SOLUTION: 1. use the proof method (M9) to construct a formal proof to demonstrate that the following argument is valid: ~(T v U), S, R ≡ ~S /.: ~(U v R) 2. use the proof method (M9)

Question 1210240: 1. use the proof method (M9) to construct a formal proof to demonstrate that the following argument is valid:
~(T v U), S, R ≡ ~S /.: ~(U v R)
2. use the proof method (M9) to construct a formal proof to demonstrate that the following argument is valid:
S v (~R • T), R ⊃ ~S /.: ~R
3. use the proof method (M9) to construct a formal proof to demonstrate that the following argument is valid:
S v (T ⊃ R), S ⊃ T, ~(T ⊃ R) /.: T

First, copy the argument above and paste it into the text box. Second, using the spacebar, set up your proof into two columns. Third, type or copy and paste symbols as required to complete your proof. For an assumed premise, use '→' before the line number. For the vertical line of a subproof, use '|' before the line number. For the horizontal line of a subproof, simply use the underline edit button (click on the "Show more buttons" button to see it). You can use the spacebar to align everything near perfectly. Don't worry about the double space between lines

Answer by Edwin McCravy(20054) (Show Source): You can put this solution on YOUR website!

We don't have access to your material, so we have no idea what your material
creator meant by proof method "M9".  I'll just prove whatever method works best,
You can put arrows before line numbers.  I don't do that.  We don't know
anything about "Show more buttons".
 
~(T v U), S, R ≡ ~S   /.: ~(U v R)

1.  ~(T v U)
2.   S
3.   R ≡ ~S   /.: ~(U v R)
4.  ~T • ~U                 1, DeMorgan's law
5.  ~U • ~T                 4, Commutation
6.  ~U                      5, Simplification
7.  (R ⊃ ~S) • (~S ⊃ R)    3, Material equivalence
8.  R ⊃ ~S                 7, Simplification
 9.  ~~S ⊃ ~R               8, Transposition
10.  S ⊃ ~R                 9, Double negation
11.  ~R                 10, 2, Modus tollens
12.  ~U • ~R            6, 11, Conjunction
13.  ~(U v R)              12, DeMorgan's law              

--------------------------------------------------

1.  S v (~R • T)
2.  R ⊃ ~S         /.: ~R
                |3.  ~~R   Assumption for Indirect Proof
                |4.  R           3, Double Negation
                |5.  ~S        2,4, Modus Ponens
                |6.  ~R • T    1,5, Disjunctive Syllogism
                |7.  ~R          6, Simplification
                |8.  R • ~R    4,7, Conjunction
9. ~R         Lines 3-8 for Indirect proof.

-------------------------------------------------- 

1.  S v (T ⊃ R)
2.  S ⊃ T
3.  ~(T ⊃ R)              /.: T
                |4.  ~T              Assumption for Indirect Proof
                |5.  ~T ⊃ ~S               2, Transposition
                |6.  ~S                  5,4, Modus ponens
                |7.  T ⊃ R               1,6, Disjunctive Syllogism
                |8.  (T ⊃ R) • ~(T ⊃ R)  7,3, Conjunction
9.  T         Lines 4-8 for Indirect Proof.

Edwin

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