SOLUTION: I need help finishing this proof. (~K -> K) -> ~L, ~(~L -> ~M) -> L, M :. K <-> ~L 1. (~K -> K) -> ~L 2. ~(~L -> ~M) -> L 3. M

Algebra ->  Proofs -> SOLUTION: I need help finishing this proof. (~K -> K) -> ~L, ~(~L -> ~M) -> L, M :. K <-> ~L 1. (~K -> K) -> ~L 2. ~(~L -> ~M) -> L 3. M      Log On


   

Question 196024: I need help finishing this proof.
(~K -> K) -> ~L, ~(~L -> ~M) -> L, M :. K <-> ~L
1. (~K -> K) -> ~L
2. ~(~L -> ~M) -> L
3. M
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
1.   (~K -> K) -> ~L
2.   ~(~L -> ~M) -> L 
3.   M                  :. K <-> ~L
--------------------------
4.   (~~K v K) -> ~L                    1    Material Implication
5.   (K v K) -> ~L                      4    Double Negation
6.   K -> ~L                            5    Tautology
-----
7.   ~~(~L -> ~M) v L                   2    Material Implication
8.   (~L -> ~M) v L                     7    Double Negation
9.   (~~L v ~M) v L                     8    Material Implication
10.  (L v ~M) v L                       9    Double Negation
11.  L v (L v ~M)                      10    Commutation
12.  (L v L) v ~M                      11    Association
13.  L v ~M                            12    Tautology
14.  ~M v L                            13    Commutation
15.  M -> L                            14    Material Implication
-----
16.  M v K                              3    Addition
17.  K v M                             16    Commutation
18.  ~~K v M                           17    Double Negation
19.  ~K -> M                           18    Material Implication
20.  ~K -> L                        19,15    Hypothetical Syllogism
21.  ~L -> ~~K                         20    Transposition
22.  ~L -> K                           21    Double Negation
23.  (K -> ~L) . (~L -> K)           6,22    Conjunction
24.  K <-> ~L                          23    Material Equivalence