Question

Determine the truth of the following statement: (p∧q)≡∼p⇒∼q?

Select the correct answer below:


TRUE

FALSE

Answer 1

Answer: False

Reason: The truth table is shown below. The columns highlighted in yellow show the truth values for p ^ q and ∼p⇒∼q. The two columns do not match up perfectly. We need a perfect match to be able to say that p ^ q is logically equivalent to ∼p⇒∼q

For example, when p = true and q = false, p ^ q is false but ∼p⇒∼q is true. You could pick any of the contradicting rows to form a counterexample. Only one counterexample is needed to disprove this claim.

Answer 2

The statement (p∧q)≡∼p⇒∼q is FALSE.

Explanation:

To determine the truth of the statement, let's break it down:

(p∧q): This represents the conjunction (AND) of propositions p and q.

∼p: This represents the negation (NOT) of proposition p.

∼q: This represents the negation (NOT) of proposition q.

The implication (⇒) states that if the left side of the implication is true, then the right side must also be true.

In this case, (p∧q)≡∼p⇒∼q is saying that if the conjunction of p and q is equivalent to the negation of p, then it implies the negation of q.

However, this statement is not generally true. There is no logical equivalence between the conjunction of p and q and the negation of p that directly leads to the negation of q.

Therefore, the correct answer is FALSE.