Question 531393
<pre>
1. ~(J & K) prem
2. ~(L & M) prem
3. J v L / ~(K & M) prem/conc
4. ~J v ~K                            1    DM
5. ~L v ~M                            2    DM
6. J -> ~K                            4    MI
7. L -> ~M                            5    MI
8. (J -> ~K) & (L -> ~M)              6,7  Conj
9. ~K v ~M                            8,3  CD
10. ~(K & M)                          9    DM
</pre>


Abbreviations/Acronyms Used:


CD = Constructive Dilemma
Conj = Conjunction
MI = Material Implication
DM = De Morgan's Law


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<pre>
1.  ~P v Q prem
2.  ~R > ~Q prem
3.  ~(R & ~S) / P > S prem / conc
4.  ~~Q > ~~R                       2     Trans
5.  Q > R                           4     DN
6.  P > Q                           1     MI
7.  P > R                           6,5   HS
8.  ~R v ~~S                        3     DM
9.  ~R v S                          8     DN
10. ~~R > S                         9     MI                         
11. R > S                           10    DN
12. P > S                           7,11  HS
</pre>



Abbreviations/Acronyms Used:


Trans = Transposition
DN = Double Negation
MI = Material Implication
HS = Hypothetical Syllogism
DM = De Morgan's Law