h                                         Homework 1

1.1:

1. (g) p = Smith has installed central heating.
       q = He has sold his car.
       r = He has paid his mortgage.
p IMPLIES q OR NOT r

(j) There are many ways to interpret this one. Here's one possibility.
p = You are wearing shoes.
q = You are wearing a shirt.
r = You will get service.
NOT p OR NOT q IMPLIES NOT r

2. (b) ((p IMPLIES q) AND (NOT ((r OR p) IMPLIES q)))

(d) (p OR ((NOT q) IMPLIES (p AND r)))


1.2:

1. (m)

1. p OR q                                               premise
   ____________________________________________________
2. | r                                                | assumption
3. | (p OR q) AND r                                   | AND i 1,2
   |__________________________________________________|
4. r IMPLIES (p OR q) AND r                             IMPLIES i 2-3

(w)

1. r                                                    premise
2. p IMPLIES (r IMPLIES q)                              premise
   ____________________________________________________
3. | p                                                | assumption
4. | r IMPLIES q                                      | MP 2,3
5. | q                                                | MP 1,4
6. | q AND r                                          | AND i 1,5
   |__________________________________________________|
7. p IMPLIES q AND r



3. (a)

1. NOT p IMPLIES p                                      premise 
   ____________________________________________________
2. | NOT p                                            | assumption
3. | p                                                | MP 1,2
4. | BOTTOM                                           | NOT e 2,3
   |__________________________________________________| 
5. p                                                    PBC 2-4

(u)

1. (s IMPLIES p) OR (t IMPLIES q)                       premise
   ____________________________________________________
2. | s IMPLIES p                                      | assumption
3. | s OR NOT s                                       | LEM
   | _______________________________________________  |
4. | | s                                            | | assumption  
   | | ___________________________________________  | |
5. | | | t                                        | | | assumption
6. | | | p                                        | | | MP 2,4
   | | |__________________________________________| | |
7. | | t IMPLIES p                                  | | IMPLIES i 5-6
8. | | (s IMPLIES q) OR (t IMPLIES p)               | | OR i 7
   | |______________________________________________| |
   | _______________________________________________  |
9. | | NOT s                                        | | assumption
   | | ___________________________________________  | |
10.| | | s                                        | | | assumption
11.| | | BOTTOM                                   | | | NOT e 9,10
12.| | | q                                        | | | BOTTOM e 
   | | |__________________________________________| | |
13.| | s IMPLIES q                                  | | IMPLIES i 10-12
14.| | (s IMPLIES q) OR (t IMPLIES p)               | | OR i 13
   | |______________________________________________| |
15.| (s IMPLIES q) OR (t IMPLIES p)                   | OR e 3,4-8,9-14
   |__________________________________________________|
   ____________________________________________________
16.| t IMPLIES q                                      | assumption
steps 17-28 are similar to steps 3-14.
29.| (s IMPLIES q) OR (t IMPLIES p)                   | OR e 17,18-22,23-28
   |__________________________________________________|
30.(s IMPLIES q) OR (t IMPLIES p)                       OR e 1,2-15,16-29