Chapter 4 Problem Solutions 4.1. w 0 0 0 0

8/14/2019 Chapter 4 Problem Solutions 4.1. w 0 0 0 0

1/97

- 4.1 -

Chapter 4

Problem Solutions

4.1.

w x y z f (a) (b) (c) (d) (e) (f) (g) (h)

0 0 0 0 0 0 0 0 0 0 1 1 0

0 0 0 1 0 0 1 0 0 0 1 1 0

0 0 1 0 0 0 1 0 0 0 1 1 0

0 0 1 1 0 0 1 0 0 0 0 1 0

0 1 0 0 0 0 0 1 0 0 1 1 0

0 1 0 1 1 0 1 1 0 0 1 1 1

0 1 1 0 0 0 1 1 0 1 1 1 0

0 1 1 1 0 0 1 1 0 0 0 1 0

1 0 0 0 1 1 0 1 0 0 1 1 0

1 0 0 1 1 0 1 1 1 0 1 1 0

1 0 1 0 1 1 1 1 0 0 1 1 0

1 0 1 1 1 0 1 1 1 0 1 1 0

1 1 0 0 1 1 0 1 0 0 1 1 0

1 1 0 1 0 0 1 1 0 0 1 0 0

1 1 1 0 1 1 1 1 0 1 1 1 0

1 1 1 1 0 0 1 1 0 0 1 0 0

(a) wz-is an implicant since it implies f. However, neither

the term w nor the term z-are implicants. Therefore, wz

-

is a prime implicant (1).

(b) y+z is neither an implicant or implicate (6).

(c) w+x is an implicate since it is implied by f. However,

nether the term w nor the term x are implicates.

Therefore, w+x is a prime implicate (3).


2/97

- 4.2 -

4.1. (continued)

(d) wx-z is an implicant since it implies f. However, wx

-z

subsumes wx-which is also an implicant. Therefore, wx

-z

is an implicant, but not prime (2).

(e) xyz-is neither an implicant or implicate (6).

(f) w-+y-+z-is an implicate since it is implied by f.

However, w-+y-+z-subsumes w+y

-which is also an implicate.

Therefore, w-+y-+z-

is an implicate, but not prime (4).

(g) w-+x-+z-is an implicate since it is implied by f.

Furthermore, none of the sum terms w-+x-, w-+z-, and x-+z- are

implicates. Therefore, w-+x-+z-is a prime implicate (3).

(h) w-xy-z is an implicant since it implies f. However, none

of the product terms w-xy-, w-xz, w

-y-z, and xy

-z are

implicants. Therefore, w-xy-z is a prime implicant (1).

4.2. (a) (b)


3/97

- 4.3 -

4.2. (continued)

(c) (d)

(e) (f)

4.3. The Karnaugh map is


4/97

- 4.4 -

4.3. (continued)

The implicants correspond to all possible subcubes of 1-

cells.

one-cell subcubes

(minterms)

two-cell subcubes four-cell subcubes

w-x-y-z-

w-x-y-

wy

w-x-y-z w

-x-z-

w-x-yz-

w-y-z

w-xy-z wxy

wx-yz-

wx-y

wx-yz x

-yz-

wxyz- wyz

wxyz wyz-

The prime implicants correspond to those subcubes which are

not properly contained in some other subcube of 1-cells.

The prime implicants are shown on the map. They are wy,w-x-y-, w

-x-z-, w-y-z, and x

-yz-.


5/97

- 4.5 -

4.4. (a)

The prime implicants are xz, xy, y-z, yz

-, x-z-, and x

-y-.

There are no essential prime implicants.

(b)

The prime implicants are w-x, xz, w

-y-z, and wyz. All four

prime implicants are essential as a result of the

essential 1-cells indicated by asterisks.


6/97

- 4.6 -

4.4. (continued)

(c)

The prime implicants are w-x-, w-z__, x-z-__, and x-y __. The

essential prime implicants are underlined.

(d)

The prime implicants are y-z-, xy

-__, x

-z-

__, and w-xz ___. The



7/97

- 4.7 -

4.5. (a)

The prime implicates are x-+y+z and x+y

-+z-. Both prime

implicates are essential.

(b)

The prime implicates are w-+z, x+z, w+x+y

-, and w

-+x+y.

All three prime implicates are essential.


8/97

- 4.8 -

4.5. (continued)

(c)

The prime implicates are w-+x-, x-+z, and w-+y+z-. All three

prime implicates are essential.

(d)

The prime implicates are x+z-

___, x-+y-+z_____, w

-+x-+y-, and w

-+y-+z-.

The essential prime implicates are underlined.


9/97


10/97

- 4.10 -

4.6. (continued)

(c)

Minimal sum:

f = x-y-+ x-z + y

-z + xyz

-

Minimal product:

f = (x+y-+z)(x-+y+z)(x-+y-+z-)

(d)

Minimal sum:

f = y-z-+ x-yz

Minimal products:

f = (y+z-)(y-+z)(x

-+z-)

f = (y+z-)(y-+z)(x

-+y-)


11/97

- 4.11 -

4.7. (a)

Minimal sum:

f = x + yz-

Minimal product

f = (x+y)(x+z-)

(b)

Minimal sum:

f = x-+ y + z

-

Minimal product:

f = x-+ y + z

-


12/97

- 4.12 -

4.7. (continued)

(c)

Minimal sum:

f = x-z-+ yz

Minimal product:

f = (y+z-)(x-+z)

(d)

Minimal sum:

f = y + x-z-

Minimal product:

f = (x-+y)(y+z

-)


13/97

- 4.13 -

4.8. (a)

Minimal sum:

f = xy + w-x-y- + x-y-z-

Minimal products:

f = (x-+y)(x+y

-)(w-+y+z

-)

f = (x-+y)(x+y

-)(w-+x+z

-)

(b)

Minimal sum:

f = xz-+ wz + x

-yz

Minimal products:

f = (x+z)(w+x-+z-)(w+x+y)

f = (x+z)(w+x-+z-)(w+y+z

-)


14/97

- 4.14 -

4.8. (continued)

(c)

Minimal sum:

f = wz + x-z + yz + w

-xz-

Minimal product:

f = (w-+z)(x+z)(w+x

-+y+z

-)

(d)

Minimal sums:

f = w-x-z-+ w-xy + wxz + wx

-y-

f = x-y-z-+ wy

-z + xyz + w

-yz-


15/97

- 4.15 -

4.8. (continued)

Minimal products:

f = (w+x+z-)(w+x-+y)(w-+x-+z)(w-+x+y-)

f = (x-+y+z)(w+y+z

-)(x+y

-+z-)(w-+y-+z)

(e)

Minimal sum:

f = x-z- + yz + wy-

Minimal products:

f = (w+y+z-)(x

-+y-+z)(w+x

-+y)

f = (w+y+z-)(x

-+y-+z)(w+x

-+z)


16/97

- 4.16 -

4.8. (continued)

(f)

Minimal sum:

f = w-x-+ y-z + wz + x

-y

Minimal product:

f = (x-+z)(w+x

-+y-)(w-+y+z)

(g)

Minimal sum:

f = yz + x-z + xy

-z-

Minimal product:

f = (x+z)(y-+z)(x

-+y+z

-)


17/97

- 4.17 -

4.8. (continued)

(h)

Minimal sums:

f = xy-+ xz + w

-x-y + wyz

-

f = xy-+ wx + w

-yz + x

-yz-

Minimal product:

f = (x+y)(w-+x+z-)(w+x-+y-+z)


18/97

- 4.18 -

4.8. (continued)

(i)

Minimal sum:

f = wx- + x-z- + x-y- + w-y-z- + wy-z

Minimal product:

f = (x-+y-)(w+x

-+z-)(w+y

-+z-)(w-+x-+z)

(j)

Minimal sum:

f = wx- + x-y + wyz + w-xz-

Minimal products:

f = (w+x+y)(w+x-+z-)(w-+x-+z)(w

-+x-+y)

f = (w+x+y)(w+x-+z-)(w-+x-+z)(x

-+y+z

-)


19/97

- 4.19 -

4.9. (a)

Minimal sums:

f = w-x-z- + xyz + x-yz- + wx-y-z + w-xy

f = w-x-z-+ xyz + x

-yz-+ wx

-y-z + w

-yz-

Minimal products:

f = (x-+y)(w

-+y+z)(x+y

-+z-)(w-+x-+z)(w+x+z

-)

f = (x-+y)(w

-+y+z)(x+y

-+z-)(w-+x-+z)(w+y+z

-)

(b)

Minimal sum:

f = x-y-+ w-x + w

-z-

Minimal product:

f = (w-+x-)(w-+y-)(x+y

-+z-)


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- 4.20 -

4.9. (continued)

(c)

Minimal sum:

f = x-y-z-+ xyz + w

-xy + w

-y-z

Minimal products:

f = (x+y-)(x-+y+z)(w

-+y+z

-)(w-+y-+z)

f = (x+y-)(x-+y+z)(w

-+y+z

-)(w-+x-+z)

(d)

Minimal sum:

f = y-z + w

-z + x

-z + w

-xy-

Minimal product:

f = (x+z)(w-+z)(y

-+z)(w

-+x-+y-)


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- 4.21 -

4.9. (continued)

(e)

Minimal sums:

f = xyz + xy-z-+ wy

-z-+ w-x-y-z + wxy

f = xyz + xy-z-+ wy

-z-+ w-x-y-z + wxz

-

Minimal products:

f = (x+y-)(x-+y+z

-)(w+x+z)(w+y

-+z)(w

-+x+z

-)

f = (x+y-)(x-+y+z

-)(w+x+z)(w+y

-+z)(w

-+y+z

-)

(f)

Minimal sums:

f = y-z-+ wz + w

-y

f = w-z-+ wy

-+ yz


22/97

- 4.22 -

4.9. (continued)

Minimal product:

f = (w+y+z-)(w-+y-+z)

(g)

Minimal sum:

f = wx + y-z + wy

-+ xz

Minimal product:

f = (w+z)(x+y-)


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- 4.23 -

4.9. (continued)

(h)

Minimal sums:

f = x-z-+ xz + wy

-+ yz

-

f = x-z-+ xz + wy

-+ xy

Minimal product:

f = (w+x+z-)(x+y

-+z-)(w+x

-+y+z)

(i)

Minimal sum:

f = x-z + wz

-

Minimal product:

f = (x-+z-)(w+z)


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- 4.24 -

4.9. (continued)

(j)

Minimal sums:

f = x-y-+ w-y-

+ wxz + xyz-

f = x-y-+ y-z + w

-xz-+ wxy

Minimal product:

f = (x+y-)(w+y

-+z-)(w-+x-+y+z)


25/97

- 4.25 -

4.10. (a)

The prime implicants are x-z-

__, w-y-, xz __, and wy. The


The prime implicates are x-+z and x+z

-. Both prime

implicates are essential.


26/97

- 4.26 -

4.10. (continued)

(b)

The prime implicants are x, w-z, and wz-. All three

prime implicants are essential.

The prime implicates are w+z___, w-+x+z

-_____, and w

-+y-+z-. The

essential prime implicates are underlined.


27/97

- 4.27 -

4.10. (continued)

(c)

The prime implicants are wx__, yz-, w-y __, and xy. The


The prime implicates are w+y___, x+y, w-+x ___, and x+z. The

essential prime implicates are underlined.


28/97

- 4.28 -

4.11. (a)

Minimal sum:

f = x-z- + xyz + w-y-z

Minimal product:

f = (x-+z)(w

-+y+z

-)(x+y

-+z-)

(b)

Minimal sum:

f = wy-+ x-yz-

Minimal products:

f = (w+y)(x-+y-)(w+z

-)

f = (w+y)(x-+y-)(y-+z-)


29/97

- 4.29 -

4.11. (continued)

(c)

Minimal sum:

f = y-z + xz + wz-

Minimal product:

f = (w+z)(x+y-+z-)

(d)

Minimal sums:

f = wx + wz + yz

f = wx + wz + xz

f = wx + wz + xy

Minimal product:

f = (x+z)(w+y)


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- 4.30 -

4.11. (continued)

(e)

Minimal sums:

f = x-z- + yz-

f = x-z-+ w-z-

Minimal products:

f = z-(x-+y)

f = z-(w-+x-)

(f)

Minimal sum:

f = wx-+ x-z + w

-xz-+ wy

-

Minimal product:

f = (w+x+z)(w+x-+z-)(w-+x-+y-)


31/97

- 4.31 -

4.11. (continued)

(g)

Minimal sum:

f = x-z- + wy- + w-xy

Minimal product:

f = (w+y)(w-+x-+y-)(x+y

-+z-)

(h)

Minimal sum:

f = w-y + xy

-+ y-z + xz

Minimal products:

f = (x+y+z)(w-+x+y

-)(x-+y-+z)

f = (x+y+z)(w-+x+y

-)(w-+y-+z)


32/97

- 4.32 -

4.11. (continued)

(i)

Minimal sum:

f = w-y-+ y-z + wyz

-+ w-x

Minimal product:

f = (w-+y+z)(w+x+y

-)(w-+y-+z-)

(j)

Minimal sums:

f = w-z + wz

-+ w-xy-

f = w-z + wz

-+ xy

-z-


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- 4.33 -

4.11. (continued)

Minimal products:

f = (w-+z-)(y

-+z)(w+x+y)

f = (w-+z-)(y

-+z)(w+x+z)

4.12. (a)

Minimal sums:

f = w-x + wx

-y + wz

f = w-x + wx-y + y-z

Minimal products:

f = (y+z)(w+x)(w-+x-+y-)

f = (y+z)(w+x)(w-+x-+z)


34/97

- 4.34 -

4.12. (continued)

(b)

Minimal sums:

f = wx-y- + w-y-z + wxy

f = wx-y-+ w

-y-z + xyz

-

Minimal products:

f = (x+y-)(w+z)(w

-+x-+y)(y

-+z-)

f = (x+y-)(w+z)(w

-+x-+y)(w+y

-)

(c)

Minimal sum:

f = x-z-+ w-y-

+ wxz

Minimal product:

f = (x+z-)(w-+x-+z)(w+x

-+y-)


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- 4.35 -

4.12. (continued)

(d)

Minimal sums:

f = w-x-z + w

-xz-+ wxy

-

f = w-x-z + w

-xz-+ xy

-z-

Minimal product:

f = (x+z)(x-+z-)(w-+y-)

(e)


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- 4.36 -

4.12. (continued)

Minimal sums:

f = w-y-+ y-z + w

-x-

f = w-y-+ y-z + x

-y

Minimal product:

f = (w-+z)(x

-+y-)

(f)

Minimal sum:

f = xz-+ w-y + x

-y-z

Minimal product:

f = (x-+z-)(x+y+z)(w

-+x+y

-)


37/97

- 4.37 -

4.12. (continued)

(g)

Minimal sum:

f = wx + w-x-y + wy

-z-+ xz

-

Minimal products:

f = (w+y)(w+x-+z-)(w-+x+y

-)(w-+x+z

-)

f = (w+y)(w+x-+z-)(w-+x+y

-)(x+y+z

-)

(h)

Minimal sum:

f = w-y + wz

-+ w-xz


38/97

- 4.38 -

4.12. (continued)

Minimal products:

f = (w-+z-)(w+y+z)(w+x+y)

f = (w-+z-)(w+y+z)(x+y+z

-)

(i)

Minimal sum:

f = xy + xz + wyz + wy-z-

Minimal products:

f = (w+x)(w+y+z)(x+y+z-)(w-+y-+z)

f = (w+x)(w+y+z)(x+y+z-)(x+y

-+z)

(j)


39/97

- 4.39 -

4.12. (continued)

Minimal sum:

f = xy + wz-+ w-y-z

Minimal product:

f = (w+z)(x+y-)(w-+y+z-)

4.13.

To construct f2:

If g=1 and f1=1, then f

2=1.


2=-.


2=0.

(Note: g=1 and f1=0 can not occur if the problem is

solvable.)

Using the above rules, the map for f2is:


40/97

- 4.40 -

4.13. (continued)

Minimal sum:

f2= xy

-+w-z

Minimal products:

f2 = (x+z)(y-+z)(w-+x) (Shown above)

Other possible minimal products:

f2= (x+z)(y

-+z)(w

-+y-)

f2= (x+z)(x

-+y-)(w-+x)

f2= (x+z)(x

-+y-)(w-+y-)

4.14. (a)

Minimal sums:

f = v-y-z + vwy + vxy

-+ v-wx-z

f = v-y-z + vwy + vxy

-+ wx

-yz


41/97

- 4.41 -

4.14. (continued)

Minimal product:

f = (w+y-)(v+z)(v

-+x+y)(v+x

-+y-)

(b)

Minimal sum:

f = yz + v-wy-+ vw

-y-z-+ vwxy


42/97

- 4.42 -

4.14. (continued)

Minimal products (shown above):

f = (v-+w-+y)(v+y

-+z)(w+y

-+z)(x+y

-+z)(v+w+y)(v

-+y+z

-)

f = (v-+w-+y)(v+y

-+z)(w+y

-+z)(x+y

-+z)(v+w+y)(w+y+z

-)

Another minimal product (not shown):

f = (v-+w-+y)(v+y

-+z)(w+y

-+z)(x+y

-+z)(w+y+z

-)(v+w+z)

(c)

Minimal sums (shown above):

f = vwz-+ w

-x-z + vy

-z-+ v-w-xy-+ v

-wxy + v

-x-yz

f = vwz-+ w-x-z + vy

-z-+ v-w-xy-+ v

-wxy + v

-wyz


43/97

- 4.43 -

4.14. (continued)

Another minimal sum (not shown):

f = vwz-+ w

-x-z + vy

-z-+ v-w-xy-

+ v-wyz + wxyz

-

Minimal product:

f = (v+x+z)(v+w-+y)(v

-+w-+z-)(w+x

-+y-)(v-+x-+z-)(w+y

-+z)

(d)

Minimal sum:

f = w-z + y

-z + wx

-y + vwy

-


44/97

- 4.44 -

4.14. (continued)

Minimal product:

f = (w+z)(w-+x-+y-)(v+y+z)

4.15.


45/97

- 4.45 -

4.15. (continued)

Minimal sums:

f = v-wz-+ uvw

-+ v-w-x + uvz

-

f = v-wz-+ uvw

-+ v-w-x + uwz

-

Minimal product:

f = (u+v-)(w-+z-)(v+w+x)


46/97

- 4.46 -

4.16.

x y z f

0 0 0 0

0 0 1 0

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 1

1 1 0 1

1 1 1 1

Minimal sum (cost=9): f = xy + xz + yz

Minimal product (cost=9): f = (x+y)(x+z)(y+z)

Realization using the minimal sum:


47/97

- 4.47 -

4.17.

Minimal sum (cost=20):

p = wz + xy-z + xyz- + x-yz + w-x-y-z-

Minimal product (cost=20):

p = (w-+z)(x

-+y+z)(x

-+y-+z-)(x+y

-+z)(w+x+y+z

-)

Comparing Tables 3.10 and 3.12, it is seen that if the

don't-cares for wxyz=1010, 1100, and 1111 are assigned

values of logic-1, the two tables become identical.

Since Table 3.10 is describable by the expression

p=(w-rx)r(yrz), this same expression can be used for

Table 3.12.

4.18. Truth table:

xi

yi

ci

ci+1

si

0 0 0 0 0

0 0 1 0 1

0 1 0 0 1

0 1 1 1 0

1 0 0 0 1

1 0 1 1 0

1 1 0 1 0

1 1 1 1 1


48/97

- 4.48 -

4.18. (continued)

For the ci+1

output:


ci+1

= xiyi+ x

ici+ y

ici

Minimal product (cost=9)

ci+1

= (xi+yi)(x

i+ci)(y

i+ci)

For the sioutput:


si= x-iy-ici

+ x-iyic-i+ x

iy-ic-i+ x

iyici

Minimal product (cost=16)

si= (x

i+yi+c

i)(x

i+y-i+c-i)(x-i+y

i+c-i)(x-i+y-i+ c

i)

However,

si= x-

iy-ici+ x-

iyic-i+ x

iy-ic-i

+ xiyici

= x-i(y-ici+y

ic-i) + x

i(y-ic-i+ y

ici)

= x-i(yirc

i) + x

i(yirc

i)

_______

= xir (y

irc

i)


49/97

- 4.49 -

4.18. (continued)

Realization:

4.19. Truth table:

x

i

y

i

b

i

b

i+1

d

i

0 0 0 0 0

0 0 1 1 1

0 1 0 1 1

0 1 1 1 0

1 0 0 0 1

1 0 1 0 0

1 1 0 0 0

1 1 1 1 1


50/97

- 4.50 -

4.19. (continued)

For the bi+1

output:


bi+1

= x-iyi+ x-ibi+ y

ibi


bi+1

= (x-i+yi)(x-

i+bi)(y

i+bi)

For the dioutput:


di= x-iy-ibi

+ x-iyib-i+ x

iy-ib-i+ x

iyibi


di= (x

i+yi+b

i)(x

i+y-i+b-i)(x-i+y

i+b-i)(x-i+y-i+bi)

However,

di= x-

iy-ibi+ x-

iyib-i+ x

iy-ib-i

+ xiyibi

= x-(y-ibi+yib-i) + x

i(y-ib-i+yibi)

= x-i(yirb

i) + x

i(yirb

i)

_______

= xir (y

irb

i)


51/97

- 4.51 -

4.19. (continued)

Realization:


52/97

- 4.52 -

4.20.

8 4 2 1

w x y z f

0 0 0 0 0

0 0 0 1 0

0 0 1 0 0

0 0 1 1 0

0 1 0 0 0

0 1 0 1 0

0 1 1 0 0

0 1 1 1 0

1 0 0 0 0

1 0 0 1 0

1 0 1 0 1

1 0 1 1 1

1 1 0 0 1

1 1 0 1 1

1 1 1 0 1

1 1 1 1 1


f = wx + wy


f = w(x+y)

Realization:


53/97

- 4.53 -

4.21. Let:

F=1 when the pressure in the fuel tank is equal to or above

a required minimum.

F=0 when the pressure in the fuel tank is below a required

minimum.

X=1 when the oxidizer tank pressure is equal to or above a

required minimum.

X=0 when the oxidizer tank pressure is below a required

minimum.

T=1 when there are 10 min. or less to lift off.

T=0 when there are more than 10 min. to lift off.

P=1 when the panel light is on.

P=0 when the panel light is off.

From the problem statement we can write the equation for

the panel light to be on:

P = FXT + F-XT-+ X-T-


P = X-T-+ F-T-+ FXT


P = (F+T-)(X+T-)(F-+X-+T)


54/97

- 4.54 -

4.21. (continued)

Realization using the minimal sum:

4.22. Let w, x, y, and z denote the bits of the 2421 code groups

and A, B, C, and D denote the bits of the 8421 code groups

where w and A are the most significant bits of their

respective code groups.

w x y z A B C D

0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 1

0 0 1 0 0 0 1 0

0 0 1 1 0 0 1 1

0 1 0 0 0 1 0 0

0 1 0 1 - - - -

0 1 1 0 - - - -

0 1 1 1 - - - -

1 0 0 0 - - - -

1 0 0 1 - - - -

1 0 1 0 - - - -

1 0 1 1 0 1 0 1

1 1 0 0 0 1 1 0

1 1 0 1 0 1 1 1

1 1 1 0 1 0 0 0

1 1 1 1 1 0 0 1


55/97

- 4.55 -

4.22. (continued)

A = xy B = xy-+ wx

-(cost=6)

B = (w+x)(x-+y-) (cost=6)

C = wy-+ w-y (cost=6) D = z

C = (w+y)(w-+y-) (cost=6)


56/97

- 4.56 -

4.23.

7 5 3 6-

w x y z f

0 0 0 0 1

0 0 0 1 -

0 0 1 0 1

0 0 1 1 -

0 1 0 0 0

0 1 0 1 -

0 1 1 0 0

0 1 1 1 1

1 0 0 0 0

1 0 0 1 1

1 0 1 0 -

1 0 1 1 0

1 1 0 0 -

1 1 0 1 0

1 1 1 0 -

1 1 1 1 0


57/97

- 4.57 -

4.23. (continued)


f = w-x-+ w-z + x

-y-z

Minimal products (cost=12):

f = (x-+z)(w-+z)(w-+y-)(w-+x-)

f = (x-+z)(w

-+z)(w

-+y-)(x-+y)

Realization:


58/97

- 4.58 -

4.24.

Inputs Outputs

8 4 2 1

w x y z a b c d e f g

0 0 0 0 1 1 1 1 1 1 0

0 0 0 1 0 1 1 0 0 0 0

0 0 1 0 1 1 0 1 1 0 1

0 0 1 1 1 1 1 1 0 0 1

0 1 0 0 0 1 1 0 0 1 1

0 1 0 1 1 0 1 1 0 1 1

0 1 1 0 1 0 1 1 1 1 1

0 1 1 1 1 1 1 0 0 0 0

1 0 0 0 1 1 1 1 1 1 1

1 0 0 1 1 1 1 1 0 1 1

1 0 1 0 - - - - - - -

1 0 1 1 - - - - - - -

1 1 0 0 - - - - - - -

1 1 0 1 - - - - - - -

1 1 1 0 - - - - - - -

1 1 1 1 - - - - - - -

a = w + y + xz + x-z-

b = x-+ y-z-+ yz


59/97

- 4.59 -

4.24. (continued)

c = x + y-+ z d = w + yz

-+ x-y + x

-z-+ xy

-z

e = x-z-+ yz

-f = w + y

-z-+ xy

-+ xz

-


60/97

- 4.60 -

4.24. (continued)

g = w + xy-+ x

-y + yz

-or g = w + xy

-+ x-y + xz

-

4.25. (a)

wxyz wxyz wxyz

0 0000 T 0,2 00-0 T 0,2,8,10 -0-0 A

2 0010 T 0,4 0-00 T 0,4,8,12 --00 B

4 0100 T 0,8 -000 T 8,10,12,14 1--0 C

8 1000 T 2,3 001- D

3 0011 T 2,10 -010 T

10 1010 T 4,12 -100 T

12 1100 T 8,10 10-0 T

13 1101 T 8,12 1-00 T

14 1110 T 10,14 1-10 T

12,13 110- E

12,14 11-0 T

The prime implicants are:

A: x-z-

B: y-z-

C: wz-

D: w-x-y

E: wxy-


61/97

- 4.61 -

4.25. (continued)

(b)

vwxyz vwxyz vwxyz

4 00100 T 4,5 0010- T 4,5,6,7 001-- A

5 00101 T 4,6 001-0 T 10,14,26,30 -1-10 B

6 00110 T 5,7 001-1 T

9 01001 E 6,7 0011- T

10 01010 T 6,14 0-110 C

7 00111 T 10,14 01-10 T

14 01110 T 10,26 -1010 T

19 10011 F 14,30 -1110 T

26 11010 T 26,30 11-10 T

30 11110 T 30,31 1111- D

31 11111 T


A: v-w-x

B: wyz-

C: v-xyz-

D: vwxy

E: v-wx-y-z

F: vw-x-yz


62/97

- 4.62 -

4.25. (continued)

(c)

wxyz wxyz wxyz

4 0100 T 4,12 -100 C 9,11,13,15 1--1 A

9 1001 T 9,11 10-1 T 12,13,14,15 11-- B

12 1100 T 9,13 1-01 T

7 0111 T 12,13 110- T

11 1011 T 12,14 11-0 T

13 1101 T 7,15 -111 D

14 1110 T 11,15 1-11 T

15 1111 T 13,15 11-1 T

14,15 111- T


A: wz

B: wx

C: xy-z-

D: xyz

(d)

wxyz wxyz wxyz

0 0000 T 0,1 000- T 0,1,2,3 00-- A

1 0001 T 0,2 00-0 T 1,3,5,7 0--1 B

2 0010 T 1,3 00-1 T 2,3,6,7 0-1- C

3 0011 T 1,5 0-01 T

5 0101 T 1,9 -001 D

6 0110 T 2,3 001- T

9 1001 T 2,6 0-10 T

10 1010 T 2,10 -010 E

12 1100 F 3,7 0-11 T

7 0111 T 5,7 01-1 T

6,7 011- T


63/97

- 4.63 -

4.25. (continued)


A: w-x-

B: w-z

C: w-y

D: x-y-z

E: x-yz-

F: wxy-z-

4.26. (a)

vwxyz vwxyz vwxyz

1 00001 T 1,3 000-1 T 1,3,17,19 -00-1 A

3 00011 T 1,17 -0001 T 6,14,22,30 --110 B

6 00110 T 3,11 0-011 D 10,11,14,15 01-1- C

10 01010 T 3,19 -0011 T

12 01100 T 6,14 0-110 T

17 10001 T 6,22 -0110 T

20 10100 T 10,11 0101- T

24 11000 G 10,14 01-10 T

11 01011 T 12,14 011-0 E

14 01110 T 17,19 100-1 T

19 10011 T 20,22 101-0 F

22 10110 T 11,15 01-11 T

15 01111 T 14,15 0111- T

29 11101 H 14,30 -1110 T

30 11110 T 22,30 1-110 T


64/97

- 4.64 -

4.26. (continued)

The prime implicates are:

A: w+x+z-

B: x-+y-+z

C: v+w-+y-

D: v+x+y-+z-

E: v+w-+x-+z

F: v-+w+x

-+z

G: v-+w-+x+y+z

H: v-+w-+x-+y+z

-

(b)

wxyz wxyz wxyz

0 0000 T 0,2 00-0 T 0,2,8,10 -0-0 A

2 0010 T 0,4 0-00 T 0,4,8,12 --00 B

4 0100 T 0,8 -000 T 4,5,12,13 -10- C

8 1000 T 2,3 001- D

3 0011 T 2,10 -010 T

5 0101 T 4,5 010- T

10 1010 T 4,12 -100 T

12 1100 T 8,10 10-0 T

13 1101 T 8,12 1-00 T

5,13 -101 T

12,13 110- T

The prime implicates are:

A: x+z

B: y+z

C: x-+y

D: w+x+y-


65/97

- 4.65 -

4.27. (a)

wxyz wxyz

4 0100 T 4,5 010- A

5 0101 T 4,12 -100 B

12 1100 T 5,7 01-1 C

7 0111 T 12,14 11-0 D

14 1110 T 7,15 -111 E

15 1111 T 14,15 111- F

m4

m5

m7

m12

m14

m15

A: w-xy-

B: xy-z-

C: w-xz

D: wxz-

E: xyz

F: wxy

p = (A+B)(A+C)(C+E)(B+D)(D+F)(E+F)

= (B+AD)(C+AE)(F+DE)

= (BC+ABE+ACD+ADE)(F+DE)

= BCF + BCDE + ABEF + ABDE + ACDF + ACDE + ADEF + ADE

= BCF + ADE + BCDE + ABEF + ACDF

There are five irredundant disjunctive normal formulas:

f1= B + C + F = xy

-z-+ w-xz + wxy

f2= A + D + E = w

-xy-+ wxz

-+ xyz

f3 = B + C + D + E = xy-z- + w-xz + wxz- + xyz

f4= A + B + E + F = w

-xy-+ xy

-z-

+ xyz + wxy

f5= A + C + D + F = w

-xy-+ w-xz + wxz

-+ wxy

The costs of f1and f

2are 12; while the costs of f

3,

f4, and f

5are 16. Therefore, f

1and f

2are minimal

sums.


66/97

- 4.66 -

4.27. (continued)

(b)

wxyz wxyz wxyz

0 0000 T 0,4 0-00 T 0,4,8,12 --00 A

4 0100 T 0,8 -000 T 4,5,12,13 -10- B

8 1000 T 4,5 010- T 8,9,10,11 10-- C

3 0011 T 4,12 -100 T 8,9,12,13 1-0- D

5 0101 T 8,9 100- T

9 1001 T 8,10 10-0 T

10 1010 T 8,12 1-00 T

12 1100 T 3,7 0-11 E

7 0111 T 3,11 -011 F

11 1011 T 5,7 01-1 G

13 1101 T 5,13 -101 T

9,11 10-1 T

9,13 1-01 T

10,11 101- T

12,13 110- T

m4

m5

m8

m9

m12

m13

A: y-z-

B: xy-

C: wx-

D: wy-

G: w-xz

(Note: Prime implicants E and F do not appear in the

prime-implicant table since they do not cover any of

the minterms having functional values of 1.)


67/97

- 4.67 -

4.27. (continued)

p = (A+B)(B+G)(A+C+D)(C+D)(A+B+D)(B+D)

= (A+B)(B+G)(C+D)(B+D)

= (B+AG)(D+BC)

= BD + BC + ADG + ABCG

= BD + BC + ADG

There are three irredundant disjunctive normal

formulas:

f1= B + D = xy

-+ wy

-

f2= B + C = xy

-+ wx

-

f3= A + D + G = y

-z-+ wy

-+ w-xz

The costs of f1

and f2are 6 and the cost of f

3is 10.

Therefore, f1and f

2are minimal sums.

4.28. (a)

wxyz wxyz wxyz

4 0100 T 4,5 010- B 9,11,13,15 1--1 A

8 1000 T 8,9 100- C

3 0011 T 3,11 -011 D

5 0101 T 5,13 -101 E

9 1001 T 9,11 10-1 T

11 1011 T 9,13 1-01 T

13 1101 T 11,15 1-11 T

14 1110 T 13,15 11-1 T

15 1111 T 14,15 111- F


68/97

- 4.68 -

4.28. (continued)

M3

M4

M5

M8

M9

M11

M13

M14

M15

A: w-+z-

B: w+x-+y

C: w-+x+y

D: x+y-+z-

E: x-+y+z

-

F: w-+x-+y-

p = DB(B+E)C(A+C)(A+D)(A+E)F(A+F)

= BCDF(A+E)= ABCDF + BCDEF

There are two irredendant conjunctive normal formulas:

f1= ABCDF = (w

-+z-)(w+x

-+y)(w

-+x+y)(x+y

-+z-)(w-+x-+y-)

f2= BCDEF = (w+x

-+y)(w

-+x+y)(x+y

-+z-)(x-+y+z

-)(w-+x-+y-)

The cost of f1is 19 and the cost of f

2is 20.

Therefore, f1 is the minimal product.

(b)

wxyz wxyz wxyz

0 0000 T 0,8 -000 B 5,7,13,15 -1-1 A

8 1000 T 8,9 100- C

5 0101 T 5,7 01-1 T

6 0110T

5,13 -101T

9 1001 T 6,7 011- D

7 0111 T 9,13 1-01 E

13 1101 T 7,15 -111 T

15 1111 T 13,15 11-1 T


69/97

- 4.69 -

4.28. (continued)

M0

M6

M7

M8

M9

M13

A: x-+z-

B: x+y+z

C: w-+x+y

D: w+x-+y-

E: w-+y+z

-

p = BD(A+D)(B+C)(C+E)(A+E)

= BD(C+E)(A+E)

= BD(E+AC)

= BDE + ABCD

There are two irredundant conjunctive normal formulas:

f1= BDE = (x+y+z)(w+x

-+y-)(w-+y+z

-)

f2= ABCD = (x

-+z-)(x+y+z)(w

-+x+y)(w+x

-+y-)

The cost of f1is 12 and the cost of f

2is 15.

Therefore, f1is the minimal product.

4.29. (a) Referring to Table P4.29a, c2dominates c

1, c

6

dominates c1, c6 dominates c2, c3 dominates c7, and c5dominates c

7. Deleting the dominating columns c

2, c

3,

c5, and c

6, the table reduces to

c1

c4

c7

cost

r1

3

r2

4

r4

4

r5 4

r6

6

r7

6

(Note: r3no longer appears since it has no crosses in

the remaining columns.)


70/97

- 4.70 -

4.29. (continued)

r1equals r

5, r

2equals r

7, and r

4equals r

6. Deleting

those rows with the higher costs, i.e., r5, r

6, and r

7,

the table becomes

c1

c4

c7

cost

**r1

3

**r2

4

**r4

4

Since only single crosses exist in each column, the

minimal cover consists of rows r1, r

2, and r

4.

(b) Referring to Table P4.29b, r3is an essential prime

implicant since c11

has a single cross. Therefore, r3

must be selected and r3and c

11can be deleted from the

table. Also, c1dominates c

8, c

2dominates c

5, c

9

dominates c3, and c

10dominates c

6. Thus, columns c

1,

c2, c

9, and c

10are deleted. The table now reduces to

c3

c4

c5

c6

c7

c8

cost

r1

3

r2

3

r4

4

r5

4

r6

5

r7 6

r8

7

r9

7

Minimal cover: {r3,...}


71/97

- 4.71 -

4.29. (continued)

r1now dominates r

5, r

1dominates r

6, r

8dominates r

2,

r7dominates r

4, and r

5dominates r

6. Rows r

5and r

6

are deleted since they have a higher cost than their

dominating rows. However, r2 and r4 cannot be deleted

since they have a lower cost than their dominating

rows. The resulting table is

c3

c4

c5

c6

c7

c8

cost

**r1

3

r2

3

r4

4

r7

6

r8

7

r9

7

Minimal cover: {r3,...}

Since column c4 has a single cross, row r1 is selected

to appear in the minimal cover. After deleting row r1

and columns c4, c

5, and c

7, the table reduces to

c3

c6

c8

cost

r2

3

r4

4

r7 6

r8

7

r9

7

Minimal cover: {r1,r3,...}


72/97

- 4.72 -

4.29. (continued)

r4equals r

7and the cost of r

4is less than the cost

of r7. Therefore, delete r

7. Also, r

8dominates r

9.

r9is deleted since it has equal cost. The table

becomes

c3

c6

c8

cost

r2

3

**r4

4

**r8

7

Minimal cover: {r1,r3,...}

Finally, since columns c3and c

8have single crosses,

rows r4and r

8must appear in the minimal cover. This

completes the covering of the prime-implicant table.

The minimal cover consists of rows r1, r

3, r

4, and r

8.

(c) Referring to Table P4.29c, c

5

dominates c

1

, c

2

equals

c6, and c

3dominates c

7. Deleting c

3, c

5, and c

6, the

table reduces to

c1

c2

c4

c7

cost

r1

3

r2

4

r3

4

r4

5

r5

5

r6

6

r7

7


73/97

- 4.73 -

4.29. (continued)

r6dominates r

1, but cost prevents deleting r

1.

However, since r2dominates r

5, r

2dominates r

7, r

3

dominates r7, and r

4dominates r

5, and since the

dominated rows have a greater or equal cost than the

dominating rows, delete r5and r

7. The table now

becomes

c1

c2

c4

c7

cost

r1

3

r2

4

r3

4

r4

5

r6

6

The resulting table cannot be reduced any further.

Petrick's method should be applied.

p = (r1+r6)(r

3+r4+r6)(r

2+r3)(r

2+ r

4)

= (r6+r1r3+r1r4)(r2+r3r4)= r

2r6+ r

3r4r6+ r

1r2r3+ r

1r3r4+ r

1r2r4+ r

1r3r4

= r2r6+ r

3r4r6+ r

1r2r3+ r

1r3r4+ r

1r2r4

Irredundant

cover

cost

r2, r

610

r3, r

4, r

615

r1, r

2, r

311

r1, r

3, r

412

r1, r

2, r

412

The minimal cost cover consists of rows r2and r

6.


74/97

- 4.74 -

4.30.

wxyz wxyz

2 0010 T 2,3 001- A

4 0100 T 2,10 -010 B

3 0011 T 4,5 010- C

5 0101 T 4,12 -100 D

10 1010 T 3,7 0-11 E

12 1100 T 5,7 01-1 F

7 0111 T 10,14 1-10 G

14 1110 T 12,14 11-0 H

15 1111 T 7,15 -111 I

14,15 111- J

m3

m4

m5

m7

m10

m12

m14

m15

cost

A 4

B 4

C 4

D 4

E 4

F 4

G 4

H 4

I 4

J 4

Row E dominates row A and row G dominates row B. Since

all rows have equal cost, delete rows A and B. The

reduced table is


75/97

- 4.75 -

4.30. (continued)

m3

m4

m5

m7

m10

m12

m14

m15

cost

C 4

D 4

**E 4

F 4

**G 4

H 4

I 4

J 4

Select prime implicants E and G for a minimal sum since

columns m3and m

10each have a single cross. Delete

rows E and G as well as the columns having crosses in

these rows. The reduced table becomes

m4

m5

m12

m15

cost

C 4

D 4

F 4

H 4

I 4

J 4

Minimal sum: f = E + G +...

Row C dominates row F, row D dominates row H, and rows

I and J are equal. Since all rows have the same cost,

delete rows F, H, and J. The resulting table is


76/97

- 4.76 -

4.30. (continued)

m4

m5

m12

m15

cost

**C 4

**D 4

**I 4

Minimal sum: f = E + G + ...

Since columns m5, m

12, and m

15each have a single

cross, rows C, D, and I are selected for a minimal sum.

Furthermore, by selecting these rows, all the columns

of the above table are covered. The minimal sum is

f = C + D + E + G + I

= w-xy-+ xy

-z-+ w-yz + wyz

-+ xyz

(Note: If row I was deleted since it was equal to row

J, then the minimal cover would consist of rows C,

D, E, G, and J. Applying Petrick's method to the

original prime-implicant table would result in 16

minimal sums of which only 2 are readily found when

using table reduction procedures.)


77/97

- 4.77 -

4.31.

1 T 1,3 (2) C 8,9,12,13 (1,4) A

8 T 1,9 (8) D 8,10,12,14 (2,4) B

3 T 8,9 (1) T

6 T 8,10 (2) T

9 T 8,12 (4) T

10 T 3,7 (4) E

12 T 6,7 (1) F

7 T 6,14 (8) G

13 T 9,13 (4) T

14 T 10,14 (4) T

12,13 (1) T

12,14 (2) T

Prime implicants:

wxyz

A: 1-0- 6 wy-

B: 1--0 6 wz-

C: 00-1 6 w-x-z

D: -001 6 x-y-z

E: 0-11 6 w-yz

F: 011- 6 w-xy

G: -110 6 xyz-


78/97

- 4.78 -

4.31. (continued)

m1

m3

m6

m8

m9

m10

m12

m14

cost

A 3

*B 3

C 4

D 4

E 4

F 4

G 4

B is an essential prime implicant. Selecting row B and

deleting the columns having crosses in row B, the table

reduces to

m1

m3

m6

m9

cost

A 3

C 4

D 4

E 4

F 4

G 4

Minimal sum: f = B + ...

Even though row D dominates row A, row A cannot be deleted

because of cost. Row C dominates row E and rows F and G

are equal. Since these rows have the same cost, delete

rows E and G. The table becomes


79/97

- 4.79 -

4.31. (continued)

m1

m3

m6

m9

cost

A 3

**C 4

D 4

**F 4

Minimal sum: f = B + ...

Since columns m3and m

6have single crosses, rows C and F

must be selected. After deleting rows C and F and the

columns having crosses in these rows, the reduced table is

m9

cost

A 3

D 4

Minimal sum: f = B + C + F + ...

Row D can be eliminated since it is equal to row A and has

a higher cost. This results in the table

m9

cost

**A 3

Minimal sum: f = B + C + F + ...

Selecting row A, the minimal sum is

f = A + B + C + F = wy-+ wz

-+ w-x-z + w

-xy

(Note: At an earlier point, row F could have been deleted

since it was equal to row G. In this case, a minimal

sum is

f = A + B + C + G = wy-+ wz

-+ w

-x-z + xyz

-)


80/97

- 4.80 -

4.32.

xyz xyz

0 000 f1_

f3

F 0,2 (2) 0-0_ _

f3A

2 010_

f2f3T 2,3 (1) 01-

_ _f3B

3 011 f1_

f3

G 2,6 (4) -10_

f2f3C

5 101 f1f2_

H 3,7 (4) -11 f1_ _

D

6 110_

f2f3T 5,7 (2) 1-1 f

1_ _

E

7 111 f1_ _

T

f1

f2

f3

m0 m3 m5 m7 m2 m5 m6 m0 m2 m3 m6

A: x-z-

B: x-y

C: yz-

D: yz

E: xz

F: x-y-z-

G: x

-

yz

H: xy-z

p = F1(D1+G1)(E

1+H1)(D

1+E1)C2H2C2(A3+F

3)(A

3+B3+C3)(B

3+G3)C3

= F1C2H2C3(D1+G1)(E

1+H1)(D

1+E1)(A

3+F3)(B

3+G3)

= F1C2H2C3(D1+G1)(E

1+D

1H1)(A

3+F3)(B

3+G3)

= F1C2H2C3(D1E1+D1H1+E1G1+D1G1H1)(A

3B3+A3G3+B3F3+F3G3)

= F1C2H2C3(D1E1A3B3+D1E1A3G3+D1E1B3F3+D1E1F3G3+D1H1A3B3

+D1H1A3G3+D1H1B3F3+D1H1F3G3+E1G1A3B3+E1G1A3G3

+E1G1B3F3+E1G1F3G3)


81/97

- 4.81 -

4.32. (continued)

= A3B3C2C3D1E1F1H2+ A

3C2C3D1E1F1G3H2+ B

3C2C3D1E1F1F3H2

+ C2C3D1E1F1F3G3H2+ A

3B3C2C3D1F1H1H2

+ A3C2C3D1F1G3H1H2+ B

3C2C3D1F1F3H1H2

+ C2C3D1F1F3G3H1H2 + A3B3C2C3E1F1G1H2

+ A3C2C3E1F1G1G3H2+ B

3C2C3E1F1F3G1H2

+ C2C3E1F1F3G1G3H2

Product terms Distinct terms Input terminals (E"i+G$

j)

A3B3C2C3D1E1F1H2

7 8 + 16 = 24

A3C2C3D1E1F1G3H2

7 8 + 17 = 25

B3C2C3D1E1F1F3H2 6 8 + 14 = 22

C2C3D1E1F1F3G3H2

6 8 + 15 = 23

A3B3C2C3D1F1H1H2

6 8 + 14 = 22

A3C2C3D1F1G3H1H2

6 8 + 15 = 23

B3C2C3D1F1F3H1H2

5 * 8 + 12 = 20 *

C2C3D1F1F3G3H1H2

5 * 8 + 13 = 21

A3B3C2C3E1F1G1H2

7 8 + 17 = 25

A3C2C3E1F1G1G3H2 6 8 + 15 = 23

B3C2C3E1F1F3G1H2

6 8 + 15 = 23

C2C3E1F1F3G1G3H2

5 * 8 + 13 = 21

There are three multiple-output minimal sums when the cost

is based on number of distinct terms:

f1= yz + x

-y-z-+ xy

-z f

1= yz + x

-y-z-+ xy

-z

f2= yz

-+ xy

-z f

2= yz

-+ xy

-z

f3= x-y + yz- + x-y-z- f

3= yz- + x-y-z- + x-yz

f1= xz + x

-y-z-+ x-yz

f2= yz

-+ xy

-z

f3= yz

-+ x-y-z-

+ x-yz


82/97

- 4.82 -

4.32. (continued)

There is only one multiple-output minimal sum when the cost

is based on number of input terminals in the realization:

f1= yz + x

-y-z-+ xy

-z

f2 = yz- + xy-z

f3= x-y + yz

-+ x-y-z-

4.33. (a)

xyz xyz xyz

0 000 f1f2_

T 0,2 (2) 0-0 f1f2_

B 0,2,4,6 (2,4) --0 f1_ _

A

2 010 f1f2_

T 0,4 (4) -00 f1_ _

T

4 100 f1_

f3T 2,3 (1) 01- f1_ _

C

3 011 f1_

f3F 2,6 (4) -10 f

1_ _

T

5 101_

f2f3G 4,5 (1) 10-

_ _f3D

6 110 f1_

f3T 4,6 (2) 1-0 f

1_

f3E

f1

f2

f3

m0

m2

m3

m4

m6

m0

m2

m5

m3

m4

m5

m6

cost

A: z- 1

*2 B: x-z-

3,4

C: x-y 3

D: xy-

3

*3 E: xz-

3,4

*3 F: x-yz 4,5

*2 G: xy-z 4,5

f1= ... f

2= x-z-+ xy

-z f

3= xz

-+ x-yz + ...


83/97

- 4.83 -

4.33. (continued)

Reducing the table:

f

1

f

3m0

m2

m3

m4

m6

m5

cost

A: z-

1

B: x-z-

1

C: x-y 3

D: xy-

3

E: xz-

1

F: x-yz 1

G: xy-z 1

f1= ... f

2= x-z-+ xy

-z f

3= xz

-+ x-yz + ...

After deleting dominated and equal rows B, E, and D,

the table becomes

f1 f3

m0

m2

m3

m4

m6

m5

cost

*1 A: z-

1

C: x-y 3

F: x-yz 1

*3 G: xy-z 1

f1 = z-+ ... f2 = x

-z-+ xy

-z f3 = xz

-+ x-yz + xy

-z


84/97

- 4.84 -

4.33. (continued)

Reducing the table:

f

1m3

cost

C: x-y 3

F: x-yz 1

f1= z-+ ... f

2= x-z-+ xy

-z f

3= xz

-+ x-yz + xy

-z

To cover the remain column, select F. The multiple-

output minimal sum is

f1= z-+ x-yz

f2= x-z-+ xy

-z

f3= xz

-+ x-yz + xy

-z

(b)

xyz xyz xyz

0 000_ _

f

3

T 0,1 (1) 00-_ _

f

3

C 1,3,5,7 (2,4) --1_

f

2

_A

1 001 f1f2f3J 0,4 (4) -00

_ _f3D 4,5,6,7 (1,2) 1--

_f2_

B

2 010 f1_ _

T 1,3 (2) 0-1_

f2_

T

4 100_

f2f3T 1,5 (4) -01 f

1f2_

E

3 011_

f2_

T 2,6 (4) -10 f1_ _

F

5 101 f1f2_

T 4,5 (1) 10-_

f2_

T

6 110 f1f2f3K 4,6 (2) 1-0

_f2f3G

7 111 f

1

f

2

_T 3,7 (4) -11

_f

2

_T

5,7 (2) 1-1 f1f2_

H

6,7 (1) 11- f1f2_

I


85/97

- 4.85 -

4.33. (continued)

f1

f2

f3

m1

m2

m5

m3

m5

m6

m7

m1

m4

m6

cost

*2 A: z 1

B: x 1

C: x-y-

3

D: y-z-

3

E: y-z 3,4

*1 F: yz-

3

G: xz-

3,4

H: xz 3,4

I: xy 3

J: x-y-z 4,5

K: xyz-

4,5

f1= yz

-+ ... f

2= z + ... f

3= ...

Reducing the table:

f

1

f

2

f

3m1

m5

m6

m1

m4

m6

cost

B: x 1

C: x-y-

3

D: y-z-

3

E: y-z 3

G: xz-

3,4

H: xz 3

I: xy 3

J: x-y-z 4,5

K: xyz-

4,5

f1= yz

-+ ... f

2= z + ... f

3= ...


86/97

- 4.86 -

4.33. (continued)

After deleting equal row I and dominated rows D, H, and

K, the table becomes

f1

f2

f3

m1

m5

m6

m1

m4

m6

cost

B: x 1

C: x-y-

3

*1 E: y-z 3

*3 G: xz-

3,4

J: x-y-z 4,5

f1= yz

-+ y

-z f

2= z + ... f

3= xz

-+ ...

Reducing the table:

f2

f3

m6

m1

cost

B: x 1

C: x

-

y

-

3

G: xz-

1

J: x-y-z 4,5

f1= yz

-+ y-z f

2= z + ... f

3= xz

-+ ...

After deleting equal rows G and J, the table becomes

f2

f3

m6

m1

cost

*2 B: x 1

*3 C: x-y-

3

f1= yz

-+ y

-z f

2= z + x f

3= xz

-+ x-y-


87/97

- 4.87 -

4.34.

w x y z (a) (b) (c) (d) (e) (f)

0 0 0 0 0 1 0 0 0 1

0 0 0 1 0 0 1 0 0 0

0 0 1 0 1 0 0 0 0 1

0 0 1 1 1 1 1 0 1 0

0 1 0 0 1 1 0 1 1 0

0 1 0 1 1 1 0 0 1 1

0 1 1 0 0 0 0 0 0 0

0 1 1 1 0 0 0 1 1 0

1 0 0 0 0 1 1 1 1 1

1 0 0 1 0 1 1 0 0 1

1 0 1 0 1 0 1 0 0 1

1 0 1 1 0 1 1 0 1 1

1 1 0 0 1 1 1 1 1 0

1 1 0 1 1 1 0 1 1 1

1 1 1 0 0 0 1 0 0 0

1 1 1 1 0 0 1 1 1 1

(a)

f = x-yz-+ xy

-+ w

-x-y


88/97

- 4.88 -

4.34. (continued)

f = (w-+x+z

-)(x+y)(x

-+y-) f = (w

-+y-+z-)(x+y)(x

-+y-)

(b)

f = x-yz + y

-z-+ xy

-+ wy

-

f = (y-+z)(w+x+y+z

-)(x-+y-)

(c)

f = x-z + wz

-+ wy


89/97

- 4.89 -

4.34. (continued)

f = (w+z)(x-+y+z

-)(w+x

-)

(d)

f = xyz + xy-z-+ wy

-z-+ wxy

-f = xyz + xy

-z-+ wy

-z-

+ wxz

f = (y-+z)(x+z

-)(w+y+z

-)(w+x)

(e)

f = yz + wy-z-+ xy

-


90/97

- 4.90 -

4.34. (continued)

f = (y-+z)(x+y+z

-)(w+x+z) f = (y

-+z)(x+y+z

-)(w+x+y)

(f)

f = wz + xy-z + x

-z-

f = (x-+z)(w+x+z-)(w+x-+y-) f = (x-+z)(w+x+z-)(w+y-+z-)


91/97

- 4.91 -

4.35.

w x y z (a) (b) (c) (d) (e)

0 0 0 0 - - 0 0 0

0 0 0 1 0 1 1 0 0

0 0 1 0 1 0 - 0 1

0 0 1 1 1 - - - 1

0 1 0 0 - - 0 0 1

0 1 0 1 1 1 1 1 0

0 1 1 0 0 1 - 1 0

0 1 1 1 0 1 1 1 -

1 0 0 0 - 0 0 - 0

1 0 0 1 0 1 0 - -

1 0 1 0 - 0 1 0 1

1 0 1 1 - 1 1 0 -

1 1 0 0 1 1 0 1 0

1 1 0 1 0 1 - 1 1

1 1 1 0 1 0 0 1 1

1 1 1 1 0 0 0 0 1

(a)

f = wz-+ x-y + w

-xy-


92/97

- 4.92 -

4.35. (continued)

f = (w-+z-)(x+y)(w+x

-+y-)

(b)

f = x-z + w-x + xy-

f = (x+z)(w-+x-+y-)


93/97

- 4.93 -

4.35. (continued)

(c)

f = w-z + x

-y

f = (w+z)(w-+x-)(w-+y)

f = (y+z)(w-+x-)(w-+y)


94/97

- 4.94 -

4.35. (continued)

(d)

f = w-xz + xyz

-+ wy

-

f = x(w+y+z)(w-+y-+z-)

(e)

f = wz + w-xy-z-+ wy + x

-y


95/97

- 4.95 -

4.35. (continued)

f = (w+x-+z-)(w-+y+z)(x+y)(w+x

-+y-)

f = (w+y+z-)(w-+y+z)(x+y)(w+x

-+y-)

4.36. (a)

f = Ax-z + Bxz

-+ x-yz


96/97

- 4.96 -

4.36. (continued)

(b)

f = Ay-+ B-x-z + Bx

-y + xy

-z-

(c)

f = ABz + Ayz + A-x-y

4.37. (a)

f = yz-+ w-y + xz

-+ wx


97/97

4.37. (continued)

(b)

f = v-wy-z-+ vxyz

-+ w-xz + v

-xy-

+ v-wx

(c)

f = x-yz + vz

-+ wy

-z

Chapter 4 Problem Solutions 4.1. w 0 0 0 0

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Transcript of Chapter 4 Problem Solutions 4.1. w 0 0 0 0