Question

Consider the following assertion.

~(~p V q) => p V q
(a) Find a statement that is one step forward from the given.
(b) Find a statement that is one step backward from the goal. (Use the addition rule—in reverse—to find a statement from which the goal will follow.)
(c) Give a proof sequence for the assertion.
(d) Is your proof reversible? Why or why not?

Answer 1

(a) One step forward from the given assertion is ~p V q.

(b) One step backward from the goal is ~(~p V q) => p V q.

(c) A proof sequence for the assertion could be:

Assume ~p V q.
By definition of disjunction, either ~p is true or q is true.
If ~p is true, then p is false, which means that p V q is true.
If q is true, then p V q is true.
Therefore, in either case, p V q is true.
Hence, ~(~p V q) => p V q is proven.
(d) The proof is not reversible because it relies on the assumption ~p V q, which is the statement being proven

Answer 2

Answer:(a)

One step forward from the given statement could be to simplify it to just "p V q".

(b)

One step backward from the goal statement "p V q" could be to add the assumption that "(~p V q) => p V q" to get "(~p V q) => (p V q)".

(c)

Proof sequence:

Given: (~p V q) => p V q

To prove: p V q

Assume ~p V q is true (as the antecedent of the implication)

Since ~p V q is true, either ~p is true or q is true.

If ~p is true, then p is false, but the disjunction p V q is true.

If q is true, then the disjunction p V q is true.

In either case, p V q is true.

Therefore, p V q is true under the assumption that ~p V q is true.

Since the assumption was arbitrary, we conclude that p V q is true regardless of the truth value of ~p V q.

(d)

The proof is not reversible because the contrapositive of the statement "p V q => (~p V q)" is not logically equivalent to the original statement "~p V q => p V q".

Explanation:

(a)

One step forward from the given statement could be to simplify it to just "p V q".

(b)

One step backward from the goal statement "p V q" could be to add the assumption that "(~p V q) => p V q" to get "(~p V q) => (p V q)".

(c)

Proof sequence:

Given: (~p V q) => p V q

To prove: p V q

Assume ~p V q is true (as the antecedent of the implication)

Since ~p V q is true, either ~p is true or q is true.

If ~p is true, then p is false, but the disjunction p V q is true.

If q is true, then the disjunction p V q is true.

In either case, p V q is true.

Therefore, p V q is true under the assumption that ~p V q is true.

Since the assumption was arbitrary, we conclude that p V q is true regardless of the truth value of ~p V q.

(d)

The proof is not reversible because the contrapositive of the statement "p V q => (~p V q)" is not logically equivalent to the original statement "~p V q => p V q".