Exercise Set 7 (Propositional Logic) - uta.edu Set... · Exercise Set 7 (Propositional Logic) Keith Burgess-Jackson 13 September 2017 Exercises I. ... (5) subcontrariety, (6) subalternation

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Exercise Set 7 (Propositional Logic) Keith Burgess-Jackson 13 September 2017

Exercises I. If A, B, and C are true propositions and X, Y, and Z are false propositions, which of the following are true?

1. ~A Ú B

2. ~B Ú X

3. ~Y Ú C

4. ~Z Ú X

5. (A • X) Ú (B • Y)

6. (B • C) Ú (Y • Z)

7. ~(C • Y) Ú (A • Z)

8. ~(A • B) Ú (X • Y)

9. ~(X • Z) Ú (B • C)

10. ~(X • ~Y) Ú (B • ~C)

11. (A Ú X) • (Y Ú B)

12. (B Ú C) • (Y Ú Z)

13. (X Ú Y) • (X Ú Z)

14. ~(A Ú Y) • (B Ú X)

15. ~(X Ú Z) • (~X Ú Z)

16. ~(A Ú C) Ú ~(X • ~Y)

17. ~(B Ú Z) • ~(X Ú ~Y)

18. ~[(A Ú ~C) Ú (C Ú ~A)]

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19. ~[(B • C) • ~(C • B)]

20. ~[(A • B) Ú ~(B • A)]

21. [A Ú (B Ú C)] • ~[(A Ú B) Ú C]

22. [X Ú (Y • Z)] Ú ~[(X Ú Y) • (X Ú Z)]

23. [A • (B Ú C)] • ~[(A • B) Ú (A • C)]

24. ~{[(~A • B) • (~X • Z)] • ~[(A • ~B) Ú ~(~Y • ~Z)]}

25. ~{~[(B • ~C) Ú (Y • ~Z)] • [(~B Ú X) Ú (B Ú ~Y)]}

II. If A and B are known to be true and X and Y are known to be false, but the truth values of P and Q are not known, of which of the following propositions can you determine the truth values?

1. A Ú P

2. Q • X

3. Q Ú ~X

4. ~B • P

5. P Ú ~P

6. ~P Ú (Q Ú P)

7. Q • ~Q

8. P • (~P Ú X)

9. ~(P • Q) Ú P

10. ~Q • [(P Ú Q) • ~P]

11. (P Ú Q) • ~(Q Ú P)

12. (P • Q) • (~P Ú ~Q)

13. ~P Ú [~Q Ú (P • Q)]

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14. P Ú ~(~A Ú X)

15. P • [~(P Ú Q) Ú ~P]

16. ~(P • Q) Ú (Q •P)

17. ~[~(~P Ú Q) Ú P] Ú P

18. (~P Ú Q) • ~[~P Ú (P • Q)]

19. (~A Ú P) • (~P Ú Y)

20. ~[P Ú (B • Y)] Ú [(P Ú B) • (P Ú Y)]

21. [P Ú (Q • A)] • ~[(P Ú Q) • (P Ú A)]

22. [P Ú (Q • X)] • ~[(P Ú Q) • (P Ú X)]

23. ~[~P Ú (~Q Ú X)] Ú [~(~P Ú Q) Ú (~P Ú X)]

24. ~[~P Ú (~Q Ú A)] Ú [~(~P Ú Q) Ú (~P Ú A)]

25. ~[(P • Q) Ú (Q • ~P)] • ~[(P • ~Q) Ú (~Q • ~P)]

III. If A, B, and C are true propositions and X, Y, and Z are false propositions, which of the following are true?

1. A כ B

2. A כ X

3. B כ Y

4. Y כ Z

5. (A כ B) כ Z

6. (X כ Y) כ Z

7. (A כ B) כ C

8. (X כ Y) כ C

9. A כ (B כ Z)

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10. X כ (Y כ Z)

11. [(A כ B) כ C] כ Z

12. [(A כ X) כ Y] כ Z

13. [A כ (X כ Y)] כ C

14. [A כ (B כ Y)] כ X

15. [(X כ Z) כ C] כ Y

16. [(Y כ B) כ Y] כ Y

17. [(A כ Y) כ B] כ Z

18. [(A • X) כ C] כ [(A כ C) כ X]

19. [(A • X) כ C] כ [(A כ X) כ C]

20. [(A • X) כ Y] כ [(X כ A) כ (A כ Y)]

21. [(A • X) Ú (~A • ~X)] כ [(A כ X) • (X כ A)]

22. {[A כ (B כ C)] כ [(A • B) כ C]} כ [(Y כ B) כ (C כ Z)]

23. {[(X כ Y) כ Z] כ [Z כ (X כ Y)]} כ [(X כ Z) כ Y]

24. [(A • X) כ Y] כ [(A כ X) • (A כ Y)]

25. [A כ (X • Y)] כ [(A כ X) Ú (A כ Y)]

IV. If A and B are known to be true and X and Y are known to be false, but the truth values of P and Q are not known, of which of the following propositions can you determine the truth values?

1. P כ A

2. X כ Q

3. (Q כ A) כ X

4. (P • A) כ B

5. (P כ P) כ X

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6.(X כ Q) כ X

7. X כ (Q כ X)

8. (P • X) כ Y

9. [P כ (Q כ P)] כ Y

10. (Q כ Q) כ (A כ X)

11. (P כ X) כ (X כ P)

12. (P כ A) כ (B כ X)

13. (X כ P) כ (B כ Y)

14. [(P כ B) כ B] כ B

15. [(X כ Q) כ Q] כ Q

16. (P כ X) כ (~X כ ~P)

17. (X כ P) כ (~X כ Y)

18. (P כ A) כ (A כ ~B)

19. (P כ Q) כ (P כ Q)

20. (P כ ~~P) כ (A כ ~B)

21. ~(A • P) כ (~A Ú ~P)

22. ~(P • X) כ ~(P Ú ~X)

23. ~(X Ú Q) כ (~X • ~Q)

24. [P כ (A Ú X)] כ [(P כ A) כ X]

25. [Q Ú (B • Y)] כ [(Q Ú B) • (Q Ú Y)] V. Use truth tables to characterize the following propositional forms as (1) tautologous, (2) self-contradictory, (3) contingent, or (4) self-consistent. More than one of these terms may apply to a given propositional form, so you will need to check for each of them.

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1. [p כ (p כ q)] כ q

2. p כ [(p כ q) כ q]

3. (p • q) • (p כ ~q)

4. p כ [~p כ (q Ú ~q)]

5. p כ [p כ (q • ~q)]

6. (p כ p) כ (q • ~q)

7. [p כ (q כ r)] כ [(p כ q) כ (p כ r)]

8. [p כ (q כ p)] כ [(q כ q) כ ~(r כ r)]

9. {[(p כ q) • (r כ s)] • (p Ú r)} כ (q Ú s)

10. {[(p כ q) • (r כ s)] • (q Ú s)} כ (p Ú r) VI. Use truth tables to determine whether the following pairs of propositional forms exhibit (1) logical implication (if so, in which direction), (2) logical equivalence, (3) contradictoriness, (4) contrariety, (5) subcontrariety, (6) subalternation (if so, in which direction), (7) independence, (8) consistency, or (9) inconsistency. More than one of these terms may apply to a given pair, so you will need to check for each of them.

1. ~(p • q) | ~p Ú ~q

2. ~(p Ú q) | ~p • ~q

3. p Ú q | ~p Ú q

4. ~p • q | ~q Ú p

5. p Ú q | q Ú p

6. p • ~p | p

7. p כ q | p • ~q

8. p º q | p • q

9. p Ú (q Ú r) | (p Ú q) Ú r

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10. p • (q Ú r) | (p • q) Ú (p • r)

11. (q כ ~r) • s | s º (q • r)

12. q Ú p | ~q כ ~p

13. p • q | ~p Ú ~q

14. p כ q | ~q כ ~p

15. p º q | (p כ q) • (q כ p)

16. q כ p | q • p

17. ~p • q | ~q • p

18. (p • q) כ r | p כ (q כ r)

19. p | p Ú p

20. p Ú ~p | p

21. t º u | t Ú u

22. ~(p Ú q) | ~p Ú ~q

23. (p • q) כ r | p Ú (q כ r)

24. q Ú p | ~q • ~p

25. p • q | ~p כ ~q

26. p | p º q

27. p | q

VII. Use truth tables to determine whether the following argument forms are valid.

1. p כ q \ ~q כ ~p

2. p כ q \ ~p כ ~q

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3. p • q \ p

4. p \ p Ú q

5. p \ p כ q

6. p כ q \ p כ (p • q)

7. (p Ú q) כ (p • q) \ (p כ q) • (q כ p)

8. p כ q ~p \ ~q

9. p כ q ~q \ ~p

10. p q \ p • q

11. p כ q p כ r \ q Ú r

12. p כ q q כ r \ r כ p

13. p כ (q כ r) p כ q \ p כ r

14. p כ (q • r) (q Ú r) כ ~p \ ~p

15. p כ (q כ r)

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q כ (p כ r) \ (p Ú q) כ r

16. (p כ q) • (r כ s) p Ú r \ q Ú s

17. (p כ q) • (r כ s) ~q Ú ~s \ ~p Ú ~s

18. p כ (q כ r) q כ (r כ s) \ p כ s

19. p כ (q כ r) (q כ r) כ s \ p כ s

20. (p כ q) • [(p • q) כ r] p כ (r כ s) \ p כ s

21. (p Ú q) כ (p • q) ~(p Ú q) \ ~(p • q)

22. (p Ú q) כ (p • q) p • q \ p Ú q

23. (p • q) כ (r • s) \ (p • q) כ [(p • q) • (r • s)]

24. (p כ q) • (r כ s) \ p כ q

VIII. Use truth tables to determine whether the following argument forms are valid.

1. (A Ú B) כ (A • B) A Ú B \ A • B

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2. (C Ú D) כ (C • D) C • D \ C Ú D

3. E כ F F כ E \ E Ú F

4. (G Ú H) כ (G • H) ~(G • H) \ ~(G Ú H)

5. (I Ú J) כ (I • J) ~(I Ú J) \ ~(I • J)

6. K Ú L K \ ~L

7. M Ú (N • ~N) M \ ~(N • ~N)

8. (O Ú P) כ Q Q כ (O • P) \ (O Ú P) כ (O • P)

9. (R Ú S) כ T T כ (R • S) \ (R • S) כ (R Ú S)

10. U כ (V Ú W) (V • W) כ ~U \ ~U

IX. For each of the following elementary valid argument forms, state the implication rule (MP, MT, HS, DS, CD, Simp, Conj, or Add) by which its conclusion follows from its premise or premises.

1. (D Ú E) • (F Ú G) \ D Ú E

2. H כ I

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\ (H כ I) Ú (H כ ~I)

3. ~(J • K) • (L כ ~M) \ ~(J • K)

4. [N כ (O • P)] • [Q כ (O • R)] N Ú Q \ (O • P) Ú (O • R)

5. (X Ú Y) כ ~(Z • ~A) ~~(Z • ~A) \ ~(X Ú Y)

6. (S º T) Ú [(U • V) Ú (U • W)] ~(S º T) \ (U • V) Ú (U • W)

7. ~(B • C) כ (D Ú E) ~(B • C) \ D Ú E

8. (F º G) כ ~(G • ~F) ~(G • ~F) כ (G כ F) \ (F º G) כ (G כ F)

9. ~(H • ~I) כ (H כ I) (I º H) כ ~(H • ~I) \ (I º H) כ (H כ I)

10. (A כ B) כ (C Ú D) A כ B \ C Ú D

11. [E כ (F º ~G)] Ú (C Ú D) ~[E כ (F º ~G)] \ C Ú D

12. (C Ú D) כ [(J Ú K) כ (J • K)] ~[(J Ú K) כ (J • K)] \ ~(C Ú D)

13. ~[L כ (M כ N)] כ ~(C Ú D) ~[L כ (M כ N)]

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\ ~(C Ú D)

14. (J כ K) • (K כ L) L כ M \ [(J כ K) • (K כ L)] • (L כ M)

15. N כ (O Ú P) Q כ (O Ú R) \ [Q כ (O Ú R)] • [N כ (O Ú P)]

16. (W • ~X) º (Y כ Z) \ [(W • ~X) º (Y כ Z)] Ú (X º ~Z)

17. [(H • ~I) כ C] • [(I • ~H) כ D] (H • ~I) Ú (I • ~H) \ C Ú D

18. [(O כ P) כ Q] כ ~(C Ú D) (C Ú D) כ [(O כ P) כ Q] \ (C Ú D) כ ~(C Ú D)

X. Each of the following is a formal proof of validity for the indicated argument. State the justification for each line that is not a premise. Use only the eight implication rules (MP, MT, HS, DS, CD, Simp, Conj, and Add).

1. 1. A • B 2. (A Ú C) כ D / \ A • D 3. A 4. A Ú C 5. D 6. A • D

2. 1. (E Ú F) • (G Ú H) 2. (E כ G) • (F כ H) 3. ~G / \ H 4. E Ú F 5. G Ú H 6. H

3. 1. I כ J 2. J כ K 3. L כ M 4. I Ú L / \ K Ú M

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5. I כ K 6. (I כ K) • (L כ M) 7. K Ú M

4. 1. Q כ R 2. ~S כ (T כ U) 3. S Ú (Q Ú T) 4. ~S / \ R Ú U 5. T כ U 6. (Q כ R) • (T כ U) 7. Q Ú T 8. R Ú U

5. 1. (A Ú B) כ C 2. (C Ú B) כ [A כ (D º E)] 3. A • D / \ D º E 4. A 5. A Ú B 6. C 7. C Ú B 8. A כ (D º E) 9. D º E

6. 1. F כ ~G 2. ~F כ (H כ ~G) 3. (~I Ú ~H) כ ~~G 4. ~I / \ ~H 5. ~I Ú ~H 6. ~~G 7. ~F 8. H כ ~G 9. ~H

7. 1. (L כ M) כ (N º O) 2. (P כ ~Q) כ (M º ~Q) 3. {[(P כ ~Q) Ú (R º S)] • (N Ú O)} כ [(R º S) כ (L כ M)] 4. (P כ ~Q) Ú (R º S) 5. N Ú O / \ (M º ~Q) Ú (N º O) 6. [(P כ ~Q) Ú (R º S)] • (N Ú O) 7. (R º S) כ (L כ M) 8. (R º S) כ (N º O) 9. [(P כ ~Q) כ (M º ~Q)] • [(R º S) כ (N º O)] 10. (M º ~Q) Ú (N º O)

XI. For each of the following, adding just two propositions to the premises will produce a formal proof of validity. Use only the

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eight implication rules (MP, MT, HS, DS, CD, Simp, Conj, and Add).

1. 1. A 2. B / \ (A Ú C) • B

2. 1. D כ E 2. D • F / \ E

3. 1. G 2. H / \ (G • H) Ú I

4. 1. J כ K 2. J / \ K Ú L

5. 1. M Ú N 2. ~M • ~O / \ N

6. 1. P • Q 2. R / \ P • R

7. 1. S כ T 2. ~T • ~U / \ ~S

8. 1. V Ú W 2. ~V / \ W Ú X

9. 1. Y כ Z 2. Y / \ Y • Z

10. 1. D כ E 2. (E כ F) • (F כ D) / \ D כ F

XII. Construct a formal proof of validity for each of the following arguments. Use only the eight implication rules (MP, MT, HS, DS, CD, Simp, Conj, and Add).

1. 1. A כ B 2. A Ú (C • D) 3. ~B • ~E / \ C

2. 1. (F כ G) • (H כ I) 2. J כ K 3. (F Ú J) • (H Ú L) / \ G Ú K

3. 1. (~M • ~N) כ (O כ N)

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2. N כ M 3. ~M / \ ~O

4. 1. (K Ú L) כ (M Ú N) 2. (M Ú N) כ (O • P) 3. K / \ O

5. 1. (Q כ R) • (S כ T) 2. (U כ V) • (W כ X) 3. Q Ú U / \ R Ú V

XIII. For each of the following elementary valid argument forms, state the replacement rule (DM, Com, Assoc, Dist, DN, Trans, MI, ME, Exp, or Taut) by which its conclusion follows from its premise.

1. (A כ B) • (C כ D) \ (A כ B) • (~D כ ~C)

2. (E כ F) • (G כ ~H) \ (~E Ú F) • (G כ ~H)

3. [I כ (J כ K)] • (J כ ~I) \ [(I • J) כ K] • (J כ ~I)

4. [L כ (M Ú N)] Ú [L כ (M Ú N)] \ L כ (M Ú N)

5. O כ [(P כ Q) • (Q כ P)] \ O כ (P º Q)

6. ~(R Ú S) כ (~R Ú ~S) \ (~R • ~S) כ (~R Ú ~S)

7. (T Ú ~U) • [(W • ~V) כ ~T] \ (T Ú ~U) • [W כ (~V כ ~T)]

8. (X Ú Y) • (~X Ú ~Y) \ [(X Ú Y) • ~X] Ú [(X Ú Y) • ~Y]

9. Z כ (A כ B) \ Z כ (~~A כ B)

10. [C • (D • ~E)] • [(C • D) • ~E]

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\ [(C • D) • ~E] • [(C • D) • ~E] XIV. Each of the following is a formal proof of validity for the indicated argument. State the justification for each line that is not a premise. Use all 18 rules of inference: the eight implication rules and the 10 replacement rules.

1. 1. A כ B 2. C כ ~B / \ A כ ~C 3. ~~B כ ~C 4. B כ ~C 5. A כ ~C

2. 1. (D • E) כ F 2. (D כ F) כ G / \ E כ G 3. (E • D) כ F 4. E כ (D כ F) 5. E כ G

3. 1. (H Ú I) כ [J • (K • L)] 2. I / \ J • K 3. I Ú H 4. H Ú I 5. J • (K • L) 6. (J • K) • L 7. J • K

4. 1. (M Ú N) כ (O • P) 2. ~O / \ ~M 3. ~O Ú ~P 4. ~(O • P) 5. ~(M Ú N) 6. ~M • ~N 7. ~M

5. 1. (Q Ú ~R) Ú S 2. ~Q Ú (R • ~Q) / \ R כ S 3. (~Q Ú R) • (~Q Ú ~Q) 4. (~Q Ú ~Q) • (~Q Ú R) 5. ~Q Ú ~Q 6. ~Q 7. Q Ú (~R Ú S) 8. ~R Ú S

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9. R כ S

6. 1. T • (U Ú V) 2. T כ [U כ (W • X)] 3. (T • V) כ ~(W Ú X) / \ W º X 4. (T • U) כ (W • X) 5. (T • V) כ (~W • ~X) 6. [(T • U) כ (W • X)] • [(T • V) כ (~W • ~X)] 7. (T • U) Ú (T • V) 8. (W • X) Ú (~W • ~X) 9. W º X

7. 1. Y כ Z 2. Z כ [Y כ (R Ú S)] 3. R º S 4. ~(R • S) / \ ~Y 5. (R • S) Ú (~R • ~S) 6. ~R • ~S 7. ~(R Ú S) 8. Y כ [Y כ (R Ú S)] 9. (Y • Y) כ (R Ú S) 10. Y כ (R Ú S) 11. ~Y

8. 1. A כ B 2. B כ C 3. C כ A 4. A כ ~C / \ ~A • ~C 5. A כ C 6. (A כ C) • (C כ A) 7. A º C 8. (A • C) Ú (~A • ~C) 9. ~A Ú ~C 10. ~(A • C) 11. ~A • ~C

9. 1. (I Ú ~~J) • K 2. [~L כ ~(K • J)] • [K כ (I כ ~M)] / \ ~(M • ~L) 3. [(K • J) כ L] • [K כ (I כ ~M) 4. [(K • J) כ L] • [(K • I) כ ~M] 5. (I Ú J) • K 6. K • (I Ú J) 7. (K • I) Ú (K • J) 8. (K • J) Ú (K • I) 9. L Ú ~M 10. ~M Ú L 11. ~M Ú ~~L 12. ~(M • ~L)

XV. For each of the following, adding just two propositions to the

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premises will produce a formal proof of validity. Use all 18 rules of inference: the eight implication rules and the 10 replacement rules.

1. 1. A כ ~A / \ ~A

2. 1. B • (C • D) / \ C • (D • B)

3. 1. E / \ (E Ú F) • (E Ú G)

4. 1. H Ú (I • J) / \ H Ú I

5. 1. ~K Ú (L כ M) / \ (K • L) כ M

6. 1. Q כ [R כ (S כ T)] 2. Q כ (Q • R) / \ Q כ (S כ T)

7. 1. U כ ~V 2. V / \ ~U

8. 1. W כ X 2. ~Y כ ~X / \ W כ Y

9. 1. Z כ A 2. ~A Ú B / \ Z כ B

10. 1. C כ ~D 2. ~E כ D / \ C כ ~~E

11. 1. F º G 2. ~(F • G) / \ ~F • ~G

12. 1. H כ (I • J) 2. I כ (J כ K) / \ H כ K

13. 1. (L כ M) • (N כ M) 2. L Ú N / \ M

14. 1. (O Ú P) כ (Q Ú R) 2. P Ú O / \ Q Ú R

15. 1. (S • T) Ú (U • V) 2. ~S Ú ~T / \ U • V

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XVI. Construct a formal proof of validity for each of the following arguments. Use all 18 rules of inference: the eight implication rules and the 10 replacement rules.

1. 1. ~A / \ A כ B

2. 1. C / \ D כ C

3. 1. E כ (F כ G) / \ F כ (E כ G)

4. 1. H כ (I • J) / \ H כ I

5. 1. K כ L / \ K כ (L Ú M) Solutions I. If A, B, and C are true propositions and X, Y, and Z are false propositions, which of the following are true?

1. ~A Ú B FT T T

2. ~B Ú X FT F F

3. ~Y Ú C TF T T

4. ~Z Ú X TF T F

5. (A • X) Ú (B • Y) T F F F T F F

6. (B • C) Ú (Y • Z) T T T T F F F

7. ~(C • Y) Ú (A • Z) T T F F T T F F

8. ~(A • B) Ú (X • Y) F T T T F F F F

9. ~(X • Z) Ú (B • C) T F F F T T T T

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10. ~(X • ~Y) Ú (B • ~C) T F F TF T T F FT

11. (A Ú X) • (Y Ú B) T T F T F T T

12. (B Ú C) • (Y Ú Z) T T T F F F F

13. (X Ú Y) • (X Ú Z) F F F F F F F

14. ~(A Ú Y) • (B Ú X) F T T F F T T F

15. ~(X Ú Z) • (~X Ú Z) T F F F T TF T F

16. ~(A Ú C) Ú ~(X • ~Y) F T T T T T F F TF

17. ~(B Ú Z) • ~(X Ú ~Y) F T T F F F F T TF

18. ~[(A Ú ~C) Ú (C Ú ~A)] F T T FT T T T FT

19. ~[(B • C) • ~(C • B)] T T T T F F T T T

20. ~[(A • B) Ú ~(B • A)] F T T T T F T T T

21. [A Ú (B Ú C)] • ~[(A Ú B) Ú C] T T T T T F F T T T T T

22. [X Ú (Y • Z)] Ú ~[(X Ú Y) • (X Ú Z)] F F F F F T T F F F F F F F

23. [A • (B Ú C)] • ~[(A • B) Ú (A • C)] T T T T T F F T T T T T T T

24. ~{[(~A • B) • (~X • Z)] • ~[(A • ~B) Ú ~(~Y • ~Z)]} T FT F T F TF F F F T T F FT F F TF T TF

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25. ~{~[(B • ~C) Ú (Y • ~Z)] • [(~B Ú X) Ú (B Ú ~Y)]} F T T F FT F F F TF T FT F F T T T TF

II. If A and B are known to be true and X and Y are known to be false, but the truth values of P and Q are not known, of which of the following propositions can you determine the truth values?

1. A Ú P = True

2. Q • X = False

3. Q Ú ~X = True

4. ~B • P = False

5. P Ú ~P = True

6. ~P Ú (Q Ú P) = True

7. Q • ~Q = False

8. P • (~P Ú X) = False

9. ~(P • Q) Ú P = True

10. ~Q • [(P Ú Q) • ~P] = False

11. (P Ú Q) • ~(Q Ú P) = False

12. (P • Q) • (~P Ú ~Q) = False

13. ~P Ú [~Q Ú (P • Q)] = True

14. P Ú ~(~A Ú X) = True

15. P • [~(P Ú Q) Ú ~P] = False

16. ~(P • Q) Ú (Q • P) = True

17. ~[~(~P Ú Q) Ú P] Ú P = True

18. (~P Ú Q) • ~[~P Ú (P • Q)] = False

22

19. (~A Ú P) • (~P Ú Y) = False

20. ~[P Ú (B • Y)] Ú [(P Ú B) • (P Ú Y)] = True

21. [P Ú (Q • A)] • ~[(P Ú Q) • (P Ú A)] = False

22. [P Ú (Q • X)] • ~[(P Ú Q) • (P Ú X)] = False

23. ~[~P Ú (~Q Ú X)] Ú [~(~P Ú Q) Ú (~P Ú X)] = True

24. ~[~P Ú (~Q Ú A)] Ú [~(~P Ú Q) Ú (~P Ú A)] = True

25. ~[(P • Q) Ú (Q • ~P)] • ~[(P • ~Q) Ú (~Q • ~P)] = False III. If A, B, and C are true propositions and X, Y, and Z are false propositions, which of the following are true?

1. A כ B = True

2. A כ X = False

3. B כ Y = False

4. Y כ Z = True

5. (A כ B) כ Z = False

6. (X כ Y) כ Z = False

7. (A כ B) כ C = True

8. (X כ Y) כ C = True

9. A כ (B כ Z) = False

10. X כ (Y כ Z) = True

11. [(A כ B) כ C] כ Z = False

12. [(A כ X) כ Y] כ Z = False

13. [A כ (X כ Y)] כ C = True

14. [A כ (B כ Y)] כ X = True

15. [(X כ Z) כ C] כ Y = False

23

16. [(Y כ B) כ Y] כ Y = True

17. [(A כ Y) כ B] כ Z = False

18. [(A • X) כ C] כ [(A כ C) כ X] = False

19. [(A • X) כ C] כ [(A כ X) כ C] = True

20. [(A • X) כ Y] כ [(X כ A) כ (A כ Y)] = False

21. [(A • X) Ú (~A • ~X)] כ [(A כ X) • (X כ A)] = True

22. {[A כ (B כ C)] כ [(A • B) כ C]} כ [(Y כ B) כ (C כ Z)] = False

23. {[(X כ Y) כ Z] כ [Z כ (X כ Y)]} כ [(X כ Z) כ Y] = False

24. [(A • X) כ Y] כ [(A כ X) • (A כ Y)] = False

25. [A כ (X • Y)] כ [(A כ X) Ú (A כ Y)] = True

IV. If A and B are known to be true and X and Y are known to be false, but the truth values of P and Q are not known, of which of the following propositions can you determine the truth values?

1. P כ A = True

2. X כ Q = True

3. (Q כ A) כ X = False

4. (P • A) כ B = True

5. (P כ P) כ X = False

6.(X כ Q) כ X = False

7. X כ (Q כ X) = True

8. (P • X) כ Y = True

9. [P כ (Q כ P)] כ Y = False

10. (Q כ Q) כ (A כ X) = False

11. (P כ X) כ (X כ P) = True

24

12. (P כ A) כ (B כ X) = False

13. (X כ P) כ (B כ Y) = False

14. [(P כ B) כ B] כ B = True

15. [(X כ Q) כ Q] כ Q = True

16. (P כ X) כ (~X כ ~P) = True

17. (X כ P) כ (~X כ Y) = False

18. (P כ A) כ (A כ ~B) = False

19. (P כ Q) כ (P כ Q) = True

20. (P כ ~~P) כ (A כ ~B) = False

21. ~(A • P) כ (~A Ú ~P) = True

22. ~(P • X) כ ~(P Ú ~X) = False

23. ~(X Ú Q) כ (~X • ~Q) = True

24. [P כ (A Ú X)] כ [(P כ A) כ X] = False

25. [Q Ú (B • Y)] כ [(Q Ú B) • (Q Ú Y)] = True

V. Use truth tables to characterize the following propositional forms as (1) tautologous, (2) self-contradictory, (3) contingent, or (4) self-consistent. More than one of these terms may apply to a given propositional form, so you will need to check for each of them.

1. [p כ (p כ q)] כ q = contingent and self-consistent (four rows)

2. p כ [(p כ q) כ q] = tautologous and self-consistent (four rows)

3. (p • q) • (p כ ~q) = self-contradictory (four rows)

4. p כ [~p כ (q Ú ~q)] = tautologous and self-consistent (four rows)

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5. p כ [p כ (q • ~q)] = contingent and self-consistent (four rows)

6. (p כ p) כ (q • ~q) = self-contradictory (four rows)

7. [p כ (q כ r)] כ [(p כ q) כ (p כ r)] = tautologous and self-consistent (eight rows)

8. [p כ (q כ p)] כ [(q כ q) כ ~(r כ r)] = self-contradictory (eight rows)

9. {[(p כ q) • (r כ s)] • (p Ú r)} כ (q Ú s) = tautologous and self-consistent (16 rows)

10. {[(p כ q) • (r כ s)] • (q Ú s)} כ (p Ú r) = contingent and self-consistent (16 rows)

VI. Use truth tables to determine whether the following pairs of propositional forms exhibit (1) logical implication (if so, in which direction), (2) logical equivalence, (3) contradictoriness, (4) contrariety, (5) subcontrariety, (6) subalternation (if so, in which direction), (7) independence, (8) consistency, or (9) inconsistency. More than one of these terms may apply to a given pair, so you will need to check for each of them.

1. ~(p • q) | ~p Ú ~q Logical implication (mutual); logical equivalence; consistency

2. ~(p Ú q) | ~p • ~q Logical implication (mutual); logical equivalence; consistency

3. p Ú q | ~p Ú q Subcontrariety; consistency

4. ~p • q | ~q Ú p Contradictoriness; inconsistency

5. p Ú q | q Ú p Logical implication (mutual); logical equivalence; consistency

6. p • ~p | p

26

Logical implication (left to right); contrariety; subalternation (left to right); inconsistency

7. p כ q | p • ~q Contradictoriness; inconsistency

8. p º q | p • q Logical implication (right to left); subalternation (right to left); consistency

9. p Ú (q Ú r) | (p Ú q) Ú r Logical implication (mutual); logical equivalence; consistency

10. p • (q Ú r) | (p • q) Ú (p • r) Logical implication (mutual); logical equivalence; consistency

11. (q כ ~r) • s | s º (q • r) Contrariety; inconsistency

12. q Ú p | ~q כ ~p Subcontrariety; consistency

13. p • q | ~p Ú ~q Contradictoriness; inconsistency

14. p כ q | ~q כ ~p Logical implication (mutual); logical equivalence; consistency

15. p º q | (p כ q) • (q כ p) Logical implication (mutual); logical equivalence; consistency

16. q כ p | q • p Logical implication (right to left); subalternation (right to left); consistency

17. ~p • q | ~q • p Contrariety; inconsistency

18. (p • q) כ r | p כ (q כ r) Logical implication (mutual); logical equivalence; consistency

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19. p | p Ú p Logical implication (mutual); logical equivalence; consistency

20. p Ú ~p | p Logical implication (right to left); subcontrariety; subalternation (right to left); consistency

21. t º u | t Ú u Subcontrariety; consistency

22. ~(p Ú q) | ~p Ú ~q Logical implication (left to right); subalternation (left to right); consistency

23. (p • q) כ r | p Ú (q כ r) Subcontrariety; consistency

24. q Ú p | ~q • ~p Contradictoriness; inconsistency

25. p • q | ~p כ ~q Logical implication (left to right); subalternation (left to right); consistency

26. p | p º q Independence; consistency

27. p | q Independence; consistency

VII. Use truth tables to determine whether the following argument forms are valid.

1. p כ q \ ~q כ ~p Valid (4 rows)

2. p כ q \ ~p כ ~q Invalid (4 rows) (Shown by third row)

3. p • q \ p Valid (4 rows)

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4. p \ p Ú q Valid (4 rows)

5. p \ p כ q Invalid (4 rows) (Shown by second row)

6. p כ q \ p כ (p • q) Valid (4 rows)

7. (p Ú q) כ (p • q) \ (p כ q) • (q כ p) Valid (4 rows)

8. p כ q ~p \ ~q Invalid (4 rows) (Shown by third row)

9. p כ q ~q \ ~p Valid (4 rows)

10. p q \ p • q Valid (4 rows)

11. p כ q p כ r \ q Ú r Invalid (8 rows) (Shown by eighth row)

12. p כ q q כ r \ r כ p Invalid (8 rows) (Shown by fifth and seventh rows)

13. p כ (q כ r) p כ q \ p כ r

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Valid (8 rows)

14. p כ (q • r) (q Ú r) כ ~p \ ~p Valid (8 rows)

15. p כ (q כ r) q כ (p כ r) \ (p Ú q) כ r Invalid (8 rows) (Shown by fourth and sixth rows)

16. (p כ q) • (r כ s) p Ú r \ q Ú s Valid (16 rows)

17. (p כ q) • (r כ s) ~q Ú ~s \ ~p Ú ~s Valid (16 rows)

18. p כ (q כ r) q כ (r כ s) \ p כ s Invalid (16 rows) (Shown by sixth and eighth rows)

19. p כ (q כ r) (q כ r) כ s \ p כ s Valid (16 rows)

20. (p כ q) • [(p • q) כ r] p כ (r כ s) \ p כ s Valid (16 rows)

21. (p Ú q) כ (p • q) ~(p Ú q) \ ~(p • q) Valid (4 rows)

22. (p Ú q) כ (p • q) p • q \ p Ú q

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Valid (4 rows)

23. (p • q) כ (r • s) \ (p • q) כ [(p • q) • (r • s)] Valid (16 rows)

24. (p כ q) • (r כ s) \ p כ q Valid (16 rows)

VIII. Use truth tables to determine whether the following argument forms are valid.

1. (A Ú B) כ (A • B) A Ú B \ A • B Valid (4 rows)

2. (C Ú D) כ (C • D) C • D \ C Ú D Valid (4 rows)

3. E כ F F כ E \ E Ú F Invalid (4 rows) (Shown by fourth row)

4. (G Ú H) כ (G • H) ~(G • H) \ ~(G Ú H) Valid (4 rows)

5. (I Ú J) כ (I • J) ~(I Ú J) \ ~(I • J) Valid (4 rows)

6. K Ú L K \ ~L Invalid (4 rows) (Shown by first row)

7. M Ú (N • ~N) M

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\ ~(N • ~N) Valid (4 rows)

8. (O Ú P) כ Q Q כ (O • P) \ (O Ú P) כ (O • P) Valid (8 rows)

9. (R Ú S) כ T T כ (R • S) \ (R • S) כ (R Ú S) Valid (8 rows)

10. U כ (V Ú W) (V • W) כ ~U \ ~U Invalid (8 rows) (Shown by second and third rows)

IX. For each of the following elementary valid argument forms, state the implication rule (MP, MT, HS, DS, CD, Simp, Conj, or Add) by which its conclusion follows from its premise or premises.

1. (D Ú E) • (F Ú G) \ D Ú E Simp

2. H כ I \ (H כ I) Ú (H כ ~I) Add

3. ~(J • K) • (L כ ~M) \ ~(J • K) Simp

4. [N כ (O • P)] • [Q כ (O • R)] N Ú Q \ (O • P) Ú (O • R) CD

5. (X Ú Y) כ ~(Z • ~A) ~~(Z • ~A) \ ~(X Ú Y) MT

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6. (S º T) Ú [(U • V) Ú (U • W)] ~(S º T) \ (U • V) Ú (U • W) DS

7. ~(B • C) כ (D Ú E) ~(B • C) \ D Ú E MP

8. (F º G) כ ~(G • ~F) ~(G • ~F) כ (G כ F) \ (F º G) כ (G כ F) HS

9. ~(H • ~I) כ (H כ I) (I º H) כ ~(H • ~I) \ (I º H) כ (H כ I) HS

10. (A כ B) כ (C Ú D) A כ B \ C Ú D MP

11. [E כ (F º ~G)] Ú (C Ú D) ~[E כ (F º ~G)] \ C Ú D DS

12. (C Ú D) כ [(J Ú K) כ (J • K)] ~[(J Ú K) כ (J • K)] \ ~(C Ú D) MT

13. ~[L כ (M כ N)] כ ~(C Ú D) ~[L כ (M כ N)] \ ~(C Ú D) MP

14. (J כ K) • (K כ L) L כ M \ [(J כ K) • (K כ L)] • (L כ M) Conj

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15. N כ (O Ú P) Q כ (O Ú R) \ [Q כ (O Ú R)] • [N כ (O Ú P)] Conj

16. (W • ~X) º (Y כ Z) \ [(W • ~X) º (Y כ Z)] Ú (X º ~Z) Add

17. [(H • ~I) כ C] • [(I • ~H) כ D] (H • ~I) Ú (I • ~H) \ C Ú D CD

18. [(O כ P) כ Q] כ ~(C Ú D) (C Ú D) כ [(O כ P) כ Q] \ (C Ú D) כ ~(C Ú D) HS

X. Each of the following is a formal proof of validity for the indicated argument. State the justification for each line that is not a premise. Use only the eight implication rules (MP, MT, HS, DS, CD, Simp, Conj, and Add).

1. 1. A • B 2. (A Ú C) כ D / \ A • D 3. A 1, Simp 4. A Ú C 3, Add 5. D 2, 4, MP 6. A • D 3, 5, Conj

2. 1. (E Ú F) • (G Ú H) 2. (E כ G) • (F כ H) 3. ~G / \ H 4. E Ú F 2, Simp 5. G Ú H 2, 4, CD 6. H 5, 3, DS

3. 1. I כ J 2. J כ K 3. L כ M 4. I Ú L / \ K Ú M 5. I כ K 1, 2, HS

34

6. (I כ K) • (L כ M) 5, 3, Conj 7. K Ú M 6, 4, CD

4. 1. Q כ R 2. ~S כ (T כ U) 3. S Ú (Q Ú T) 4. ~S / \ R Ú U 5. T כ U 2, 4, MP 6. (Q כ R) • (T כ U) 1, 5, Conj 7. Q Ú T 3, 4, DS 8. R Ú U 6, 7, CD

5. 1. (A Ú B) כ C 2. (C Ú B) כ [A כ (D º E)] 3. A • D / \ D º E 4. A 3, Simp 5. A Ú B 4, Add 6. C 1, 5, MP 7. C Ú B 6, Add 8. A כ (D º E) 2, 7, MP 9. D º E 8, 4, MP

6. 1. F כ ~G 2. ~F כ (H כ ~G) 3. (~I Ú ~H) כ ~~G 4. ~I / \ ~H 5. ~I Ú ~H 4, Add 6. ~~G 3, 5, MP 7. ~F 1, 6, MT 8. H כ ~G 2, 7, MP 9. ~H 8, 6, MT

7. 1. (L כ M) כ (N º O) 2. (P כ ~Q) כ (M º ~Q) 3. {[(P כ ~Q) Ú (R º S)] • (N Ú O)} כ [(R º S) כ (L כ M)] 4. (P כ ~Q) Ú (R º S) 5. N Ú O / \ (M º ~Q) Ú (N º O) 6. [(P כ ~Q) Ú (R º S)] • (N Ú O) 4, 5, Conj 7. (R º S) כ (L כ M) 3, 6, MP 8. (R º S) כ (N º O) 7, 1, HS 9. [(P כ ~Q) כ (M º ~Q)] • [(R º S) כ (N º O)] 2, 8, Conj 10. (M º ~Q) Ú (N º O) 9, 4, CD

XI. For each of the following, adding just two propositions to the premises will produce a formal proof of validity. Use only the eight implication rules (MP, MT, HS, DS, CD, Simp, Conj, and Add).

35

1. 1. A 2. B / \ (A Ú C) • B 3. A Ú C 1, Add 4. (A Ú C) • B 3, 2, Conj

2. 1. D כ E 2. D • F / \ E 3. D 2, Simp 4. E 1, 3, MP

3. 1. G 2. H / \ (G • H) Ú I 3. G • H 1, 2, Conj 4. (G • H) Ú I 3, Add

4. 1. J כ K 2. J / \ K Ú L 3. K 1, 2, MP 4. K Ú L 3, Add

5. 1. M Ú N 2. ~M • ~O / \ N 3. ~M 2, Simp 4. N 1, 3, DS

6. 1. P • Q 2. R / \ P • R 3. P 1, Simp 4. P • R 3, 2, Conj

7. 1. S כ T 2. ~T • ~U / \ ~S 3. ~T 2, Simp 4. ~S 1, 3, MT

8. 1. V Ú W 2. ~V / \ W Ú X 3. W 1, 2, DS 4. W Ú X 3, Add

9. 1. Y כ Z 2. Y / \ Y • Z 3. Z 1, 2, MP

36

4. Y • Z 2, 3, Conj

10. 1. D כ E 2. (E כ F) • (F כ D) / \ D כ F 3. E כ F 2, Simp 4. D כ F 1, 3, HS

XII. Construct a formal proof of validity for each of the following arguments. Use only the eight implication rules (MP, MT, HS, DS, CD, Simp, Conj, and Add).

1. 1. A כ B 2. A Ú (C • D) 3. ~B • ~E / \ C 4. ~B 3, Simp 5. ~A 1, 4, MT 6. C • D 2, 5, DS 7. C 6, Simp

2. 1. (F כ G) • (H כ I) 2. J כ K 3. (F Ú J) • (H Ú L) / \ G Ú K 4. F כ G 1, Simp 5. (F כ G) • (J כ K) 4, 2, Conj 6. F Ú J 3, Simp 7. G Ú K 5, 6, CD

3. 1. (~M • ~N) כ (O כ N) 2. N כ M 3. ~M / \ ~O 4. ~N 2, 3, MT 5. ~M • ~N 3, 4, Conj 6. O כ N 1, 5, MP 7. ~O 6, 4, MT

4. 1. (K Ú L) כ (M Ú N) 2. (M Ú N) כ (O • P) 3. K / \ O 4. K Ú L 3, Add 5. M Ú N 1, 4, MP 6. O • P 2, 5, MP 7. O 6, Simp

5. 1. (Q כ R) • (S כ T)

37

2. (U כ V) • (W כ X) 3. Q Ú U / \ R Ú V 4. Q כ R 1, Simp 5. U כ V 2, Simp 6. (Q כ R) • (U כ V) 4, 5, Conj 7. R Ú V 6, 3, CD

XIII. For each of the following elementary valid argument forms, state the replacement rule (DM, Com, Assoc, Dist, DN, Trans, MI, ME, Exp, or Taut) by which its conclusion follows from its premise.

1. (A כ B) • (C כ D) \ (A כ B) • (~D כ ~C) Trans

2. (E כ F) • (G כ ~H) \ (~E Ú F) • (G כ ~H) MI

3. [I כ (J כ K)] • (J כ ~I) \ [(I • J) כ K] • (J כ ~I) Exp

4. [L כ (M Ú N)] Ú [L כ (M Ú N)] \ L כ (M Ú N) Taut

5. O כ [(P כ Q) • (Q כ P)] \ O כ (P º Q) ME

6. ~(R Ú S) כ (~R Ú ~S) \ (~R • ~S) כ (~R Ú ~S) DM

7. (T Ú ~U) • [(W • ~V) כ ~T] \ (T Ú ~U) • [W כ (~V כ ~T)] Exp

8. (X Ú Y) • (~X Ú ~Y) \ [(X Ú Y) • ~X] Ú [(X Ú Y) • ~Y] Dist

9. Z כ (A כ B)

38

\ Z כ (~~A כ B) DN

10. [C • (D • ~E)] • [(C • D) • ~E] \ [(C • D) • ~E] • [(C • D) • ~E] Assoc

XIV. Each of the following is a formal proof of validity for the indicated argument. State the justification for each line that is not a premise. Use all 18 rules of inference: the eight implication rules and the 10 replacement rules.

1. 1. A כ B 2. C כ ~B / \ A כ ~C 3. ~~B כ ~C 2, Trans 4. B כ ~C 3, DN 5. A כ ~C 1, 4, HS

2. 1. (D • E) כ F 2. (D כ F) כ G / \ E כ G 3. (E • D) כ F 1, Com 4. E כ (D כ F) 3, Exp 5. E כ G 4, 2, HS

3. 1. (H Ú I) כ [J • (K • L)] 2. I / \ J • K 3. I Ú H 2, Add 4. H Ú I 3, Com 5. J • (K • L) 1, 4, MP 6. (J • K) • L 5, Assoc 7. J • K 6, Simp

4. 1. (M Ú N) כ (O • P) 2. ~O / \ ~M 3. ~O Ú ~P 2, Add 4. ~(O • P) 3, DM 5. ~(M Ú N) 1, 4, MT 6. ~M • ~N 5, DM 7. ~M 6, Simp

5. 1. (Q Ú ~R) Ú S 2. ~Q Ú (R • ~Q) / \ R כ S 3. (~Q Ú R) • (~Q Ú ~Q) 2, Dist 4. (~Q Ú ~Q) • (~Q Ú R) 3, Com

39

5. ~Q Ú ~Q 4, Simp 6. ~Q 5, Taut 7. Q Ú (~R Ú S) 6, Add 8. ~R Ú S 7, 6, DS 9. R כ S 8, MI

6. 1. T • (U Ú V) 2. T כ [U כ (W • X)] 3. (T • V) כ ~(W Ú X) / \ W º X 4. (T • U) כ (W • X) 2, Exp 5. (T • V) כ (~W • ~X) 3, DM 6. [(T • U) כ (W • X)] • [(T • V) כ (~W • ~X)] 4, 5, Conj 7. (T • U) Ú (T • V) 1, Dist 8. (W • X) Ú (~W • ~X) 6, 7, CD 9. W º X 8, ME

7. 1. Y כ Z 2. Z כ [Y כ (R Ú S)] 3. R º S 4. ~(R • S) / \ ~Y 5. (R • S) Ú (~R • ~S) 4, ME 6. ~R • ~S 5, 4, DS 7. ~(R Ú S) 6, DM 8. Y כ [Y כ (R Ú S)] 1, 2, HS 9. (Y • Y) כ (R Ú S) 8, Exp 10. Y כ (R Ú S) 9, Taut 11. ~Y 10, 7, MT

8. 1. A כ B 2. B כ C 3. C כ A 4. A כ ~C / \ ~A • ~C 5. A כ C 1, 2, HS 6. (A כ C) • (C כ A) 5, 3, Conj 7. A º C 6, ME 8. (A • C) Ú (~A • ~C) 7, ME 9. ~A Ú ~C 4, MI 10. ~(A • C) 9, DM 11. ~A • ~C 8, 10, DS

9. 1. (I Ú ~~J) • K 2. [~L כ ~(K • J)] • [K כ (I כ ~M)] / \ ~(M • ~L) 3. [(K • J) כ L] • [K כ (I כ ~M) 2, Trans 4. [(K • J) כ L] • [(K • I) כ ~M] 3, Exp 5. (I Ú J) • K 1, DN 6. K • (I Ú J) 5, Com 7. (K • I) Ú (K • J) 6, Dist 8. (K • J) Ú (K • I) 7, Com

40

9. L Ú ~M 4, 8, CD 10. ~M Ú L 9, Com 11. ~M Ú ~~L 10, DN 12. ~(M • ~L) 11, DM

XV. For each of the following, adding just two propositions to the premises will produce a formal proof of validity. Use all 18 rules of inference: the eight implication rules and the 10 replacement rules.

1. 1. A כ ~A / \ ~A 2. ~A Ú ~A 1, MI 3. ~A 2, Taut

2. 1. B • (C • D) / \ C • (D • B) 2. (C • D) • B 1, Com 3. C • (D • B) 2, Assoc

3. 1. E / \ (E Ú F) • (E Ú G) 2. E Ú (F • G) 1, Add 3. (E Ú F) • (E Ú G) 2, Dist

4. 1. H Ú (I • J) / \ H Ú I 2. (H Ú I) • (H Ú J) 1, Dist 3. H Ú I 2, Simp

5. 1. ~K Ú (L כ M) / \ (K • L) כ M 2. K כ (L כ M) 1, MI 3. (K • L) כ M 2, Exp

6. 1. Q כ [R כ (S כ T)] 2. Q כ (Q • R) / \ Q כ (S כ T) 3. (Q • R) כ (S כ T) 1, Exp 4. Q כ (S כ T) 2, 3, HS

7. 1. U כ ~V 2. V / \ ~U 3. ~~V 2, DN 4. ~U 1, 3, MT

8. 1. W כ X 2. ~Y כ ~X / \ W כ Y 3. X כ Y 2, Trans 4. W כ Y 1, 3, HS

9. 1. Z כ A

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2. ~A Ú B / \ Z כ B 3. A כ B 2, MI 4. Z כ B 1, 3, HS

10. 1. C כ ~D 2. ~E כ D / \ C כ ~~E 3. ~D כ ~~E 2, Trans 4. C כ ~~E 1, 3, HS

11. 1. F º G 2. ~(F • G) / \ ~F • ~G 3. (F • G) Ú (~F • ~G) 1, ME 4. ~F • ~G 3, 2, DS

12. 1. H כ (I • J) 2. I כ (J כ K) / \ H כ K 3. (I • J) כ K 2, Exp 4. H כ K 1, 3, HS

13. 1. (L כ M) • (N כ M) 2. L Ú N / \ M 3. M Ú M 1, 2, CD 4. M 3, Taut

14. 1. (O Ú P) כ (Q Ú R) 2. P Ú O / \ Q Ú R 3. O Ú P 2, Com 4. Q Ú R 1, 3, MP

15. 1. (S • T) Ú (U • V) 2. ~S Ú ~T / \ U • V 3. ~(S • T) 2, DM 4. U • V 1, 3, DS

XVI. Construct a formal proof of validity for each of the following arguments. Use all 18 rules of inference: the eight implication rules and the 10 replacement rules.

1. 1. ~A / \ A כ B 2. ~A Ú B 1, Add 3. A כ B 2, MI

2. 1. C / \ D כ C 2. C Ú ~D 1, Add

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3. ~D Ú C 2, Com 4. D כ C 3, MI

3. 1. E כ (F כ G) / \ F כ (E כ G) 2. (E • F) כ G 1, Exp 3. (F • E) כ G 2, Com 4. F כ (E כ G) 3, Exp

4. 1. H כ (I • J) / \ H כ I 2. ~H Ú (I • J) 1, MI 3. (~H Ú I) • (~H Ú J) 2, Dist 4. ~H Ú I 3, Simp 5. H כ I 4, MI

5. 1. K כ L / \ K כ (L Ú M) 2. (K כ L) Ú M 1, Add 3. (~K Ú L) Ú M 2, MI 4. ~K Ú (L Ú M) 3, Assoc 5. K כ (L Ú M) 4, MI