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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
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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
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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
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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
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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
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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).
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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
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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)
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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)
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\ 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
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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
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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
