Question

Create a truth table and prove that for any statement p, ~(~p) equals p.

Answer 1

`p=T\implies\sim p=F\implies\sim(\sim p)=T`

`p=F\implies\sim p=T\implies\sim(\sim p)=F`

In either case, we have
`p\equiv\sim(\sim p)`, so the statement is always true (a tautology).

Answer 2

The required truth table is

p ~p ~(~p)

T F T

F T F

Explanation:

If p is true, then ~p is false. The sign ~(~p) means the statement "~p is false" is false. So, we can say that ~(~p) means p is true.

If p is false, then ~p is true. The sign ~(~p) means the statement "~p is true" is false. So, we can say that ~(~p) means p is false.

The required truth table is

p ~p ~(~p)

T F T

F T F

From the above truth table it is clear that

~(~p) = p

Hence proved.