Question

Give a proof sequence for the assertion. Consider the following assertion.¬(¬p∨q)⇒p∨q

Answer 1

To prove the implication ¬(¬p∨q)⇒p∨q, we can use a proof sequence that involves assuming the negation of the hypothesis, applying De Morgan's law, simplifying through distribution and elimination, and finally arriving at the conclusion p∨q.

Step-by-step explanation:

We can prove the given assertion using a proof sequence.

1. To prove the implication, we assume the negation of the hypothesis, which is ¬(¬p∨q).
2. Using De Morgan's law, we can rewrite the negation as ¬¬p∧¬q. Applying double negation, we have p∧¬q.
3. Next, we use the laws of distribution to simplify p∧¬q as (p∧¬q)∨(q∧¬q). Since q∧¬q is always false, we can eliminate it and get the final form of our hypothesis as p∨¬q.
4. Now, we can use our hypothesis to prove the conclusion p∨q. By applying the law of disjunction elimination, we have two cases: p is true, and q is true.
5. In the case where p is true, p∨q is true, and our conclusion holds.
6. In the case where q is true, p∨q is true, and our conclusion holds.
7. Therefore, in both cases, p∨q is true, and the implication ¬(¬p∨q)⇒p∨q is proven.