# Sample_Test_II_3_Solution ```Quiz 3 solution
50 minutes
All problems worth 20 points.
1. For the following function, show as many minimum sum of products expressions as you
can. The minimum is 5 terms and 13 literals. No credit for any solution that uses terms
that are not prime implicants. (Four copies of the map are shown for your convenience.)
First solution: 8 points
Next two: 3 points each
Additional ones: 2 points each (up to a maximum of 26 points)
There are three groups of four 1's. The first group chosen covers four 1's. The next can
only cover two. That leaves five 1's. The other group of four only covers one and leaves a
pattern requiring three more terms.
ab
ab
00 01 11 10
cd
00
1
01
1
1
1
00
1
01
1
11
10
1
1
1
1
00 01 11 10
cd
1
1
1
10
1
1
1
11
1
1
1
1
1
1
c d' + a' d' + a b' c' +
{b c' d + a b c} or {b c' d + a b d} or {a b d + a' b' c}
ab
cd
00
ab
00 01 11 10
1
01
1
1
1
00
1
01
1
11
10
1
cd
1
1
1
00 01 11 10
1
1
1
10
1
1
1
11
1
1
1
1
1
1
c d' + b' d' + a' b c' +
{a c' d + a b c} or {a c' d + a b d} or {a b d + a b' c'}
ab
ab
00 01 11 10
cd
1
00
01
1
1
1
cd
1
00
1
01
1
11
1
10
1
00 01 11 10
1
1
1
1
10
1
1
1
11
1
1
1
1
1
1
a' d' + b' d' + a b c +
{b c' d + a b' c'} or {b c' d + a c' d} or {a c' d + a' b c'}
2. For the following function
a) List all of the prime implicants and indicate which are essential.
b) Find all 3 minimum sum of product expressions.
Four copies of the map are shown for your convenience.
wx
yz
wx
00 01 11 10
00
1
X
X
01
1
X
X
11
10
1
y' z'
w' y' * x y'
wy
x z' *
yz
1
00
1
X
X
01
1
X
X
1
X
11
1
1
10
wx
w z'
00 01 11 10
1
1
1
X
1
1
f = w' y' + x z' + w y + w z'
wx
wx
00 01 11 10
yz
00
1
X
X
01
1
X
X
11
10
1
00 01 11 10
yz
1
00
1
X
X
01
1
X
X
1
X
11
1
1
10
f = w' y' + x z' + w y + y' z'
1
1
1
X
1
1
f = f = w' y' + x z' + w z' + w x
3. For the following function, show BOTH the minimum sum of products expression AND
the minimum product of sums expression.
f (a, b, c, d) = ∑ m (4, 10, 11, 12, 13, 14, 15)
Two blank maps are shown for your use.
ab
cd
ab
00 01 11 10
00 01 11 10
1
00
1
01
1
01
1
1
11
1
1
11
1
1
10
1
1
10
1
1
00
1
cd
f = a b + a c + b c' d'
1
1
f' = b' c' + a' d + a' c
f = (b + c) (a + d') (a + c')
4. For the following 5-variable function, find BOTH minimum sum of product solutions. (5
terms, 16 literals)
A
0
BC
DE
00 01 11 10
DE
00
01
1
1
BC
1
00 01 11 10
00
1
1
01
1
1
1
1
1
1
11
1
1
11
10
1
1
10
1
1
F = A' D' E + A' C D + A B' C D' + A C D' + {A B C or B C D}
5. The following function is the only minimum sum of products expression. Implement it
using only two-input NAND gates. No gate may be used as a NOT gate. Show both the
block diagram and the equation. Assume that all inputs are available both
uncomplemented and complemented. (Full credit for 15 gates; partial credit if you use one
three-input gate.)
F = B' C' D' E + B C D' + A' B' C + A' B C' + A' B' E' + A D E'
= B' (C' D' E + A' C) + B (C D' + A' C') + E' (A' B' + A D')
= B' (C + D' E) (C' + A') + B (C + A') (C' + D') + E' (A' B' + A D')
= [B + (C + D' E) (C' + A')] [B' + (C + A') (C' + D')] + E' (A' B' + A D')
D'
E
C'
B'
A
C
A
C'
F
B
C
D
E'
A'
B'
A
D'
```