4

4.8 a) F = Sum(0,1,2,3,5,6,7,10,11,15)

`G0`	`0000`	yes
`G1`	0001 `0010`	yes `yes`
`G2`	0011 `0101 0110 1010`	yes `yes yes yes`
`G3`	`0111 1011`	yes `yes`
`G4`	1111	yes

`G0`	000- `00-0`	yes `yes`
`G1`	00-1 `0-01 -010 0-10 001-`	yes `yes yes yes yes`
`G2`	0-11 `-011 01-1 011- 101-`	yes `yes yes yes yes`
`G3`	-111 `1-11`	yes `yes`

`G0`	00--	no	P0	a'b'
`G1`	0--1 `-01- 0-1-`	no `no no`	P1 `P2 P3`	a'd `b'c a'c`
`G2`	--11	no	P4	cd

	`m0`	m1	m2	m3	m5	m6	m7	m10	m11	m15
`P0`	x	x	x	x
`P1`		x		x	x		x
`P2`			x	x				x	x
`P3`			x	x		x	x
`P4`				x			x		x	x

All PI are Essential Prime Implicates
F = P0 + P1 + P2 + P3 + P4
= a'b' + a'd + b'c + a'c + cd

4.8 b) F = Sum(0,1,2,3,6,7,8,9,12,13,14,15)

`G0`	0000	yes
`G1`	0001 `0010 1000`	yes `yes yes`
`G2`	0011 `0110 1001 1100`	yes `yes yes yes`
`G3`	0111 `1101 1110`	yes `yes yes`
`G4`	1111	yes

`G0`	000- `00-0 -000`	yes `yes yes`
`G1`	00-1 `-001 001- 0-10 100- 1-00`	yes `yes yes yes yes yes`
`G2`	0-11 `011- -110 1-01 110- 11-0`	yes `yes yes yes yes yes`
`G3`	-111 `11-1 111-`	yes `yes yes`

`G0`	00-- `-00-`	no `no`	P1 `P2`	a'b' `b'c'`
`G1`	0-1- `1-0-`	no `no`	P3 P4	a'c `ac'`
`G2`	-11- `11--`	no `no`	P5 `P6`	bc `ab`

m0

m1

m2

m3

m6

m7

m8

m9

m12

m13

m14

m15

P1

P2

P3

P4

P5

P6

no Essential Prime Implicates
Two partitions without overlap: P1 + P4 + P5 and P2 + P3 + P6
F = a'b' + ac' + bc
and 
= b'c' + a'c + ab

4.8 c) F = Sum(0,2,4,5,10,11,13,15)

`G0`	0000	y
`G1`	0010 `0100`	y `y`
`G2`	0101 `1010`	y `y`
`G3`	1011 `1101`	y `y`
`G4`	1111	y

`G0`	00-0 `0-00`	n `n`	P0 `P1`	a'b'd' `a'c'd'`
`G1`	-010 `010-`	n `n`	P2 `P3`	b'cd' `a'bc'`
`G2`	-101 `101-`	n `n`	P4 `P5`	bc'd `ab'c`
`G3`	1-11 `11-1`	n `n`	P6 `P7`	acd `abd`

	`m0`	m2	m4	m5	m10	m11	m13	m15
`P0`	x	x
`P1`	x		x
`P2`		x			x
`P3`			x	x
`P4`				x			x
`P5`					x	x
`P6`						x		x
`P7`							x	x

no essential prime implicates
Two partitions without overlap: P0+P3+P5+P7 and P1+P2+P4+P6
F = a'b'd' + a'bc' + ab'c + abd
and 
= a'c'd' + b'cd' + bc'd + acd

4.8 d) F = Sum(1,2,3,6,7,8,9,12,14)

`G1`	0001 `0010 1000`	y `y y`
`G2`	0011 `0110 1001 1100`	y `y y y`
`G3`	0111 `1110`	y `y`

`G1`	00-1 `-001 001- 0-10 100- 1-00`	n `n y y n n`	P0 `P1 P2 P3`	a'b'd `b'c'd ab'c' ac'd'`
`G2`	0-11 `011- -110 11-0`	y `y n n`	P4 `P5`	bcd' `abd'`

G1

0-1-

P6

a'c

	`m1`	m2	m3	m6	m7	m8	m9	m12	m14
`P0`	x		x
`P1`	x						x
`P2`						x	x
`P3`						x		x
`P4`				x					x
`P5`								x	x
`P6`		x	x	x	x

P6 is the only Essential Prime Implicate
F = P6 + P5 + P2 + (P0 or P1)
and
= P6 + P1 + P3 + (P4 or P5)
F = a'c + abd' + ab'c' + a'b'd
= a'c + abd' + ab'c' + b'c'd
= a'c + b'c'd + ac'd' + bcd'
= a'c + b'c'd + ac'd' + abd'

4.10)

4-19) Hazard Free Design

4) F = S m(0,3,7,11,13,15), D = S m(8,10)

		cd
ab		00	01	11	10
	00	1	0	1	0
	01	0	0	1	0
	11	0	1	1	0
	10	-	0	1	-

minimal cover = cd + abd + b'c'd'
a) Implement with 8:1 mux.
Choose bcd as selector lines:

	a
bcd	0	1	L(a)
000	1	-	1	=L0
001	0	0	0	=L1
010	0	-	0	=L2
011	1	1	1	=L3
100	0	0	0	=L4
101	0	1	a	=L5
110	0	0	0	=L6
111	1	1	1	=L7

b) Implement with 4:1 mux.
Choose cd as selector lines:

	ab
cd	00	01	10	11	L(ab)
00	1	0	-	0	b'	=L0
01	0	0	0	1	ab	=L1
10	0	0	-	0	0	=L2
11	1	1	1	1	1	=L3

4c) Implement with 2:1 muxes.
Choose d as selector line:

	abc
d	000	001	010	011	100	101	110	111	L(abc)
0	1	0	0	0	-	-	0	0	b'c'	=L0
1	0	1	0	1	0	1	1	1	c + ab	=L1

Choose c as selector line for L1.

4d) Implement with 3:8 decoder and two-level AND-OR network.
Use abc as selector: