2014.3.10 @stu.hirosaki-u.ac.jp

1 1 1.1 2 3 ( 1) x ( ) 0 1 ( 2)NOT 0 NOT 1 1 NOT 0 ( 3)AND 1 AND 1 3 AND 0 ( 4)OR 0 OR 0 3 OR 1 0 1 x NOT x x AND x x OR x + 1 1 0 x x 1 x 0 x 0 x 1 1.2 n ( ) 1 ( ) n x 1, x 2,..., x n {0, 1} = f(x 1, x 2,..., x n ) 1 AND x

2 1 0, 1 2 n 1.1 3 1.1: 3 (a) NOT (b) AND (c) OR x = x 0 1 1 0 x 1 x 2 = x 1 x 2 0 0 0 0 1 0 1 0 0 1 1 1 x 1 x 2 = x 1 + x 2 0 0 0 0 1 1 1 0 1 1 1 1 = x 1 x 2 x 1 x 2 = x 1 x 2 0 0 0 0 1 0 1 0 1 1 1 0 x 1 x 2 x 1, x 2 1.3 NOT, AND, OR 1.1 x (a)not (b)and (c)or 1.1: 1.2:

1.4. 3 = x 1 x 2 1.2 ( ) NOT x 0 1 = x ( ) AND x 1 x 2 = x 1 x 2 0 0 0 1 1 0 1 1 ( ) OR x 1 x 2 = x 1 + x 2 0 0 0 1 1 0 1 1 ( ) 1.2 x 1 x 2 = x 1 x 2 0 0 0 1 1 0 1 1 1.4 NOT, AND, OR 3 3 AND ( = x 1 x 2 x 3 = (x 1 x 2 ) x 3 ) 2 AND ( 1.3) NAND NOR 1.3(b)(c) exclusive OR( = x 1 x 2 + x 1 x 2 ) (d) 1.7 NAND NOT AND OR NOR NOT AND OR NAND NOR 1

4 1 x3 x3 (a)3 AND (b)nand (c)nor (d)exclusive OR 1.3: ( ) 3 AND 1.3(a) x 1 x 2 x 3 = x 1 x 2 x 3 = (x 1 x 2 ) x 3 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 ( ) NAND 1.3(b) ( ) NOR 1.3(c) ( ) exclusive OR 1.3(d)

1.5. 5 1.5 1.4 NOT AND OR 1 High( ) 0 Low( ) x (a) NOT (b) AND (c) OR 1.4: - 2 NOT (TI SN7404) 1.5 SN74LS04 SN74LS04 High 2.4V Low 0.4V 2.0V High 0.8V Low ( 1.2) Vo, output voltage, V Guaranteed output range for logical 1 2.4 Guaranteed output range for logical 0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.4 0.5 Permissible input range for logical 0 Measured transfer characteristics for a tpical gate at Vcc=5.0 and fan-out=10 0.4 0.8 1.2 1.6 2.0 2.4 2.8 Vi, input voltage, V Permissible input range for logical 1 1.5: NOT SN7404 2 1.4

6 1 1.2: NOT SN74LS04 ( ) MIN NOM MAX UNIT TEST CONDITIONS V CC, Suppl voltage 4.5 5 5.5 V I OH, High-level output current 400 µa I OL, Low-level output current 8 ma V IH, High-level input voltage 2 V V IL, Low-level input voltage 0.8 V V OH, High-level output voltage 2.7 3.4 V V CC =MIN, V IL = V IL Max, I OH =MAX V OL, Low-level output voltage 0.4 V V CC =MIN, V IH =2V, I OL =4mA I IH, High-level input current 20 µa V CC =MAX, V IH =2.7V I IL, Low-level input current 0.4 ma V CC =MAX, V IL =0.4V ( ) NOT (SN74LS04) 0 1 ( ) 1.4(a) (c) 0,1 ( ) 1.4(a) NOT 1.5 1.6 IC(Integrated Circuit: ) 6 NOT 4 AND 4 OR 1 IC IC SSI(Small Scale Integrated circuit) IC MSI, LSI, LSI ( 1.3) Pentium III 950 Pentium IV 4200 LSI 1.3: IC IC SSI MSI LSI LSI 1 IC 30 30 1000 1000 10 10 1.7 1.4

1.7. 7 1.4: 1 + x = 1 1 x = x 0 + x = x 0 x = 0 x + x = x x (x + ) = x x + = + x x = x x + ( + z) = (x + ) + z x ( z) = (x ) z x ( + z) = x + x z x + z = (x + ) (x + z) x + x = x x x = x x + x = 1 x x = 0 x = x x + = x x = x + (a)nand (b)nor 1.6: NAND 1.6(a) NOR 1.6(b) 1.7 NAND NOR NOT NAND NOT, AND, OR NOR NOT, AND, OR x x x x (a)nand NOT x x (b)nor NOT 1.7: NAND NOR NOT 1.8 NOT, AND, OR 1.9 ( ) 1.4

8 1 (a)not: x (b)and: x 1 x 2 (c)or: x 1 + x 2 1.8: (a)x 1 + x 2 (b)x 1 + x 2 (c)x 1 (d)x 2 (e)x 1 x 2 1.9: ( ) NAND NOR exclusive OR ( ) 1.6(a) (b) ( ) NAND NOT, AND, OR ( ) NOR NOT, AND, OR ( ) x + = x x = x + 1.6 AND OR ( ) ( )

9 2 ( 2.1) 2.1: (1) (2) (3) (4) 2.1 = f(x 1, x 2,..., x n ) x 1 = 0 x 1 = 1 1.4 f(x 1, x 2,..., x n ) = (x 1 + x 1 ) f(x 1, x 2,..., x n ) = x 1 f(x 1, x 2,..., x n ) + x 1 f(x 1, x 2,..., x n ) = x 1 f(0, x 2,..., x n ) + x 1 f(1, x 2,..., x n ) 1 f(x 1 ) = (x 1 + x 1 ) f(x 1 ) = x 1 f(x 1 ) + x 1 f(x 1 ) = x 1 f(0) + x 1 f(1)

10 2 2 f(x 1, x 2 ) = (x 1 + x 1 ) f(x 1, x 2 ) = x 1 f(x 1, x 2 ) + x 1 f(x 1, x 2 ) = x 1 f(0, x 2 ) + x 1 f(1, x 2 ) = x 1 (x 2 f(0, 0) + x 2 f(0, 1)) + x 1 (x 2 f(1, 0) + x 2 f(1, 1)) = x 1 x 2 f(0, 0) + x 1 x 2 f(0, 1) + x 1 x 2 f(1, 0) + x 1 x 2 f(1, 1) f(x 1, x 2,..., x n ) = x 1 x 2... x n f(0, 0,..., 0) +x 1 x 2... x n f(0, 0,..., 1)... +x 1 x 2... x n f(1, 1,..., 1) 2 n f(0, 0,..., 0), f(0, 0,..., 1), f(1, 1,..., 1) {0, 1} f(...) = 0 1. 1 2. 0 x i 1 x i 3. 2.2(a) (b) 1.2 x 1 x 2 x 3 1( ) 0( ) = 1 x 1 = 0, x 2 = 0, x 3 = 1 = 1 OK = 1 1 3 1. = 1 x 1 = 0, x 2 = 0, x 3 = 1 x 1 = 1, x 2 = 0, x 3 = 0

2.1. 11 x 1 x 2 x 3 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 (a) (b) 2.2: x 1 = 1, x 2 = 0, x 3 = 1 x 1 = 1, x 2 = 1, x 3 = 0 x 1 = 1, x 2 = 1, x 3 = 1 2. x i = 0 x i x i = 1 x i x 1 x 2 x 3, x 1 x 2 x 3, x 1 x 2 x 3, x 1 x 2 x 3, x 1 x 2 x 3 3. ( ) = x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 2.3 ( ) x 1 x 2 0 0 0 0 1 1 1 0 1 1 1 0 (1) (2)

12 2 x3 x3 x3 2.3: = f(x 1, x 2,..., x n ) x 1 = 0 x 1 = 1 f(x 1, x 2,..., x n ) = x 1 x 1 + f(x 1, x 2,..., x n ) = (x 1 + f(x 1, x 2,..., x n )) ( x 1 + f(x 1, x 2,..., x n )) = (x 1 + f(0, x 2,..., x n )) (x 1 + f(1, x 2,..., x n )) 1 f(x 1 ) = x x 1 + f(x 1 ) = (x 1 + f(x 1 )) (x 1 + f(x 1 )) = (x 1 + f(0)) (x 1 + f(1)) 2 f(x 1, x 2 ) = x 1 x 1 + f(x 1, x 2 ) = (x 1 + f(x 1, x 2 )) (x 1 + f(x 1, x 2 )) = (x 1 + f(0, x 2 )) (x 1 + f(1, x 2 )) = (x 1 + (x 2 + f(0, 0)) (x 2 + f(0, 1))) (x 1 + (x 2 + f(1, 0)) (x 2 + f(1, 1))) = (x 1 + x 2 + f(0, 0)) (x 1 + x 2 + f(0, 1)) (x 1 + x 2 + f(1, 0)) (x 1 + x 2 + f(1, 1)) f(x 1, x 2,..., x n ) = (x 1 + x 2 +... + x n + f(0, 0,..., 0))

2.1. 13 (x 1 + x 2 +... + x n + f(0, 0,..., 1))... (x 1 + x 2 +... + x n + f(1, 1,..., 1)) 2 n f(0, 0,..., 0), f(0, 0,..., 1), f(1, 1,..., 1) {0, 1} f(...) = 1 1. 0 2. 0 x i 1 x i 3. x 1 x 2 x 3 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 3 1. = 0 x 1 = 0, x 2 = 0, x 3 = 0 x 1 = 0, x 2 = 1, x 3 = 0 x 1 = 0, x 2 = 1, x 3 = 1 2. x i = 0 x i x i = 1 x i x 1 + x 2 + x 3, x 1 + x 2 + x 3, x 1 + x 2 + x 3 ( ) 3. = (x 1 + x 2 + x 3 ) (x 1 + x 2 + x 3 ) (x 1 + x 2 + x 3 )

14 2 x3 x3 x3 2.4: 2.4 ( ) x 1 x 2 0 0 0 0 1 1 1 0 1 1 1 0 (1) (2) 2.2 2.2(a) ( )

2.2. 15 x 1 x 2 x 3 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 1* 1 0 1 1 1 1 0 1 1 1 1 1 2 1.4 x 1 x 2 x 3 + x 1 x 2 x 3 = (x 1 + x 1 ) x 2 x 3 = x 2 x 3 4 x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 =... = x 1 1.4 x 1 x 2 x 3 = x 1 x 2 x 3 + x 1 x 2 x 3 = x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 = (x 1 x 2 x 3 + x 1 x 2 x 3 ) + (x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 ) = x 2 x 3 + x 1 (Karnaugh) ( 2.5) = 1 x 2 = 0, x 3 = 1 x 1 = 1 x 1 {}}{ x 1 x 2 00 01 11 10 x 3 0 1 1 { x 3 1 1 1 1 } {{ } x 2 2.5: 2.2(a) (1)

16 2 (2) 1 2 n (3) 1 (2) (4) 2.5 4 1 2 1 x 1 x 2 x 3 = x 1 + x 2 x 3 2.6 2.2(a) 2.3 2.4 x3 2.6: 2.2(a) 4 3.11 6 Quine-McCluske ( ) 2.6 2.2(a) 2.3 (Half Adder) (Full Adder) 2.7(a) S = A B + A B C = A B 2.7(b) XOR(eXclusive OR) 2.7(c) 2.8(a)

2.3. 17 A B S C 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 (a) C A B (b)not AND OR A B (c)exclusive OR C S S 2.7: S = A B C i + A B + C i + A B C i + A B C i C o = A B C i + A B C i + A B C i + A B C i S C o C o = A B + B C i + C i A 2.8(b) exclusive OR 8 1 7 2.9 2 8 A 7...A 1 A 0 B 7...B 1 B 0 S 7...S 1 S 0 ( ) 2.7(c) 1 ( ) 2.8(b) 1 ( ) 2.9 2 2

18 2 A B C i S C o 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 (a) A B Ci (b) Co S 2.8: 2.9: 8 8 1 0 ( 2.10) 2.10:

2.3. 19 n 2 n 8 3 ( ) 3 8 1 000 111 ( ) 000 111 8 ( 1) 3 2.11(a)(b) 3 2.11(c)(d) x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 1 2 3 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 (a) x3x4 x5x6x7x8 (b) 1 12 23 3 1 2 3 z1 z2 z3 z4 z5 1 2 3 z 1 z 2 z 3 z 4 z 5 z 6 z 7 z 8 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 1 (c) 1 2 3 (d) z6 z7 z8 2.11: ( ) 2 (1) 2 (2) 2 (3)

20 2 7 7 LED 7 0 9 ( 2.12) a f g b e c d 2.12: 7 LED 3 (7 ) 1 2 3 a b c d e f g 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 1 1 0 1 1 0 1 2 0 1 1 1 1 1 1 0 0 1 3 1 0 0 0 1 1 0 0 1 1 4 1 0 1 1 0 1 1 0 1 1 5 1 1 0 1 0 1 1 1 1 1 6 1 1 1 1 1 1 0 0 1 0 7 a d 2.13 ( ) 2.13 e g e g 2.14(a) ( ) 2 (7 ) (1) 1, 2 a g (2) a g 1, 2 (3) (4)

2.3. 21 1 {}}{ 1 2 3 00 01 11 10 { 0 1 1 1 3 1 1 1 1 }{{} 2 a = 2 + 1 3 + 1 3 1 {}}{ 1 2 3 00 01 11 10 { 0 1 1 1 3 1 1 1 1 }{{} 2 b = 1 + 2 3 + 2 3 1 {}}{ 1 2 3 00 01 11 10 { 0 1 1 1 3 1 1 1 1 1 }{{} 2 c = 1 + 2 + 3 1 {}}{ 1 2 3 00 01 11 10 { 0 1 1 1 3 1 1 1 }{{} 2 d = 1 2 + 1 3 + 2 3 + 1 2 3 2.13: 7 (a d)

22 2 1 12 23 3 a b c d e 1 2 3 f g 1 2 3 a b c d e f g e f d g a c b (a) (a d) (b) 2.14: 3-7 1 2 a b c d e f g a f g b e c d 2.15: 2-7

23 3 3.1 1 ( ) n 2 n RS 3.1(a) RS (b) (c) (d) R n = 1, S n = 1 (a)rs (b) (c) ( ) R n S n Q n 0 0 Q n 1 0 1 1 1 0 0 1 1 (d) 3.1: RS ( ) 3.1(b) RS (c)

24 3 JK 3.2(a) JK Q T J K 3.2(b) J K Q n+1 0 0 Q n 0 1 0 1 0 1 1 1 Q n (a)jk (b) 3.2: JK ( ) JK

3.1. 25 J K Q n Q n+1 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 3.2(b) T 3.3(a) T (b) JK (c) (d) Q ( ) 2 ( 2 1) T (a)t (b) (c) ( ) Q n Q n+1 0 1 1 0 (d) 3.3: T ( ) 3.3(b) T (c) D 3.4(a) D (b) JK (c) (d) D ( ) ( ) 3.4(b) D (c)

26 3 (a)d (b) (c) ( ) D n Q n+1 0 0 1 1 (d) 3.4: D 3.2 T n 2 n 10 2 n ( ) 3.5(a) 4 10Hz ( ) T Q A, Q B 2.15 2-7 LED (a)4 (b) ( ) 3.5: 7 LED 3.6 ( 2 ) LED LED a g 7 7 D

3.2. 27 LED1 a g T1 LED2 a g T2 D T Q a g LED 10 7 10 17 LED LED 7 10=70 a b c d e f g a f g b e c d a f g b e c d 3.6: 100 ( ) 100 ( ) ( ) 2.11 2 n n 128 7 - - 3.7 ( ) 3.8(a) 4 (b) -

28 3 3.7: - (a)4 (b) ( ) 3.3 3.8: - 3.9(a) n 2 n 3.9(a) (b)(c) (c) (b)

3.3. 29 (a) (b) (c) 3.9:...150...50 100 ( ) 3.10 S 0 : S 1 : 50 S 2 : 100 x 1, x 2 / 1, 2 : / x 1 : 100 x 2 : 50 1 : 2 : 3.10: 3.10 3.1(a) ( ) 3 2bit 2 D

30 3 Q 1, Q 2 S 0 = 00, S 1 = 01, S 2 = 10( S i = Q 1 Q 2 ) 3.9 3.1(a) 1, 2 D 1, D 2 4 3.1(a) (b) D 1 D 2 / 1 2 3.1: (a) (b) x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 00 01 10 11 00 01 10 11 Q 1 Q 2 S 0 S 0 /0,0S 1 /0,0S 2 /0,0 00 00/00 01/00 10/00 S 1 S 1 /0,0S 2 /0,0S 0 /1,0 01 01/00 10/00 00/10 S 2 S 2 /0,0S 0 /1,0S 0 /1,1 10 10/00 00/10 00/11 3.2 Q 1 Q 2 = 11 ( ) x 1 x 2 = 11 ( ) 0 1 (Don t Care Event) 0 1 0 D 1 = Q 1 Q 2 x 1 x 2 + Q 1 Q 2 x 1 x 2 + Q 1 Q 2 x 1 x 2 D 2 = Q 1 Q 2 x 1 x 2 + Q 1 Q 2 x 1 x 2 1 = Q 1 Q 2 x 1 x 2 + Q 1 Q 2 x 1 x 2 + Q 1 Q 2 x 1 x 2 2 = Q 1 Q 2 x 1 x 2 3.11 1 D 1 = Q 2 x 2 + Q 1 Q 2 x 1 + Q 1 x 1 x 2 D 2 = Q 1 Q 2 x 2 + Q 2 x 1 x 2 1 = Q 1 x 2 + Q 2 x 1 + Q 1 x 1 2 = Q 1 x 1 3.12

3.3. 31 3.2: Q 1 Q 2 x 1 x 2 D 1 D 2 1 2 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 1 1 1 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 ( ) 3.1 ( A) B F ( S i Q 1 Q 2 ) S 0 S 1 S 2 A( 3.1) 00 01 10 B 00 01 11 C 00 10 01 D 00 10 11 E 00 11 01 F 00 11 10 G 01 00 10 H 01 00 11 I 10 00 01 J 10 00 11

32 3 x 1 x 2 00 01 11 10 Q 1 Q 2 00 0 0 1 01 0 1 0 11 10 1 0 0 x 1 x 2 00 01 11 10 Q 1 Q 2 00 0 1 0 01 1 0 0 11 10 0 0 0 D 1 = Q 2 x 2 + Q 1 Q 2 x 1 + Q 1 x 1 x 2 D 2 = Q 1 Q 2 x 2 + Q 2 x 1 x 2 (1) D 1 (2) D 2 x 1 x 2 00 01 11 10 Q 1 Q 2 00 0 0 0 01 0 0 1 11 10 0 1 1 x 1 x 2 00 Q 1 Q 2 01 11 10 00 0 0 0 01 0 0 0 11 10 0 0 1 1 = Q 1 x 2 + Q 2 x 1 + Q 1 x 1 2 = Q 1 x 1 (3) 1 (4) 2 3.11: ( ) 3.10

3.3. 33 Q 1Q 1Q 2Q 2 1 2 D 1 D 2 3.12:

35 A1

36 3 A2