# 1
1. ~B v [(C -> D) & (E -> D)]
2. B & (C v E) /therefore D
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3. (C v E) & B 2 Commutation
4. B 2 Simplification
5. C v E 3 Simplification
6. ~~B 4 Double Negation
7. (C -> D) & (E -> D) 1,6 Disjunctive Syllogism
8. D v D 7,5 Constructive Dilemma
9. D 8 Tautology
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# 2
1. W & (A & M)
2. (A & W) -> [N v (R v H)]
3. ~N & ( ~P & ~H ) /therefore, R
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4. (W & A) & M 1 Association
5. W & A 4 Simplification
6. A & W 5 Commutation
7. N v (R v H) 2,6 Modus Ponens
8. ~N 3 Simplification
9. ( ~P & ~H ) & ~N 3 Commutation
10. ~P & ~H 9 Simplification
11. ~H & ~P 10 Commutation
12. ~H 11 Simplification
13. R v H 7,8 Disjunctive Syllogism
14. H v R 13 Commutation
15. R 14,12 Disjunctive Syllogism
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# 3
1. (O & T)>(S & M)
2. R -> ~M
3. T & R
4. O & S /therefore, V
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5. R & T 3 Commutation
6. R 5 Simplification
7. O 4 Simplification
8. T 3 Simplification
9. O & T 7,8 Conjunction
10. S & M 1,9 Modus Ponens
11. M & S 10 Commutation
12. M 11 Simplification
13. ~~M 12 Double Negation
14. ~R 2,13 Modus Tollens
15. R & ~R 6,14 Conjunction
16. F 15 Contradiction
17. F v V 16 Addition
18. V 17 Tautology
note: the truth value of R & ~R is ALWAYS false (since R can't be both true and false at the same time). I'm denoting 'false' as the letter 'F'. Also, the truth value of F v V is dependent on the truth value of V (since F is a constant). So F v V is equivalent to V
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# 4
1. F -> W /therefore, (F & S) -> W
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2. ~F v W 1 Material Implication
3. (~F v W) v ~S 2 Addition
4. ~F v (W v ~S) 3 Association
5. ~F v (~S v W) 4 Commutation
6. (~F v ~S) v W 5 Association
7. ~(F & S) v W 6 De Morgan's Law
8. (F & S) -> W 7 Material Implication
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