Do the following proofs using only inference rules and replacement rules.

 

1. [ solution ]


1. p->~p /.:~p

 

2. [ solution ]

1. ~avy
2. (zvw)->(a&~y) /.:~z

 

 

3. [ solution ]

1. a->b /.:(c->a)->(c->b)

 

4. [ solution ]

1. (a->b)&(c->d) /.:(~dv~b)->(~cv~a)

 

 

5. [ solution ]

1. a->b
2. c->~b
3. d->(a&c) /.:~d

 

 

 

6. [ solution ]

1. ~(avb)->e
2. (~av~b)->~(avb)
3. ~a
4. ~a->~b
5. e->s /.:s

 

7. [ solution ]

1. ~pvq
2. (avq)->r
3. (~pvo)->(u&a) /.:r

 

8. [ solution ]

1. ~(avb)
2. a<->c
3. {(dva)<->e}->(cvb)
4. e->(avc) /.:d

 

9. [ solution ]

1. ~s->~k
2. l
3. a&(h&b)
4. {(jvh)&l}->k /.:s

 

10. [ solution ]

1. (~z&d)<->b
2. ~{s&(b&~z)}
3. ~a->(zvg)
4. a->(s&q) /.:b->g

 

 

11. [ solution ]

1. (avb)->~c
2. d->(~f&~g) /.:(avd)->~(c&f)