How do i Prove the following through natural deduction

Question

Using conjunction elimination/introduction; disjunction elimination/introduction; contidional elimination/introduction, negation elimination/introduction; biconditional elimination/introduction.

 

1.

A -->(B-->C)
____________

(A-->(D&(E&B)))-->(A-->C)

 

 

 

2.

A-->B
~A-->~C

_________
C-->B

 

3.

C v ~C

C --> ~B

~C --> ~N

___________

~B v ~N

 

 

4.

F-->(GvH)
G

_________
F-->H

 

Answer 1

PROBLEM 1

1. A -->(B-->C)

2 |(A-->(D&(E&B))) ACP

3 || A ACP

4 ||B-->C ->E 1,2

5 ||D&(E&B) ->E 2,3

6 || E&B &E5

7 || B &E6

8 || C ->E 4,7

9 | A-->C ->I 3-8

10 (A-->(D&(E&B)))-->(A-->C) ->I 2-9

 

 PROBLEM 2

1 A-->B

2 ~A-->~C

3 |C ACP

4 || ~A ACP

5 || ~C -> E 2,4

6 || ⊥ ⊥I 3, 5

7 |A ~I

8 C --> A ->I 3-7

9 | C ACP

10| A ->E 8,9

11| B ->E 1, 10

__________________

12 C -->B ->I 9-11

 

PROBLEM 3

1 C v ~C

2 C --> ~B

3 ~C --> ~N

4| C ACP

5| ~B ->E 2,4

6| ~Bv~N vI 5

7 C --> (~Bv~N) ->I 4-6

8| ~C ACP

9| ~N ->E 3,8

10| ~Bv~N vI 9

11 ~C --> (~Bv~N) ->I 8-10

__________________________

12 ~Bv~N vE 1, 7, 11

 

 

PROBLEM 4: This is not a valid argument. Might you have typed it incorrectly?

F-->(GvH)

G

_________

F-->H