , C , D
)
Minimize (SOP):
F16(A , B , C , D)=∑(0,1,3,5,7,9,11 ,13 ,15)
Minimize (SOP):
F17(A , B , C , D)=∑(0,2,4,6,8,10 ,12 ,14)
Minimize (SOP):
F18(A , B ,C , D)=∑(0,4,5,7,8,12 ,13 ,15)
Minimize (POS):
F19(A , B ,C , D)=Π(0,1,2,3,4,5,6,7,8,9,10 ,11)
Minimize (POS):
F20(A , B , C , D)=Π(5,7,13 ,15)
Minimize (SOP):
F21(A , B ,C , D)=∑(1,5,6,7,10 ,11 ,12 ,13 ,14 ,15 )
Minimize (SOP +
d
):
F22(A , B ,C , D)=∑(0,1,2,3)+d(5,7,9,11)
Minimize (SOP +
d
):
F23(A , B ,C , D)=∑(1,4,6,9,11 ,12,14)+d(0,5,8,10)
Minimize (POS +
d
):
F24 (A , B , C , D )=Π(0,2,4,6)⋅d(8,10 ,12 ,14)
Minimize (POS +
d
):
F25(A , B ,C , D)=Π(1,3,5,7,13 ,15 )⋅d(9,11)
Minimize (SOP):
F26(A , B , C , D)=∑(4,5,6,7,12 ,13 ,14 ,15)
Minimize (SOP):
F27(A , B , C , D)=∑(0,1,3,4,5,7,9,11 ,12 ,13 ,15 )
Minimize (POS):
F28(A , B , C , D)=Π(0,2,4,6,8,10 ,12 ,14 ,15)
Minimize (SOP +
d
):
F29(A , B ,C , D)=∑(0,1,2,3,5,7)+d(11 ,15)
Minimize (POS +
d
):
F30(A , B ,C , D)=Π(0,4,8,12)⋅d(1,5,9,13)
Solutions and Detailed Explanations
Part I: 2-Variable Solutions
F1(A , B)=∑(0,3)
Minimal SOP:
A B+AB
(Exclusive NOR )
Rationale:
m0
and
m3
are diagonal and cannot be grouped. They form two separate
prime implicants.
F2(A , B)=∑(1,2,3)
Minimal SOP:
A+B
Rationale: Group
m1,m3
(giving
B
). Group
m2,m3
(giving
A
). The two pairs cover all
minterms.
F3(A , B)=Π(0,1)
Minimal POS:
A
Rationale: Grouping the two s (
M0, M1
) eliminates
B
and results in the Maxterm
A
.
F4(A , B )=∑(0)+d(2,3)
Minimal SOP:
B
Rationale: Use
d2
as and
d3
as . Group
m0
with
d2
(which is
A B
and
A B
) to form the
largest group. The optimal grouping is
m0
and
d2
to form
B
(Quad covering
m0, m2
).
Correction:
m0, d2, d3
cannot form a quad. Use
d2
and
d3
as 1s. Group
m0
and
d2
as
B
.
The remaining
d3
is not covered. The simplest solution is
B+A
. The minimal solution
is
B
by using
d2
to form a pair. Optimal:
B
(if we use
d2
as a 1).
A B
(if
d2, d3
are
ignored). Lets use
d2
as a to form
B
.
Minimal SOP (Corrected):
B
F5(A , B)=Π(3)⋅d(0,2)
Minimal POS:
(B)
Rationale: We group the s (Maxterms). Use
d0
and
d2
as s. The s are
M0, M 2, M3
.
Grouping
M2, M 3
results in
(A+B)
. The minimal grouping is
M0, M2
to form
(B)
.
Part II: 3-Variable Solutions
F6(A , B , C)=∑(0,1,4,5)
Minimal SOP:
C
Rationale: These four minterms form two horizontal pairs that wrap vertically (0-4
and 1-5). The result is
C
.
F7(A , B , C )=∑(1,2,3,6,7)
Minimal SOP:
B+C A
Rationale: Quad
m2,m3, m6, m7
(wrapping) gives
B
. Remaining
m1
is covered by
m1,m3
pair (
A C
). The minimal solution is
B+A C
.
F8(A , B , C )=Π(0,2,4,6)
Minimal POS:
C
Rationale: Grouping the four s (
M0, M 2, M 4, M 6
) forms a Quad, eliminating
A
and
B
,
resulting in the Maxterm
C
.
F9(A , B , C)=Π(1,3,5)
Minimal POS:
(A+C)(B+C)
Rationale: No Quad is possible. Two pairs:
M1, M3
gives
(A+C)
.
M1, M5
gives
(B+C)
.
Both are essential.
F10(A , B ,C )=∑(0,1,2,6)
Minimal SOP:
A B+A C
Rationale: EPIs:
m0, m1
gives
A B
.
m2,m6
gives
A C
.
F11 (A , B , C)=∑(2,3,4)+d(0,6)
Minimal SOP:
B C+A C
Rationale: Use
d0
as a to group with
m4
(giving
A C
). Use
d6
as a to group with
m2,m3
(giving
B C
). The groups are
m2,m3, d6
(
B C
) and
m4, d0
(
A C
). Minimal SOP is
BC+A C
. The minimal solution is
A C+BC
.
Minimal SOP (Corrected):
B C+A C
F12(A , B ,C )=∑(1,5,7)+d(3)
Minimal SOP:
C+A B D
Rationale: Use
d3
as a . The four terms
m1,d3, m5, m7
form a Quad (all
C=1
), resulting
in
C
.
F13(A , B ,C )=Π(4,5)⋅d(0,1)
Minimal POS:
A+B+C
Rationale: We group the s (Maxterms). Use
d0
and
d1
as s. The s are
M0, M 1, M4, M 5
.
This forms a Quad, simplifying to
B+C
. Minimal POS:
(B+C)
.
F14 (A , B , C )=∑(0,2,4,5,6)
Minimal SOP:
B+A C
Rationale: Quad
m0, m2, m4, m6
(all
B
) gives
B
. Remaining
m5
is covered by the pair
m4, m5
(
A C
). The final term
B+A C
.
F15(A , B ,C )=Π(0,1,2,3,4,5)
Minimal POS:
A⋅B
Rationale: The six s cover the entire
A
plane and
B
plane. The minimal groups are
Octet
M0, M 1, M2, M 3
(giving
A
) and Octet
M0, M 2, M 4, M 6
(giving
B
). Minimal POS:
A⋅B
. (Correction:
M0, M1, M 2, M 3
gives
A
.
M0, M 1, M4, M5
gives
B
. The minimal
solution is
A⋅B
).
Part III: 4-Variable Solutions
F16(A , B , C , D)=∑(0,1,3,5,7,9,11 ,13 ,15)
Minimal SOP:
D+A BC D
Rationale: The eight odd minterms form an Octet, simplifying to
D
. Remaining
m0
is
isolated. Minimal SOP:
D+A BC D
.
F17(A , B , C , D)=∑(0,2,4,6,8,10 ,12 ,14)
Minimal SOP:
D
Rationale: The eight even minterms (where
D=0
) form a single Octet, simplifying to
D
.
F18(A , B ,C , D)=∑(0,4,5,7,8,12 ,13 ,15)
Minimal SOP:
B D+CD+B C
Rationale: Quad
m0, m4,m12 , m8
(corners) gives
C D
. Quad
m5,m7, m13 , m15
gives
BD
.
Minimal SOP:
C D+BD
.
F19(A , B ,C , D )=Π(0,1,2,3,4,5,6,7,8,9,10 ,11)
Minimal POS:
A+C
Rationale: The 12 s can be covered by two Octets:
M0,... , M7
(giving
A
) and
M0, M 1, M8, M 9
(giving
C
). The minimal solution is the product of two Octets:
(A)⋅(C)
.
F20(A , B , C , D)=Π(5,7,13 ,15)
Minimal POS:
(B+D)(C+D)
Rationale: Grouping the four s forms a Quad, eliminating
A
and
C
, resulting in the
Maxterm
(B+D)
. (Correction:
M5, M 7, M13 , M15
gives
B+D
).
F21(A , B ,C , D)=∑(1,5,6,7,10 ,11 ,12 ,13 ,14 ,15 )
Minimal SOP:
AC+AB+CD+A C D
Rationale: EPIs: Quad
m6, m7, m14 , m15
(
BC
). Quad
m12 , m13 , m14 ,m15
(
AB
). Pair
m10 , m11
(
A BC
). Remaining
m1,m5
(
A C D
). Minimal SOP:
AB+BC+A CD +B C D
.
F22(A , B ,C , D)=∑(0,1,2,3)+d(5,7,9,11)
Minimal SOP:
A B
Rationale: Use
d5, d7, d9, d11
as s. The eight s (
m0, m1, m2, m3, d5, d7, d9, d11
) form an
Octet by covering the first half and third row. Minimal SOP:
A
.
F23(A , B ,C , D)=∑(1,4,6,9,11 ,12,14 )+d(0,5,8,10 )
Minimal SOP:
D+BC
Rationale: Use
d0, d4, d8, d12
as s. This forms an Octet
D
. Use
d5, d10
to cover
m6, m14
(
B D
). Minimal SOP:
D+BC
.
F24 (A , B , C , D )=Π(0,2,4,6)⋅d(8,10 ,12 ,14 )
Minimal POS:
D
Rationale: We group the s. Use
d8, d10 , d12 , d14
as s. The eight s form an Octet,
simplifying to the Maxterm
(D)
.
F25(A , B ,C , D)=Π(1,3,5,7,13 ,15)⋅d(9,11)
Minimal POS:
A
Rationale: We group the s. The s are
∑(0,2,4,6,8,10 ,12 ,14)
. This is
D
. The s are
Π(1,3,5,7,13 ,15)⋅d(9,11)
. Use
d9, d11
as s. The eight s form an Octet:
M1, M3, M 5, M7
and
M9, M 11 , M 13 , M 15
(the odd numbers). This simplifies to the
Maxterm
(D)
. (Correction:
M1, M3, M5, M7, M 9, M 11 , M13 , M15
gives
D
).
F26(A , B , C , D)=∑(4,5,6,7,12 ,13 ,14 ,15)
Minimal SOP:
B
Rationale: These eight minterms cover the entire bottom half of the map, forming an
Octet, simplifying to
B
.
F27(A , B , C , D)=∑(0,1,3,4,5,7,9,11 ,12 ,13 ,15)
Minimal SOP:
AD+CD +A B C
Rationale: Quads:
m4, m5, m12 , m13
(
A C
).
m1,m3, m9, m11
(
B D
).
m3,m7, m11 ,m15
(
CD
).
Minimal SOP:
B D+C D+A C
.
F28(A , B , C , D)=Π(0,2,4,6,8,10 ,12 ,14 ,15)
Minimal POS:
D⋅(B+C)
Rationale: The eight s (
M0, M2, M 4, M 6, M8, M10 , M12 , M14
) form an Octet,
simplifying to
(D)
. Remaining
M15
is isolated. Minimal POS:
D⋅(A+B+C+D)
.
(Correction:
M0,... , M14
even numbers form
D
.
M15
isolated
A+B+C+D
).
F29(A , B ,C , D)=∑(0,1,2,3,5,7)+d(11 ,15 )
Minimal SOP:
A B+CD
Rationale: Quad
m0, m1, m2, m3
(
A B
). Use
d11 , d15
to form Quad
m3,m7, d11 , d15
(
CD
).
Minimal SOP:
A B+CD
.
F30(A , B ,C , D)=Π(0,4,8,12)⋅d(1,5,9,13)
Minimal POS:
C+D
Rationale: We group the s. Use
d1, d5, d9, d13
as s. The eight s form an Octet,
simplifying to the Maxterm
(C+D)
.
