Concise introduction to logic 13th edition hurley solutions manual

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Exercise 7.1, I

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Chapter 7 Exercise 7.1 Part I 1. G

1, 2, MT

2. M

1, 2, MP

3. E  D

1, 2, HS

4. C

1, 2, DS

5. K

1, 3, MP

6. S

2, 3, MT

7. F  D

1, 3, HS

8. N

2, 3, DS

9. N

2, 4, MT

10.

GA

1, 4, HS

11.

A

2, 4, MP

12.

S

2,3, DS

13.

C

1, 3, MT

14.

N

2, 4, DS

15.

C  T

1, 3, HS

16.

P

1, 2, MP

17.

AC

1, 4, DS

18.

(R  M)  (M  E)

19.

(S  C) 1, 3, MT

1, 3, HS

© 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

1

Exercise 7.1, I

20.

AN

2, 4, MP

Part II 1. B

11. F  G 1, 2, DS

2. N

12. M

1, 2, MP 3. T 4. R  C

2, 4, DS 13. S

1, 2, MT

14. F  J

1, 2, HS 5. N 6. W  T

2, 4, HS 16. Z

2, 3, HS

1, 4, DS 3, 4, MP

17. H  (E • R) 2, 3, MP

18. (M • G)

1, 3, DS

3, 4, HS 2, 4, MT

19. H  G

9. S 10. A

3, 4, MT

15. S 1, 3, MT

7. Q 8. C

2, 4, HS

3, 4, MP

20. (H • A)

1, 4, MT

2, 4, MP 1, 4, DS

Part III (1) 1. C  (A  C) 2. C 3. A  C 4. A

/ A 1, 2, MP 2, 3, MT

(2) 1. F  (D  T) 2. F 3. D 4. D  T 5. T

/T 1, 2, DS 3, 4, MP

(3) 1. (K • B)  (L  E) 2 (K • B) 3 E / L 4. L  E 1, 2, DS 5. L 3, 4, MT © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

2

Exercise 7.1, III

(4) 1. P  (G  T) 2. Q  (T  E) 3. P 4. Q 5. G  T 6. T  E 7. G  E

/GE 1, 3, MP 2, 4, MP 5, 6, HS

(5) 1. W  [W  (X  W)] 2. W / X 3. W  (X  W) 1, 2, MP 4. X  W 2, 3, MP X 5.

2, 4, MT (6) 1. J  (K  L) 2. L  J 3. L 4. J 5. K  L 6. K

/ K 2, 3, DS 1, 4, MP 3, 5, MT

(7) 1. S  D 2. S  (D  K) 3. D 4. S 5. D  K 6. K

/K 1, 3, MT 2, 4, DS 3, 5, MP

(8) 1. A  (E  F) 2. H  (F  M) 3. A 4. H 5. E  F 6. F  M 7. E  M

/EM 1, 3, MP 2, 4, DS 5, 6, HS

(9) 1. G  (G  A) 2. A  (C  A) 3. G 4. G  A 5. A 6. C  A 7. C

/ C 1, 3, MP 3, 4, DS 2, 5, MP 5, 6, MT

(10) 1. N  (J  P) © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

3

Exercise 7.1, III

2. (J  P)  (N  J) 3. N /P 4. J  P 1, 3, MP 5. N  J 2, 4, MP 6. N  P 4, 5, HS 7. P 3, 6, MP (11) 1. G  [O  (G  D)] 2. O  G 3. O /D 4. G 2, 3, DS 5. O  (G  D) 1, 4, MP 6. G  D 3, 5, MP 7. D 4, 6, MP (12) 1. M  (B  T) 2. B  W 3. M 4. W 5. B  T 6. B 7. T

/ T 1, 3, DS 2, 4, MT 5, 6, DS

(13) 1. R  (G  A) 2. (G  A)  S 3. G  S 4. R 5. G  A 6. S 7. G 8. A

/ A 1, 4, MP 2, 5, MP 3, 6, MT 5, 7, DS

(14) 1. (L  N)  C 2. (L  N)  (P  E) 3. E  C 4. C / P 5. (L  N) 1, 4, MT 6. P  E 2, 5, DS 7. E 3, 4, MT 8. P 6, 7, MT (15) 1. J  [A  (D  A)] 2. J  A 3. J / D A 4.  (D  A) 1, 3, MP © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

4

Exercise 7.1, III

5. A 6. D  A 7. D

2, 3, DS 4, 5, MP 5, 6, MT

(16) 1. (B  M)  (T  S) 2. B  K 3. K  M 4. S  N /TN 5. B  M 2, 3, HS S 6. T  1, 5, MP 7. T  N 4, 6, HS (17) 1. H  (Q  F) 2. R  (Q  R) 3. R  H 4. R 5. H 6. Q  F 7. Q  R 8. Q 9. F

/F 3, 4, DS 1, 5, DS 2, 4, DS 4, 7, MT 6, 8, DS

(18) 1. A  (B  C) 2. D  (C  A) 3. D  A 4. D 5. A 6. B  C 7. C  A 8. B  A 9. B

/B 3, 4, DS 1, 5, MP 2, 4, MP 6, 7, HS 5, 8, MT

(19) 1. G  [G  (S  G)] 2. (S  L)  G 3. S  L /L G 4.

2, 3, MP 5. G  (S  G) 1, 4, MP 6. S  G 4, 5, DS 7. S 4, 6, MT 8. L 3, 7, DS (20) 1. H  [E  (C  D)] 2. D  E 3. E  H © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

5

Exercise 7.1, III

4. E 5. H 6. E  (C  D) 7. C  D 8. C  E 9. C

/ C 3, 4, DS 1, 5, MP 4, 6, MP 2, 7, HS 4, 8, MT

(21) 1. B  [(A  K)  (B  K)] 2. J  K 3. A  J 4. B / A 5. (A  K)  (B  K) 1, 4, MP 6. A  K 2, 3, HS 7. B  K 5, 6, MP 8. K 4, 7, DS 9. A 6, 8, MT (22) 1. (C  M)  (N  P) 2. (C  N)  (N  M) 3. (C  P)  M 4. C  N / C 5. N  M 2, 4, MP 6. C  M 4, 5, HS 7. N  P 1, 6, MP 8. C  P 4, 7, HS 9. M 3, 8, MP 10. C 6, 9, MT (23) 1. (R  F)  [(R  G)  (S  Q)] 2. (Q  F)  (R  Q) 3. G  F 4. Q  G /SF 5. Q  F 3, 4, HS 6. R  Q 2, 5, MP 7. R  F 5, 6, HS 8. (R  G)  (S  Q) 1, 7, MP 9. R  G 4, 6, HS 10. S  Q 8, 9, MP 11. S  F 5, 10, HS (24) 1. A  [A  (T  R)] 2. R  [R  (A  R)] 3. (T  D)  R 4. T  D 5. R

/D 3, 4, MP

© 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

6

Exercise 7.1, III

6. R  (A  R) 7. A  R 8. A 9. A  (T  R) 10. T  R 11. T 12. D

2, 5, MP 5, 6, DS 5, 7, MT 1, 8, MP 8, 9, DS 5, 10, MT 4, 11, DS

(25) 1. N  [(B  D)  (N  E)] 2. (B  E )  N 3. B  D 4. D  E / D 5. B  E 3, 4, HS 6. N 2, 5, MP 7. (B  D)  (N  E) 1, 6, MP 8. N  E 3, 7, MP 9. E 6, 8, DS 10. D 4, 9, MT

Part IV (1) 1. O  (P  T) 2. P 3. O 4. P  T 5. T

/T 1, 3, MP 2, 4, DS

(2) 1. P  L 2. S  L 3. S 4. L 5. P

/ P 2, 3, DS 1, 4, MT

(3) 1. (Q  J)  (M  D) 2. Q  M 3. M  J / Q  D 4. Q  J 2, 3, HS 5. M  D 1, 4, MP 6. Q  D 2, 5, HS (4) 1. (R  L)  (L  F) 2. F  (R  L) 3. F / R 4. R  L 2, 3, DS © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

7

Exercise 7.1, IV

5. L  F 6. L 7. R

1, 4, MP 3, 5, MT 4, 6, MT

(5) 1. C  (H  R) 2. S  (R  E) 3. C 4. S 5. H  R 6. R  E 7. H  E

/HE 1, 3, DS 2, 4, DS 5, 6, HS

(6) 1. D  (L  F) 2. D  F 3. F 4. D 5. L  F 6. L

/ L 2, 3, MT 1, 4, DS 2, 5, MT

(7) 1. H  (D  A) 2. V  (R  V) 3. R  H 4. V 5. R  V 6. R 7. H 8. D  A

/DA 2, 4, DS 4, 5, MT 3, 6, DS 1, 7, MP

(8) 1. R  B 2. (I  H)  R 3. L  (I  T) 4. T  H 5. L 6. I  T 7. I  H 8. R 9. B

/B 3, 5, MP 4, 6, HS 2, 7, MP 1, 8, DS

(9) 1. (D  C)  (N  W) 2. D  S 3. S  C 4. N /W 5. D  C 2, 3, HS 6. N  W 1, 5, MP 7. W 4, 6, DS © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

8

Exercise 7.1, IV

(10) 1. C  [C  (J  D)] 2. C  (C • U) 3. (C • U) 4. D / J 5. C 2, 3, MT 6. C  (J  D) 1, 5, MP 7. J  D 5, 6, DS 8. J 4, 7, MT

Exercise 7.2 Part I 1. B

2

6. A  N

1, 3

2. T  Q

1, 3

7. Q  K

1

3. D • W

1, 2

8. (E • G)  T

1

4. H  F

1

9. B • (F  N)

1, 2

5. R

1

10. L  M

1, 2

1. G

2, Simp 3, Add

6. N  F

1, Add 3, 4, CD

2. E

1, 2, DS 1, 3, Conj

7. F

2, 3, MT 1, 4, Conj

3. (B  N) • (K  R) 1, 3, Conj 2, 4, CD

8. E  B

1, Simp 3, 4, HS

4. T  U

1, Add 3, 4, MP

9. H • R

2, 3, MP Simp

5. S • P

2, 3, DS 4, Simp

10. M • E

1, 3, Conj 2, 4, MP

Part II

Part III (1) 1. M  Q 2. R  T © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

9

Exercise 7.2, III

3. M  R 4. (M  Q) • (R  T) 5. Q  T

/ Q  T 1, 2, Conj 3, 4, CD

(2) 1. N  (D • W) 2. D  K 3. N 4. D • W 5. D 6. K 7. N • K

/N•K 1, 3, MP 4, Simp 2, 5, MP 3, 6, Conj

(3) 1. E  (A • C) 2. A  (F • E) 3. E 4. A • C 5. A 6. F • E 7. F

/F 1, 3, MP 4, Simp 2, 5, MP 6, Simp

(4) 1. (H  B)  R 2. (H  M)  P 3. H 4. H  B 5. R 6. H  M 7. P 8. R • P

/R•P 3, Add 1, 4, MP 3, Add 2, 6, MP 5, 7, Conj

(5) 1. G  (S • T) 2. (S  T)  J 3. G 4. S • T 5. S 6. S  T 7. J

/J 1, 3, MP 4, Simp 5, Add 2, 6, MP

(6) 1. (L  T)  (B • G) 2. L • (K  R) /L•B 3. L 2, Simp 4. L  T 3, Add 5. B • G 1, 4, MP 6. B 5, Simp 7. L • B 3, 6, Conj (7) 1. (F  X)  (P  T) 2. F  P © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

10

Exercise 7.2, III

3. P 4. F 5. F  X 6. P  T 7. T

/T 2, 3, MT 4, Add 1, 5, MP 3, 6, DS

(8) 1. (N  B) • (O  C) 2. Q  (N  O) 3. Q /BC 4. N  O 2, 3, MP 5. B  C 1, 4, CD (9) 1. (U  W)  (T  R) 2. U • H 3. R • J / U • T 4. U 2, Simp 5. U  W 4, Add 6. T  R 1, 5, MP 7. R 3, Simp 8. T 6, 7, MT T 9. U • 4, 8, Conj (10) 1. (D  E)  (G • H) 2. G  D 3. D • F /M 4. D 3, Simp 5. D  E 4, Add 6. G • H 1, 5, MP 7. G 6, Simp 8. D 2, 7, MP 9. D  M 4, Add 10. M 8, 9, DS (11) 1. (B  F)  (A  G) 2. (B  F)  (G  K) 3. B • H /AK 4. B 3, Simp 5. B  F 4, Add 6. A  G 1, 5, MP 7. B  E 4, Add 8. G  K 2, 7, MP 9. A  K 6, 8, HS (12) 1. (P  R)  (M  P) 2. (P  M)  (P  R) 3. P  M

/RP

© 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

11

Exercise 7.2, III

4. P  R 5. M  P 6. (P  R) • (M  P) 7. R  P (13) 1. (C  N) • E 2. D  (N  D) 3. D 4. N  D 5. N 6. C  N 7. C 8. C  P

2, 3, MP 1, 4, MP 4, 5, Conj 3, 6, CD

/ C  P 2, 3, DS 3, 4, MT 1, Simp 5, 6, MT 7, Add

(14) 1. F  (T • A) 2. (T  G)  (H  T) 3. F • O / H • T 4. F 3, Simp 5. T • A 1, 4, MP 6. T 5, Simp 7. T  G 6, Add 8. H  T 2, 7, MP 9. H 6, 8, MT 10. H • T 6, 9, Conj (15) 1. (S  B)  (S  K) 2. (K  D)  (H  S) 3. S • W / H 4. S 3, Simp 5. S  B 4, Add 6. S  K 1, 5, MP 7. K 4, 6, DS 8. K  D 7, Add 9. H  S 2, 8, MP 10. H 4, 9, MT (16) 1. (C  G)  (P • L) 2. (P • C)  (C  D) 3. C • R /DR 4. C 3, Simp 5. C  G 4, Add 6. P • L 1, 5, MP 7. P 6, Simp 8. P • C 4, 7, Conj 9. C  D 2, 8, MP 10. D 4, 9, MP © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

12

Exercise 7.2, III

11. D  R

10, Add

(17) 1. [A  (K • J)]  (E • F) 2. M  [A • (P  R)] 3. M • U / E • A 4. M 3, Simp 5. A • (P  R) 2, 4, MP 6. A 5, Simp 7. A  (K • J) 6, Add 8. E • F 1, 7, MP 9. E 8, Simp 10. E • A 6, 9, Conj (18) 1. H  (T  R) 2. H  (E  F) 3. T  E 4. H • D 5. H 6. T  R 7. E  F 8. (T  R) • (E  F) 9. R  F (19) 1. (U • P)  Q 2. O  U 3. P  O 4. O • T 5. O 6. U 7. P 8. U • P 9. Q

/RF 4, Simp 1, 5, MP 2, 5, DS 6, 7, Conj 3, 8, CD

/Q 4, Simp 2, 5, MP 3, 5, MT 6, 7, Conj 1, 8, MP

(20) 1. (M  N)  (F  G) 2. D  C 3. C  B 4. M • H 5. D  F 6. D  B 7. M 8. M  N 9. F  G 10. (D  B) • (F  G)

/BG 2, 3, HS 4, Simp 7, Add 1, 8, MP 6, 9, Conj

© 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

13

Exercise 7.2, III

11. B  G

5, 10, CD

(21) 1. (F • M)  (S  T) 2. (S  A)  F 3. (S  B)  M 4. S • G /T S 5.

4, Simp 6. S  A 5, Add 7. F 2, 6, MP 8. S  B 5, Add 9. M 3, 8, MP 10. F • M 7, 9, Conj 11. S  T 1, 10, MP 12. T 5, 11, DS (22) 1. (K • N)  [(P  K) • (R  G)] 2. K  N 3. N • B 4. P  R /G 5. N 3, Simp 6. K 2, 5, MT K N 7.

• 5, 6, Conj 8. (P  K) • (R  G) 1, 7, MP 9. K  G 4, 8, CD 10. G 6, 9, DS (23) 1. (A  D)  (B  F) 2. (B  C)  (A  E) 3. A  B 4. A 5. A  D 6. B  F 7. B 8. B  C 9. A  E 10. (A  E) • (B  F) 11. E  F

/EF 4, Add 1, 5, MP 3, 4, DS 7, Add 2, 8, MP 8, 9, Conj 3, 10, CD

(24) 1. (J  K) • (O  P) 2. (L  J) • (M  O) 3. K  (L  M) 4. K • G / P 5. K 4, Simp © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

14

Exercise 7.2, III

6. L  M 7. J  O 8. K  P 9. P

3, 5, MP 2, 6, CD 1, 7, CD 5, 8, DS

(25) 1. (M • N)  [(M  H)  (K • L)] 2. M • (C  D) 3. N • (F  G) / K • N M 4.

2, Simp 5. N 3, Simp 6. M • N 4, 5, Conj 7. (M  H)  (K • L) 1, 6, MP 8. M  H 4, Add 9. K • L 7, 8, MP 10. K 9, Simp 11. K • N 5, 10, Conj (26) . (P  S)  (E  F) 2. (P  T)  (G  H) 3. (P  U)  (E  G) 4. P 5. P  S 6. E  F 7. P  T 8. G  H 9. P  U 10. E  G 11. (E  F) • (G  H) 12. F  H

/FH 4, Add 1, 5, MP 4, Add 2, 7, MP 4, Add 3, 9, MP 6, 8, Conj 10, 11, CD

(27) 1. (S  Q) • (Q  S) 2. S  Q 3. Q /P•R S 4. Q  1, 2, CD 5. S 3, 4, DS 6. Q 2, 5, DS 7. Q  (P • R) 6, Add 8. P • R 3, 7, DS (28) 1. (D  B) • (C  D) 2. (B  D) • (E  C) 3. B  E 4. D  C 5. B  D 6. B  D

/DB 2, 3, CD 1, 4, CD 2, Simp

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15

Exercise 7.2, III

7. D  B 8. (B  D) • (D  B) 9. D  B

1, Simp 6, 7, Conj 5, 8, CD

(29) 1. (R  H) • (S  I) 2. (H • L)  (R  S) 3. H • (K  T) 4. H  L /IM 5. H 3, Simp 6. L 4, 5, DS 7. H • L 5, 6, Conj 8. R  S 2, 7, MP 9. H  I 1, 8, CD 10. I 5, 9, DS 11. I  M 10, Add (30) 1. (W • X)  (Q  R) 2. (S  F)  (Q  W) 3. (S  G)  (Q  X) 4. Q  S 5. Q • H /R 6. Q 5, Simp 7. S 4, 6, DS 8. S  F 7, Add 9. Q  W 2, 8, MP 10. W 6, 9, DS 11. S  G 7, Add 12. Q  X 3, 11, MP 13. X 6, 12, MP 14. W • X 10, 13, Conj 15. Q  R 1, 14, MP 16. R 6, 15, DS

Part IV (1) 1. T  (Q • F) 2. T • C 3. T 4. Q • F 5. Q 6. Q  O

/QO 2, Simp 1, 3, MP 4, Simp 5, Add

(2) 1. (C • E)  (E  T) 2. C © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

16

Exercise 7.2, IV

3. E 4. C • E 5. E  T 6. T

/T 2, 3, Conj 1, 4, MP 3, 5, DS

(3) 1. (S  B)  (C  P) 2. S • R 3. (C  F) • (P  D) 4. (F  D)  E /E 5. S 2, Simp 6. S  B 5, Add 7. C  P 1, 6, MP 8. F  D 3, 7, CD 9. E 4, 8, MP (4) 1. M  P 2. (P  S)  (R • D) 3. M 4. P 5. P  S 6. R • D 7. R

/R 1, 3, DS 4, Add 2, 5, MP 6, Simp

(5) 1. H  [(I  P)  E] 2. (H  W)  E 3. H / (I  P) 4. H  W 3, Add 5. E 2, 4, MP 6. (I  P)  E 1, 3, MP (I 7.  P) 5, 6, MT (6) 1. (P  S)  [(L  R) • (I  M)] 2. (P  N)  (L  I) 3. P • W /RM 4. P 3, Simp 5. P  N 4, Add 6. L  I 2, 5, MP 7. P  S 4, Add 8. (L  R) • (I  M) 1, 7, MP 9. R  M 6, 8, CD (7) 1. (C  M)  (C  T) 2. C  T 3. C /B C M 4.  3, Add 5. C  T 1, 4, MP © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

17

Exercise 7.2, IV

6. T 7. T  B 8. T 9. B

3, 5, MP 6, Add 2, 3, DS 7, 8, DS

(8) 1. (F  A)  (H • P) 2. L  A 3. D  F 4. (D  L) • I 5. (D  F) • (L  A) 6. D  L 7. F  A 8. H • P 9. H

/H 2, 3, Conj 4, Simp 5, 6, CD 1, 7, MP 8, Simp

(9) 1. L  G 2. C  A 3. (G  A)  (L • M) 4. L  C 5. C  I 6. (L  G) • (C  A) 7. G  A 8. L • M 9. L 10. C 11. I

/I 1, 2, Conj 4, 6, CD 3, 7, MP 8, Simp 4, 9, DS 5, 10, MP

(10) 1. (V • E)  (P  E) 2. V  E 3. V • I 4. E  (P  J) 5. V 6. E 7. V • E 8. P  E 9. P 10. P  J 11. J 12. J • E

/ J • E 3, Simp 2, 5, MP 5, 6, Conj 1, 7, MP 6, 8, MT 4, 6, MP 9, 10, DS 6, 11, Conj

Exercise 7.3 Part I 1. N • G

2

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18

Exercise 7.3, I

2. (S • T)  (S • U)

3

3. (M  G)  T

1

4. A • S

3

5. R  T

1

6. H • (Z  W)

2

7. G  Q

1

8. J  (N  S)

3

9. E • (H • Q)

3

10. (R • P)

1

11. B  E

1

12.

(Q  T) • (Q  R)

3

13.

H  (L  D)

2

14.

(M • R) • T

3

15.

D  (K • W)

1

Part II 1. C  K 2. (R  N) 3. T • H 4. L • (S • F) 5. B  K 6. A 7. D • (M  N) 8. U  T

1, Com 2, 3, DS 2, DM 1, 3, MT 1, Com 2, Simp 1, Assoc 2, Simp 1, DN 2, DM 2, DN 1, 3, MT 1, Dist 2, Simp 2, Com

9. L 10. (D  N) • (D  H) 11. K  (E • G) 12. N  F 13. M  (G  T) 14. A  S 15. R  T

2, DN 1, 3, DS 1, Dist 2, Simp 1, Dist 2, 3, DS 2, Com 1, 3, CD 1, Assoc 2, 3, DS 2, DM 1, 3, CD 1, Add 2, DM

© 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

19

Exercise 7.3, I

1, 3, MP Or (T  U)  R

1, Com 2, 3, MP

Part III (1) 1. (M  P) • (N  Q) 2. (M • N) 3. M  N 4. P  Q

/PQ 2, DM 1, 3, CD

(2) 1. S 2. S  F 3. F  S 4. (F • S)

/ (F • S) 1, Add 2, Com 3, DM

(3) 1. J  (K • L) 2. K 3. (J  K) • (J  L) 4. J  K 5. K  J 6. J

/J 1, Dist 3, Simp 4, Com 2, 5, DS

(4) 1. (N • T) 2. T 3. N  T 4. T  N 5. T 6. N

/ N 2, DM 3, Com 2, DN 4, 5, DS

(5) 1. H  A 2. A 3. A 4. H 5. H • A 6. (H  A)

/ (H  A) 2, DN 1, 3, MT 3, 4, Conj 5, DM

(6) 1. R  B 2. D  R 3. B 4. B 5. R 6. R  D 7. D

/D 3, DN 1, 4, MT 2, Com 5, 6, DS

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20

Exercise 7.3, III

(7) 1. T  (B  E) 2. E • T 3. T • E 4. T 5. B  E 6. E  B 7. E 8. B

/B 2, Com 3, Simp 1, 4, MP 5, Com 2, Simp 6, 7, DS

(8) 1. (O  M)  S 2. S 3. (O  M) 4. O • M 5. M • O 6. M

/ M 1, 2, MT 3, DM 4, Com 5, Simp

(9) 1. Q  (L  C) 2. C 3. (Q  L)  C 4. C  (Q  L) 5. Q  L 6. L  Q

/LQ 1, Assoc 3, Com 2, 4, DS 5, Com

(10) 1. (K • H)  (K • L) 2. L 3. K • (H  L) 4. (H  L) • K 5. H  L 6. L  H 7. H

/H 1, Dist 3, Com 4, Simp 5, Com 2, 6, DS

(11) 1. (E • N)  T 2. G  (N  E) 3. (E  N)  T 4. (E  N)  T 5. (E  N)  T 6. (N  E)  T 7. G  T

/GT 1, DM 3, DN 4, DN 5, Com 2, 6, HS

(12) 1. H • (C • T) 2. (F • T) 3. F  T 4. F  T 5. T  F 6. (H • C) • T

/F 2, DM 3, DN 4, Com 1, Assoc

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21

Exercise 7.3, III

7. T • (H • C) 8. T 9. T 10. F

6, Com 7, Simp 8, DN 5, 9, DS

(13) 1. (E • I)  (M • U) 2. E 3. E  I 4. (E • I) 5. M • U 6. M 7. M 8. E • M 9. (E  M)

/ (E  M) 2, Add 3, DM 1, 4, DS 5, Simp 6, DN 2, 7, Conj 8, DM

(14) 1. (J  K) 2. B  K 3. S  B 4. J • K 5. K • J 6. K 7. B 8. S 9. J 10. S • J

/ S • J 1, DM 4, Com 5, Simp 2, 6, MT 3, 7, MT 4, Simp 8, 9, Conj

(15) 1. (G • H)  (M • G) 2. G  (T • A) 3. (G • H)  (G • M) 4. G • (H  M) 5. G 6. T • A 7. A • T 8. A

/A 1, Com 3, Dist 4, Simp 2, 5, MP 6, Com 7, Simp

(16) 1. (Q • N)  (N • T) 2. (Q  C)  N 3. (N • Q)  (N • T) 4. N • (Q  T) 5. N 6. N 7. (Q  C) 8. Q • C 9. Q 10. (Q  T) • N

/T 1, Com 3, Dist 4, Simp 5, DN 2, 6, MT 7, DM 8, Simp 4, Com

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22

Exercise 7.3, III

11. Q  T 12. T

10, Simp 9, 11, DS

(17) 1. (U  R) 2. (R  N)  (P • H) 3. Q  H 4. U • R 5. R • U 6. R 7. R  N 8. P • H 9. H • P 10. H 11. H 12. Q

/ Q 1, DM 4, Com 5, Simp 6, Add 2, 7, MP 8, Com 9, Simp 10, DN 3, 11, MT

(18) 1. (F • A) 2. (L  A) 3. D  (F  L) 4. L • A 5. A • L 6. A 7. F  A 8. A  F 9. F 10. L 11. F • L 12. (F  L) 13. D

/ D 2, DM 4, Com 5, Simp 1, DM 7, Com 6, 8, DS 4, Simp 9, 10, Conj 11, DM 3, 12, MT

(19) 1. [(I  M)  G]  G 2. M  G 3. (M  G)  I 4. I  (M  G) 5. (I  M)  G 6. G 7. G  M 8. M

/M 2, Add 3, Com 4, Assoc 1, 5, MP 2, Com 6, 7, DS

(20) 1. E  B 2. U  C 3. (E • U)

/ (B • C)

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23

Exercise 7.3, III

4. (E  B) • (U  C) 5. E  U 6. E  U 7. E  U 8. B  C 9. (B • C) (21) 1. (K  F) 2. F  (K  C) 3. (G  C)  H 4. K • F 5. F • K 6. F 7. K  C 8. K 9. C 10. C  G 11. G  C 12. H 13. K • H 14. (K  H)

1, 2, Conj 3, DM 5, DN 6, DN 4, 7, CD 8, DM

/ (K  H) 1, DM 4, Com 5, Simp 2, 6, MP 4, Simp 7, 8, DS 9, Add 10, Com 3, 11, MP 8, 12, Conj 13, DM

(22) 1. S  (I • J) 2. S  R 3. J  Q 4. (S  I) • (S  J) 5. (S  J) • (S  I) 6. S  J 7. (S  R) • (J  Q) 8. R  Q 9. (R • Q)

/ (R • Q) 1, Dist 4, Com 5, Simp 2, 3, Conj 6, 7, CD 8, DM

(23) 1. (J  F)  M 2. (J  M)  P 3. F 4. (F  J)  M 5. F  (J  M) 6. J  M 7. P 8. F • P 9. (F  P)

/ (F  P) 1, Com 4, Assoc 3, 5, DS 2, 6, MP 3, 7, Conj 8, DM

(24) 1. (K • P)  (K • Q) 2. P  K 3. K • (P  Q) 4. K

/QT 1, Dist 3, Simp

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24

Exercise 7.3, III

5. K 6. P 7. (P  Q) • K 8. P  Q 9. Q 10. Q  T

4, DN 2, 5, MT 3, Com 7, Simp 6, 8, DS 9, Add

(25) 1. E  (D  C) 2. (E  D)  C 3. E  (D • C) 4. (E  D) • (E  C) 5. E  D 6. C 7. (E  C) • (E  D) 8. E  C 9. C  E 10. C 11. E

/E 1, DM 3, Dist 4, Simp 2, 5, MP 4, Com 7, Simp 8, Com 6, DN 9, 10, DS

(26) 1. A • (F • L) 2. A  (U  W) 3. F  (U  X) 4. (A • F) • L 5. A • F 6. A 7. U  W 8. F • A 9. F 10. U  X 11. (U  W) • (U  X) 12. U  (W • X)

/ U  (W • X) 1, Assoc 4, Simp 5, Simp 2, 6, MP 5, Com 8, Simp 3, 9, MP 7, 10, Conj 11, Dist

(27) 1. (T • R)  P 2. (P • R) • G 3. (T  N)  H 4. P • (R • G) 5. P 6. (T • R) 7. T  R 8. R  T 9. (R • P) • G 10. R • (P • G) 11. R 12. R 13. T 14. T  N

/H 2, Assoc 4, Simp 1, 5, MT 6, DM 7, Com 2, Com 9, Assoc 10, Simp 11, DN 8, 12, DS 13, Add

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25

Exercise 7.3, III

15. H

3, 14, MP

(28) 1. P  (I • L) 2. (P  I)  (L  C) 3. (P • C)  (E • F) 4. (P  I) • (P  L) 5. P  I 6. (L  C) 7. L • C 8. L 9. (P  L) • (P  I) 10. P  L 11. L  P 12. P 13. C • L 14. C 15. P • C 16. E • F 17. F • E 18. F 19. F  D

/FD 1, Dist 4, Simp 2, 5, MP 6, DM 7, Simp 4, Com 9, Simp 10, Com 8, 11, DS 7, Com 13, Simp 12, 14, Conj 3, 15, MP 16, Com 17, Simp 18, Add

(29) 1. B  (S • N) 2. B  S 3. S  N 4. (B  S) • (B  N) 5. B  S 6. (B  S) • (S  N) 7. S  N 8. (S • N) 9. (S • N)  B 10. B 11. B  W

/BW 1, Dist 4, Simp 2, 3, Conj 5, 6, CD 7, DM 1, Com 8, 9, DS 10, Add

(30) 1. (M  E)  (S  U) 2. (Q  E)  (U  H) 3. (M  Q) 4. M • Q 5. M 6. M  E 7. S  U 8. Q • M 9. Q 10. Q  E 11. U  H

/SH 3, DM 4, Simp 5, Add 1, 6, MP 4, Com 8, Simp 9, Add 2, 10, MP

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26

Exercise 7.3, III

12. S  H (31) 1. (R  D)  (F • G) 2. (F • R)  S 3. F • S 4. S • F 5. S 6. (F • R) 7. F  R 8. F 9. F 10. R 11. R  D 12. (F • G) 13. F  G 14. G 15. S • G 16. (S  G)

7, 11, HS

/ (S  G) 3, Com 4, Simp 2, 5, MT 6, DM 3, Simp 8, DN 7, 9, DS 10, Add 1, 11, MP 12, DM 9, 13, DS 5, 14, Conj 15, DM

(32) 1. Q  (C • B) 2. T  (B • H) 3. (Q • T) 4. Q  T 5. [Q  (C • B)] • [T  (B • H)] 6. (C • B)  (B • H) 7. (B • C)  (B • H) 8. B • (C  H) 9. B (33) 1. (A • G) 2. (A • E) 3. G  E 4. A  G 5. A  E 6. (A  G) • (A  E) 7. A  (G • E) 8. A  (G  E) 9. (G  E)  A 10. (G  E) 11. A 12. A  F 13. (A • F) (34) 1. (M • N)  (O • P) 2. (N  O)  P

/B 3, DM 1, 2, Conj 4, 5, CD 6, Com 7, Dist 8, Simp

/ (A • F) 1, DM 2, DM 4, 5, Conj 6, Dist 7, DM 8, Com 3, DN 9, 10, DS 11, Add 12, DM

/N

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27

Exercise 7.3, III

3. [(M • N)  O] • [(M • N)  P] 4. (M • N)  O 5. O  (M • N) 6. (O  M) • (O  N) 7. (O  N) • (O  M) 8. O  N 9. N  O 10. P 11. [(M • N)  P] • [(M • N)  O] 12. (M • N)  P 13. P  (M • N) 14. M • N 15. N • M 16. N

1, Dist 3, Simp 4, Com 5, Dist 6, Com 7, Simp 8, Com 2, 9, MP 3, Com 11, Simp 12, Com 10, 13, DS 14, Com 15, Simp

(35) 1. (T • K)  (C • E) 2. K  E 3. E  C 4. [(T • K)  C] • [(T • K)  E] 5. [(T • K)  E] • [(T • K)  C] 6. (T • K)  E 7. E  (T • K) 8. (E  T) • (E  K) 9. (E  K) • (E  T) 10. E  K 11. K  E 12. (K  E) • (E  C) 13. E  C 14. C  E 15. (C • E) 16. (C • E)  (T • K) 17. T • K

/T•K 1, Dist 4, Com 5, Simp 6, Com 7, Dist 8, Com 9, Simp 10, Com 2, 3, Conj 11, 12, CD 13, Com 14, DM 1, Com 15, 16, DS

Part IV (1) 1. (S • D)  (S • H) 2. S  (I • R) 3. S • (D  H) 4. S 5. I • R 6. R • I 7. R 8. S • R

/S•R 1, Dist 3, Simp 2, 4, MP 5, Com 6, Simp 4, 7, Conj

(2) 1. (C • I)  (H • I) © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

28

Exercise 7.3, IV

2. B  I 3. (I • C)  (I • H) 4. I • (C  H) 5. I 6. I 7. B

/ B 1, Com 3, Dist 4, Simp 5, DN 2, 6, MT

(3) 1. (S  C) 2. (S • R)  (C  D) 3. S • C 4. S 5. S  R 6. (S • R) 7. C  D 8.  C • S 9. C 10. D

/D 1, DM 3, Simp 4, Add 5, DM 2, 6, MP 3, Com 8, Simp 7, 9, DS

(4) 1. G  (R • E) 2. (G  E)  R 3. (G  R) • (G  E) 4. (G  E) • (G  R) 5. G  E 6. R 7. G  R 8. R  G 9. G 10. G  M

/GM 1, Dist 3, Com 4, Simp 2, 5, MP 3, Simp 7, Com 6, 8, DS 9, Add

(5) 1. E • (P • B) 2. (E • B)  (P • M) 3. E • (B • P) 4. (E • B) • P 5. E • B 6. (P • M) 7. P  M 8. P • (E • B) 9. P 10. P 11. M 12. M 13. E 14. E • M

/E•M 1, Com 3, Assoc 4, Simp 2, 5, MP 6, DM 4, Com 8, Simp 9, DN 7, 10, DS 11, DN 3, Simp 12, 13, Conj

(6) 1. F  (U • R) © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

29

Exercise 7.3, IV

2. F  (P • J) 3. F  F 4. [F  (U • R)] • [F  (P • J)] 5. (U • R)  (P • S) 6. [(U • R)  P] • [(U • R)  S] 7. (U • R)  P 8. P  (U • R) 9. (P  U) • (P  R) 10. P  U 11. U  P (7) 1. R  (C  M) 2. (I  C) 3. (A  M) 4. I • C 5. A • M 6. C • I 7. C 8. M • A 9. M 10. C • M 11. (C  M) 12. R

/UP 1, 2, Conj 3, 4, CD 5, Dist 6, Simp 7, Com 8, Dist 9, Simp 10, Com

/ R 2, DM 3, DM 4, Com 6, Simp 5, Com 8, Simp 7, 9, Conj 10, DM 1, 11, MT

(8) 1. (P  U)  L 2. (I  W)  K 3. L • K 4. L 5. L 6. (P  U) 7. P • U 8. U • P 9. U 10. K • L 11. K 12. K 13. (I  W) 14. I • W 15. W • I 16. W 17. U • W 18. (U  W)

/ (U  W) 3, Simp 4, DN 1, 5, MT 6, DM 7, Com 8, Simp 3, Com 10, Simp 11, DN 2, 12, MT 13, DM 14, Com 15, Simp 9, 16, Conj 17, DM

(9) 1. (S  E) 2. (S • M)  E

/ M

© 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

30

Exercise 7.3, IV

3. S • E 4. E • S 5. E 6. (S • M) 7. S  M 8. S  M 9. S 10. M

1, DM 3, Com 4, Simp 2, 5, MT 6, DM 7, DN 3, Simp 8, 9, DS

(10) 1. E  (B • P) 2. E  (G • W) 3. P  W 4. (E  B) • (E  P) 5. (E  P) • (E  B) 6. E  P 7. (E  G) • (E  W) 8. (E  W) • (E  G) 9. E  W 10. (E  P) • (E  W) 11. E  (P • W) 12. (P • W)  E 13. (P • W) 14. E

/ E 1, Dist 4, Com 5, Simp 2, Dist 7, Com 8, Simp 6, 9, Conj 10, Dist 11, Com 3, DM 12, 13, DS

Part V (The letters used in the translations are underlined in the text.)

New Cradle Chloe, with youngster in tow, is strolling across campus when she bumps into her friend Dylan. “Hi,” she says. “You must be coming from class.”

“Yes, I am,” he says. “Social psychology. And who is the little one?”

“This is my niece Ashley. She was three last month. Ashley, say Hi to Dylan. We’re heading over to the campus day-care center. Do you want to join us?”

“Sure,” he says. “At least as far as the cafeteria.”

“You know, kids are really a kick in the pants,” she says. “That’s why my heart goes out to a same-sex couple I know.”

“And why is that?” Dylan asks.

“They’re trying to adopt a baby, but the only adoption agency in this small town refuses to help them because the agency is religiously affiliated. I’m talking about the New Cradle agency. Those people at New Cradle think same-sex marriage is sinful.”

“Well, in light of this religious freedom legislation that’s been coming down the pike lately, the people at New Cradle may be within their rights.”

“How is that?” Chloe asks. © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

31

Exercise 7.3, V

“It goes like this,” he says. “If religious freedom is a right, then the people at New Cradle have a right to practice it as they choose. And if they have a right to practice it as they choose, then if they think same-sex marriage violates their religion, then they can refuse to help same-sex couples adopt a baby. Now religious freedom is a right and those people think same-sex marriage violates their religion. Hence they can refuse to help same-sex couples adopt a baby.”

1. R  P 2. P  (T  H) 3. R • T /H 4. R 3, Simp 5. P 1, 4, MP 6. T  H 2, 5, MP 7. T • R 3, Com 8. T 7, Simp 9. H 6, 8, MP “But wait just a minute,” Chloe says. “New Cradle is engaged in a commercial activity, just like a restaurant, and some of its babies come from out of state. So I would offer you this argument. If New Cradle receives government support or is engaged in interstate commerce, then if it discriminates against same-sex couples, then it violates those couple’s civil rights. Now if New Cradle refuses to help same-sex couples adopt, then it discriminates. New Cradle is engaged in interstate commerce and it refuses to help same-sex couples adopt. Thus, New Cradle violates those couples civil rights.”

1. (R  E)  (D  V) 2. H  D 3. E • H /V 4. E 3, Simp 5. E  R 4, Add 6. R  E 5, Com 7. D  V 1, 6, MP 8. H • E 3, Com 9. H 8, Simp 10. D 2, 9, MP 11. V 7, 10, MP “I can see that New Cradle faces a dilemma,” Dylan replies. “If it helps same-sex couples adopt a baby, then it will violate its own principles; but if it does not help them, then it will violate constitutional principles. And if New Cradle abandons its own principles, then it will close up shop and lay off its employees. But New Cradle will never lay off its employees, and it will either help same-sex couples adopt a baby or not help them. Therefore, New Cradle will violate constitutional principles.”

1. (H  O) • (H  C) 2. O  (S • L) 3. L 4. H  H /C 5. O  C 1, 4, CD 6. L  S 3, Add 7. S  L 6, Com 8. (S • L) 7, DM 9. O 2, 8, MT © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

32

Exercise 7.3, V

10. C 5, 9, DS “Another point we must not forget,” Chloe adds, “is that same sex couples now have the right to marry. And if that is the case, they have a right to raise a family. And if they have a right to raise a family and they have good parenting skills, then if New Cradle can refuse to help them adopt a baby, then religion is misappropriated and the public good is subverted. Same-sex couples do have good parenting skills, and the public good will not be subverted. Therefore, New Cradle cannot refuse to help them adopt a baby.”

1. M 2. M  R 3. (R • G)  [H  (E • P)] 4. G • P / H 5. R 1, 2, MP 6. G 4, Simp 7. R • G 5, 6, Conj 8. H  (E • P) 3, 7, MP 9. P • G 4, Com 10. P 9, Simp 11. P  E 10, Add 12. E  P 11, Com 13. (E • P) 12, DM 14. H 8, 13, MT “I think you’re right, says Dylan, “that equality plays a big role in this issue. Our country is based on the principle of equality. If straight couples have a right to the services of an adoption agency, and obviously they do, then either same-sex couples have such a right or they are not equal to straight couples. And if same-sex couples have a right to the services of an adoption agency, then New Cradle must change its policy and other religiously affiliated agencies must do the same. Now if our country is based on the principle of equality, then samesex couples are equal to straight couples. Thus, as you say, New Cradle must change its policy.”

1. C 2. S 3. S  (A  E) 4. A  (N • O) 5. C  E /N 6. A  E 2, 3, MP 7. E 1, 5, MP 8. E 7, DN 9. E  A 6, Com 10. A 8, 9, DS 11. N • O 4, 10, MP 12. N 11, Simp “Yet,” Dylan continues, “up till now we’ve said nothing about the welfare of the baby who will be adopted. If New Cradle helps same-sex couples adopt a baby, then it’s important that the baby grow up well adjusted. But if the baby is to be well adjusted, then it must have both a male role model and a female role model. If the parents are both men, then the baby will not have a female role model. If they are both women, then it will not have a male role model.

If the couple is a same-sex couple, then the parents will be either both men or both women, and

© 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

33

Exercise 7.3, V

the couple is a same-sex couple. Therefore, New Cradle must not help same-sex couples adopt a baby.”

1. H  A 2. A  (M • F) 3. E  F 4. O  M 5. S  (E  O) 6. S / H 7. E  O 5, 6, MP 8. (E  F) • (O  M) 3, 4, Conj 9. F  M 7, 8, CD 10. M  F 9, Com 11. (M • F) 10, DM 12. A 2, 11, MT 13. H 1, 12, MT “I think your reasoning is a bit shortsighted,” Chloe replies. “Let’s suppose a same-sex couple adopts a baby. If a same-sex couple adopts a baby, then the parents will be either both men or they will be both women. This much I grant you. But if they are both men, then surely they will have close female friends. And if that is so, then there will be both male and female role models. If the parents are both women, then surely they will have close male friends. And if that is so, then there will be both male and female role models. If there are both male and female role models, then the baby will be well adjusted. And if the baby will be well adjusted, then New Cradle should help with the adoption. Thus, the conclusion follows, that New Cradle should help with the adoption.”

1. A 2. A  (N  O) 3. N  E 4. E  (M • F) 5. O  L 6. L  (M • F) 7. (M • F)  A 8. A  H /H 9. N  O 1, 2, MP 10. (N  E) • (O  L) 3, 5, Conj 11. E  L 9, 10, CD 12. [E  (M • F)] • [L  (M • F)] 4, 6, Conj 13. (M • F)  (M • F) 11, 12, CD 14. M • (F  F) 13, Dist 15. M 14, Simp 16. (F • M)  (M • F) 13, Com 17. (F • M)  (F • M) 16, Com 18. F • (M  M) 17, Dist 19. F 18, Simp 20. M • F 15, 19, Conj 21. A 7, 20, MP 22. H 8, 21, MP © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

34

Exercise 7.3, V

“Interesting point,” Dylan says. “But I wonder if those surrogate role models would be as effective as male and female parents. Anyway, here’s where I get off. See you again soon, I hope.”

“Yeah, me too, says Chloe.”

“Bye-bye, Ashley. It was nice meeting you.”

“G’bye,” she says, waving her hand.

Exercise 7.4 Part I 1. G  Q

3

9.

DH

1

2. (R • S)  N

1

10.

SG

3

3. P  H

2

11.

JF

2

4. B  N

1

12.

(C • H)  A

5. A

2

13.

W  T 2

6. Q  L

3

14.

(K  M)  S

7. C  F

1

15.

SD

8. G  (N  Z)

2

3

1

1

Part II 1. J  M 2. J  (F  N) 3. (C  A) • (A  C) 4. K  K 5. (G  B) • (H  C) 6. (J • M)  Q 7. H  (C  R) 8. T  G

1, Impl 2, 3, HS 1, Exp 2, 3, MP 1, 2, Conj 3, Equiv 1, 2, CD 3, Taut 1, Trans 2, 3, CD 1, Exp 2, 3, MP 1, Impl 2, Exp 1, Trans 2, 3, HS

9. (K • A)  F 10. H  H 11. S  K 12. (M • M)  D 13. (N  A) • (A  N) 14. (E • R)  (E • R) 15. Q  (G  W)

1, Exp 2, 3, MT 1, Impl 2, Taut 1, Add 2, Impl 1, Exp 2, Taut 1, Trans 2, Equiv 1, Add 2, Equiv 1, Trans 2, Exp

© 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

35

Exercise 7.4, I

Part III (1) 1. (S • K)  R 2. K 3. (K • S)  R 4. K  (S  R) 5. S  R

/SR 1, Com 3, Exp 2, 5, MP

(2) 1. T  (F  F) 2. (F • F) 3. T  F 4. F 5. T

/ T 1, Taut 2, Taut 3, 4, MT

(3) 1. G  E 2. H  E 3. E  H 4. E  H 5. G  H

/ G  H 2, Trans 3, DN 1, 4, HS

(4) 1. S  Q 2. S 3. (S  Q) • (Q  S) 4. (Q  S) • (S  Q) 5. Q  S 6. Q

/ Q 1, Equiv 3, Com 4, Simp 2, 5, MT

(5) 1. N  P 2. (N  P)  T 3. N  P 4. T

/T 1, Impl 2, 3, MP

(6) 1. F  B 2. B  (B  J) 3. (B • B)  J 4. B  J 5. F  J

/FJ 2, Exp 3, Taut 1, 4, HS

(7) 1. (B  M) • (D  M) 2. B  D 3. M  M 4. M

/M 1, 2, CD 3, Taut

(8) 1. Q  (F  A) 2. R  (A  F) © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

36

Exercise 7.4, III

/FA 3, Simp 1, 4, MP 3, Com 6, Simp 2, 7, MP 5, 8, Conj 9, Equiv

3. Q • R 4. Q 5. F  A 6. R • Q 7. R 8. A  F 9. (F  A) • (A  F) 10. F  A (9) 1. T  (T  G) 2. G 3. T  (T  G) 4. (T • T)  G 5. T  G 6. T

/ T 1, Impl 3, Exp 4, Taut 2, 5, MT

(10) 1. (B  G) • (F  N) 2. (G • N) 3. G  N 4. (G  B) • (F  N) 5. (G  B) • (N  F) 6. B  F 7. (B • F)

/ (B • F) 2, DM 1, Trans 4, Trans 3, 5, CD 6, DM

(11) 1. (J • R)  H 2. (R  H)  M 3. (P  J) 4. J  (R  H) 5. J  M 6. P • J 7. P • J 8. P 9. J • P 10. J 11. M 12. M • P

/ M • P 1, Exp 2, 4, HS 3, DM 6, DN 7, Simp 7, Com 9, Simp 5, 10, MP 8, 11, Conj

(12) 1. T 2. T  S 3. S  T 4. S  T

/ST 1, Add 2, Com 3, Impl

(13) 1. K  (B  M) 2. D  (K • M) 3. K  (M  B) 4. K  (M  B)

/ D  B 1, Trans 3, DN

© 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

37

Exercise 7.4, III

5. (K • M)  B 6. D  B

4, Exp 2, 5, HS

(14) 1. (O  C) • (S  D) 2. (E  D) • (E  C) 3. O  C 4. (S  D) • (O  C) 5. S  D 6. D  S 7. E  D 8. (E  C) • (E  D) 9. E  C 10. C  E 11. O  E 12. O  D 13. O  S

/OS 1, Simp 1, Com 4, Simp 5, Trans 2, Simp 2, Com 8, Simp 9, Trans 7, 10, HS 7, 11, HS 6, 12, HS

(15) 1. (U • W)  X 2. U  U 3. U  U 4. U 5. U  W 6. (U • W) 7. X 8. U • X 9. U • X 10. (U  X) (16) 1. T  R 2. T  R 3. R  T 4. R  T 5. T  T 6. T  T 7. T (17) 1. S  N 2. S  Q 3. N  S 4. N  S 5. S  Q 6. N  Q (18) 1. M  (U  H) 2. (H  U)  F

/ (U  X) 2, Impl 3, Taut 4, Add 5, DM 1, 6, MP 4, 7, Conj 8, DN 9, DM / T 2, Trans 3, DN 1, 4, HS 5, Impl 6, Taut /NQ 1, Com 3, Impl 2, Impl 4, 5, HS /MF

© 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

38

Exercise 7.4, III

3. (U  H)  F 4. (U  H)  F 5. M  F

2, Com 3, Impl 1, 4, HS

(19) 1. R  P 2. R  P 3. R  P 4. P  R 5. P  R 6. (R  P) • (P  R) 7. R  P

/RP 1, Impl 2, Com 4, Impl 3, 5, Conj 6, Equiv

(20) 1. H  B 2. H  D 3. (B • D) 4. B  H 5. B  H 6. D  H 7. D  H 8. B  D 9. (B  H) • (D  H) 10. H  H 11. H

/H 1, Trans 4, DN 2, Trans 6, DN 3, DM 5, 7, Conj 8, 9, CD 10, Taut

(21) 1. J  (G  L) 2. (J • G)  L 3. (G • J)  L 4. G  (J  L)

/ G  (J  L) 1, Exp 2, Com 3, Exp

(22) 1. S  (L • M) 2. M  (L  R) 3. (M • L)  R 4. (L • M)  R 5. S  R

/SR 2, Exp 3, Com 1, 4, HS

(23) 1. F  (A • K) 2. G  (A • K) 3. F  G 4. [F  (A • K)] • [G  (A • K)] 5. (A • K)  (A • K) 6. A  K (24) 1. (I  E)  C 2. C  C 3. C  C

/AK 1, 2, Conj 3, 4, CD 5, Equiv

/I 2, Impl

© 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

39

Exercise 7.4, III

4. C 5. (I  E) 6. (I  E) 7. I • E 8. I • E 9. I

3, Taut 1, 4, MT 5, Impl 6, DM 7, DN 8, Simp

(25) 1. T  G 2. S  G 3. T  G 4. S  G 5. G  T 6. G  S 7. (G  T) • (G  S) 8. G  (T • S) 9. (T • S)  G 10. (T  S)  G 11. (T  S)  G (26) 1. H  U 2. H  U 3. (H  U)  T 4. H  (U  T) 5. H  (U  T)

/ H  (U  T) 1, Impl 2, Add 3, Assoc 4, Impl

(27) 1. Q  (W • D) 2. Q  (W • D) 3. (Q  W) • (Q  D) 4. Q  W 5. Q  W (28) 1. P  (E  B) 2. (B  E) 3. (E  B) 4. (E  B) 5. (E  B) 6. P

/ (T  S)  G 1, Impl 2, Impl 3, Com 4, Com 5, 6, Conj 7, Dist 8, Com 9, DM 10, Impl

/QW 1, Impl 2, Dist 3, Simp 4, Impl

/ P 2, Com 3, DN 4, Impl 1, 5, MT

(29) 1. (G  J)  (H  Q) 2. J • Q 3. J 4. J  G 5. G  J 6. G  J 7. H  Q

/ H 2, Simp 3, Add 4, Com 5, Impl 1, 6, MP

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40

Exercise 7.4, III

8. Q • J 9. Q 10. H (30) 1. I  (N • F) 2. I  F 3. (I  N) • (I  F) 4. (I  F) • (I  N) 5. I  F 6. I  F 7. I  F 8. F  I 9. F  F 10. F  F 11. F  F 12. F (31) 1. K  R 2. K  (R  P) 3. P 4. (K • R)  (K • R) 5. (K • R)  P 6. (K • R) 7. K • R 8. R • K 9. R

2, Com 9, Simp 7, 9, MT

/F 1, Dist 3, Com 4, Simp 5, DN 6, Impl 2, Trans 7, 8, HS 9, Impl 10, DN 11, Taut

/ R 1, Equiv 2, Exp 3, 5, MT 4, 6, DS 7, Com 8, Simp

(32) 1. C  (L  Q) 2. L  C 3. Q 4. (C • L)  Q 5. (C • L) 6. C  L 7. C  L 8. C  L 9. C  C 10. C  C 11. C

/ C 1, Exp 3, 4, MT 5, DM 6, DN 7, Impl 2, 8, HS 9, Impl 10, Taut

(33) 1. (E  A) • (F  A) 2. E  G 3. F  G 4. E  G 5. E  G 6. F  G

/A 2, DN 4, Impl 3, DN

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41

Exercise 7.4, III

7. F  G 8. G  F 9. E  F 10. E  F 11. E  F 12. A  A 13. A

6, Impl 7, Trans 5, 8, HS 9, Impl 10, DN 1, 11, CD 12, Taut

(34) 1. (F • H)  N 2. F  S 3. H 4. (H • F)  N 5. H  (F  N) 6. F  N 7. N  F 8. F  S 9. F  S 10. N  S 11. N  S 12. N  S

/NS 1, Com 4, Exp 3, 5, MP 6, Trans 2, DN 8, Impl 7, 9, HS 10, Impl 11, DN

(35) 1. T  (H • J) 2. (H  N)  T 3. T  (H • J) 4. (T  H) • (T  J) 5. T  H 6. T  H 7. (H  N)  T 8. (H • N)  T 9. T  (H • N) 10. (T  H) • (T  N) 11. (H  T) • (T  N) 12. H  T 13. H  T 14. (T  H) • (H  T) 15. T  H

/TH 1, Impl 3, Dist 4, Simp 5, Impl 2, Impl 7, DM 8, Com 9, Dist 10, Com 11, Simp 12, Impl 6, 13, Conj 14, Equiv

(36) 1. T  (A  N) 2. T  N 3. T  (A  N) 4. T  (A  N) 5. T  (A • N) 6. T  (A • N) 7. (T  A) • (T  N) 8. (T  N) • (T  A)

/ T  N 1, Impl 3, Impl 4, DM 5, DN 6, Dist 7, Com

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42

Exercise 7.4, III

9. T  N 10. T  N 11. N  T 12. N  T 13. N  T 14. (T  N) • (N  T) 15. T  N (37) 1. (D  E)  (E  D) 2. (D  E)  (G • H) 3. E • G 4. E 5. E  D 6. D  E 7. D  E 8. E  D 9. (D  E) • (E  D) 10. D  E 11. (G • H) 12. G  H 13. G  H 14. G • E 15. G 16. G 17. H 18. G • H (38) 1. (O  R)  S 2. (P  R)  S 3. S  (P  R) 4. S  (P  R) 5. (O  R)  (P  R) 6. (O  R)  (P  R) 7. (O  R)  (P  R) 8. (O • R)  (P  R) 9. (O • R)  (P  R) 10. (R • O)  (P  R) 11. (R • O)  (P • R) 12. (R • O)  (P • R) 13. (R • O)  (R • P) 14. R • (O  P) 15. R

8, Simp 9, Impl 2, Com 11, DN 12, Impl 10, 13, Conj 14, Equiv

/G•H 3, Simp 4, Add 5, Com 6, Impl 1, 7, MP 7, 8, Conj 9, Equiv 2, 10, MP 11, DM 12, DN 3, Com 14, Simp 15, DN 13, 16, DS 15, 17, Conj / R 2, Trans 3, DN 1, 4, HS 5, Impl 6, Impl 7, DM 8, DN 9, Com 10, DM 11, DN 12, Com 13, Dist 14, Simp

(39) 1. (L  P)  U 2. (M  U)  I © 2018 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

43

Exercise 7.4, III

3. P 4. P  L 5. L  P 6. U 7. U  M 8. M  U 9. M  U 10. I

/I 3, Add 4, Com 1, 5, MP 6, Add 7, Com 8, Impl 2, 9, MP

(40) 1. A  W 2. A  W 3. R  A 4. (A • W)  (A • W) 5. (A • W) 6. A • W 7. A 8. R 9. W •A 10. W 11. W • R 12. (W  R) (41) 1. (S  T)  (S  T) 2. (S  T)  (T  K) 3. S  T 4. S  T 5. T  K 6. S  T 7. S  T 8. S  K 9. S  K 10. S  K (42) 1. G  M 2. G  M 3. G  (M  T) 4. (G • M)  (G • M) 5. G  M 6. G  M 7. (G • M) 8. (G • M)  (G • M) 9. G • M 10. (G • M)  T 11. T

/ (W  R) 1, Equiv 2, DM 4, 5, DS 6, Simp 3, 7, MT 6, Com 9, Simp 8, 10, Conj 11, DM

/SK 1, 3, MP 2, 4, MP 3, DN 6, Impl 5, 7, HS 8, Impl 9, DN

/T 1, Equiv 2, DN 5, DN 6, DM 4, Com 7, 8, DS 3, Exp 9, 10, MP

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44