Midterm Solution
1 Hour, 50 minutes
Do any 10 of the 12 problems. Make sure it is clear which one you do not want
graded (or I will grade the first 10 problems). All problems are worth 20 points.
1. The following function is already in minimum sum of products form. Show the equation
and a block diagram for an implementation using only 2-input NAND gates. No gate may
be used as a NOT. All variables are available both uncomplemented and complemented.
For full credit, find a solution using 14 gates; partial credit for more gates or one or two 3input gates. Bonus of up to 5 points for less.
F = A' B' C' D' + B' C D' E + B D + B C' E' + C D E' + C' D E
= B' D' (A' C' + C E) + D (C E' + C' E) + B (D + C' E')
= [D' + C E' + C' E] [B + D' (A' C' + C E)] + B (D + C' E')
= [D' + (C + E) (C' + E')] [B + D' (A' C' + C E)] + B (D + C' E')
where the underlined terms use the same gate.
This uses 12 gates.
A'
C'
B'
C
E
D'
F
D
B
C'
E'
D'
2. Show an algebraic expression for the following circuit. Reduce it to a minimum sum of
products expression (2 terms, 4 literals.)
c
a
b'
y
w
g
c'
d
z
x
w = (a b')' = a' + b
y = c w = c (a b')'
x = c' + d
z = w' + x' = a b' + c d'
g = (y + z)' = (c w + w' + x') = (c + w' + c d')'
= (c + w')' = (c + a b')' = c' (a' + b)
= a' c' + b c'
3. Convert the following to a sum of minterms. Remove any duplicates from the list.
f = (a + b' + c) (a + c + d) (b + c') (b' + d')
= (a + c + b' d) (b d' + b' c')
= a b d' + a b' c' + b c d' + b' c' d
= a b c' d' + a b c d' + a b' c' d' + a b' c' d + a' b c d' + a b c d'
= a' b' c' d + a' b c d' + a b' c' d' + a b' c' d + a b c' d' + a b c d'
4. For the following function, find THE minimum sum of products solution.
ab
cd
00
00 01 11 10
1
01
1
11
1
10
1
1
1
1
1
f = a' b' d' + a b' d + a' b c + b c' d
5. For the following function, list all of the prime implicants and find BOTH minimum sum
of product expressions. (extra maps shown for your convenience)
h (w, x, y, z) = ∑ m (2, 3, 5, 6, 7, 9, 10, 11, 13, 14)
+
wx
yz
wx
00 01 11 10
yz
00
wx
00 01 11 10
yz
00
01
1
11
1
1
10
1
1
w' y
1
1
00
1
01
1
11
1
1
1
10
1
1
y z'* x' y
00 01 11 10
1
w x' z
1
1
w y' z
1
01
1
11
1
1
1
10
1
1
x y' z
1
1
1
1
1
1
w' x z
h = y z' + w' y + x y' z + w x' z
h = y z' + x' y + w y' z + w' x z
6. For the function, g, shown, show the equations and a block diagram for a minimum
cost two-level NAND implementation a minimum two-level NOR implementation (spare
map shown). All variables are available both uncomplemented and complemented.
wx
wx
00 01 11 10
yz
00
00 01 11 10
yz
1
00
1
01
1
1
1
01
1
11
1
1
1
11
1
1
1
10
1
10
g
1
g' = w' x' + w' z' + x y' z'
g = (w + x) (w + z) (x' + y + z)
x
z
w
y
1
g'
g = x z + w x' + w y
w
x'
1
w
x
g
w
z
x'
y
z
g
7. For the function, F, shown find a minimum sum of products expression and a minimum
product of sums expression (spare map shown).
WX
WX
00 01 11 10
1
00
01
1
11
1
X
10
1
X
00 01 11 10
YZ
X
00
X
01
1
1
1
1
1
X
1
X
11
X
1
10
X
1
F'
F
F' = X Z' + W' Y' Z + W X’ Z
F = (X' + Z) (W + Y + Z') (W’ + X + Z’)
F = X' Z' + W' Y + W X Z
8. For the following function, find all 4 minimum sum of products expressions.
ab
cd
00
01
11
10
ab
00 01 11 10
X
cd
X
1
X
00
1
X
X
01
X
1
1
11
X
a b + {b c' or b d or c' d}
10
00 01 11 10
X
X
1
X
1
X
X
X
1
1
a d' + b d
X
9. For the following 5-variable function, F, find BOTH minimum sum of product solutions.
(Two copies of the map are shown.)
BC
DE
A
0
00 01 11 10
00
1
01
1
BC
DE
1
00 01 11 10
1
00
1
01
11
1
11
1
1
10
1
10
1
1
1
1
1
1
F = A' C' D' + A B D + C D' E + {A' B C' or B C' D}
10. For the following 5-variable function, F, find the minimum sum of product solution.
(Two copies of the map are shown.) (6 terms, 20 literals)
BC
DE
00
A
0
00 01 11 10
1
BC
DE
1
00 01 11 10
1
00
01
1
1
01
1
1
11
1
1
1
11
10
1
1
1
10
1
1
1
1
1
1
F = A' B' C' D' E' + A' C D + A C' E + B C' D + B C D' + B D' E
11. For the two functions shown, find a minimum two-level solution (map and equations)
corresponding to 5 AND gates and 2 OR gates with a total of 19 inputs. All inputs are
available both uncomplemented and complemented. (Show only the maps and
equations.)
wx
wx
00 01 11 10
yz
00
1
01
1
1
11
1
1
10
1
00 01 11 10
yz
00
1
1
1
01
1
1
1
1
11
1
1
1
1
10
1
g
f
f = w' x' + w x y + x z
g = x' y' z' + w x y + x z + w' z
12. For the three functions shown, find a minimum two-level solution (map and
equations) corresponding to 7 AND gates and 3 OR gates with a total of 30 inputs. All
inputs are available both uncomplemented and complemented. (Show the maps and
equations.)
ab
cd
ab
00 01 11 10
1
00
01
1
1
11
1
1
10
cd
1
00
01
1
1
1
f
1
cd
1
1
1
11
10
ab
00 01 11 10
1
1
g
f = b' c d + b c d' + a' d + a c' d'
g = b' c' d' + b c' d + b c d'
h = b c' d + a b' d' + a' d
00 01 11 10
1
00
01
1
1
11
1
1
1
10
1
h