SOLUTION: How do I solve these proofs? Number one 1.~B v[(C>D)&(E>D)] 2. B&(C v E) /therefore, D Number two 1. W&(A&M) 2. (A&W)>([(N v(R v H)] 3. ~N & (~P&~H) /therefore,

Question 242905: How do I solve these proofs?
Number one
1.~B v[(C>D)&(E>D)]
2. B&(C v E) /therefore, D

Number two
1. W&(A&M)
2. (A&W)>([(N v(R v H)]
3. ~N & (~P&~H) /therefore, R

Number three
1. (O&T)>(S&M)
2. R>~M
3. T&R
4. O&S /therefore, V

Number 4. last one
1. F>W /therefore, (F&S)>W

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
# 1

1.  ~B v [(C -> D) & (E -> D)]
2.  B & (C v E) /therefore   D
--------------------------------
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

==============================================
# 2


1.  W & (A & M)
2.  (A & W) -> [N v (R v H)]
3.  ~N & ( ~P & ~H )    /therefore, R
---------------------------------- 
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

==============================================
# 3

1.  (O & T)>(S & M)
2.  R -> ~M
3.  T & R
4.  O & S          /therefore, V 
----------------------------------
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

==============================================
# 4

1.  F -> W          /therefore, (F & S) -> W
---------------------------------------
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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