Prove that either

a) this problem is solvable

or

b) this problem is unsolvable

We have two options,a and b,and we consider them as propositions.
Let c be the proposition that this problem has a solution.

(avb)<->c

Why?:If it is solvable or unsolvable,then it has a solution,and if it has a solution,then it is solvable or unsolvable.

b<->(~c)

Why?:If it is unsolvable,then it does not have a solution,and if it does not have a solution,then it is unsolvable.

Now,we find the truth and falsity of each.

1:(avb)<->c,Given.

2:b<->(~c),Given.

3:(avb)<->(~b),Consequence of 1 and 2.

4:b->(avb),Tautology.

5:(avb)->(~b),Consequence of 3.

6:b->(~b),Consequence of 4 and 5.

7:(b->(~b))->(~b),Tautology.

8:~b,Consequence of 6 and 7.

9:c,Consequence of 2 and 8.

10:c->(avb),Consequence of 1.

11:avb,Consequence of 9 and 11.

12:(avb)&(~b),Consequence of 8 and 11.

13:((avb)&(~b))->a,Tautology.

14:a,Consequence of 12 and 13.

a=true,b=false,c=true.

Posted by Tim Axoy on 2003-06-02 15:06:09